i'-i*i i-KTii 6P W5^ QfottttU IttiOTtattg Slihratg Strata, New gnvtt BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 0M£ 0"^"' i* I -^^^ W' i'i m^ i£f^ Cornell University Library ! BF251 .W34 Psychology of sound, by Henry J. Watt 3 1924 029 086 945 olln Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924029086945 THE PSYCHOLOGY OF SOUND CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager ILonJon: FETTER LANE, E.C. ESmbntst): loo PRINCES STREET iJtefa Sotfc: G. P. PUTNAM'S SONS BntniaH, Calrattn anU ilSlattaB: MACMILI.AN AND CO., arotonto: J. M. DENT AND SONS, Ltd. Eoitao: THE MARUZEN-KABUSHIKI-KAISHA Ltd. All rights reserved THE PSYCHOLOGY OF SOUND BY HENRY J. WATT, M.A., Ph.D., D.Phil. Lecturer on Psychology in the University of Glasgow and to the Glasgow Provincial Committee for the Training of Teachers; Sometime Lecturer on Psychology in the University of Liverpool Cambridge : at the University Press 1917 PREFACE I HAVE undertaken this work in the interests of a purely psycho- logical theory of the senses. A purely psychological analysis and theory of sensory experience has seemed to me for some years to be not only ideally desirable and even necessary, but really also possible. I have made two previous statements of the case for hearing. The first, published in 1911 in The British Journal of Psychology (vol. iv.), formed an incident in a general programme and tentative sketch of this pure psychology. The second, published in the same journal in 1914 (vol. VII.), was planned to meet the numerous attempts that had appeared in the intervening years to reform the elementary psycho- logy of hearing. These attempts made strong appeals towards other lines of construction than those I had advocated ; but I had confidence enough in the inherent appropriateness or, as it might be called technically, in the phenomenological correctness, of my 'idea' to be eager to come to grips with these others both in detail and in general. These new movements have since gained in interest and weight by the fact that Stumpf, in reviewing them in 1914, has seen fit to abandon his own generally accepted position, held since 1883, and to put himself at the head of one of the movements, though rejecting the special arguments brought forward for it by its first public exponents. From my second statement it may still not have been clear to many that the ground and basis of my analysis and theory of hearing are as purely psychological as I believe them to be. Such revolutionary teaching in psychology must needs have the most explicit statement. I hesitate to say that this doctrine is fundamentally new. In philo- sophy there is nothing so new under the sun. But at least in respect of the material to which the primary general principles of science have been applied, if not also in respect of the special principles that have sprung from the new rock that has been struck, there is surely much in my doctrine that is fresh growth and that will in its time give both blossom and fruit. So I have thought it needful to make a third more vi PREFACE generally accessible statement of my analysis and theory of hearing. There can be no mistake about its general nature and purport this time. I have tried to be as clear as possible without going to the tiresome extreme of saying everything. So I have laid bare the critical structure of my scheme in two successive summaries. I have also added an account of my theory in more or less untechnical, and, I hope, more familiar, terms for those who are unaccustomed to psychological ter- minology, but desire to understand my views. I have taken some account of the fact that these readers are likely to be most interested in the musical issues of hearing. My obligations will be obvious to the expert reader.' I have not attempted to mention all theorists equally, as my aim is to expound and to prove my own theory, not to review and apportion the history and merits of all others. So I have only selected a background for my work. Besides it would be a work of supererogation to repeat what has been so well done by others. For all the simple processes of hearing I have drawn freely upon the work and observations of Stumpf. No one who follows his work closely can fail to be impressed by his meticulous concern for the true facts, and by the careful logic of his inferences. These merits have made his work and that of his col- laborators deservedly authoritative. In a sense my endeavour has been merely to subject their work and that of other workers on hearing to proper psychological methods and to make it really fruitful for theory. In dealing with binaural processes I have been greatly assisted by the excellent summary of 0. Klemm. Besides these chief sources my authorities are set out plainly as occasion arises. In a question of the foundations of a science a little difference goes a long way. And the httle difference from which I build has never definitely been held nor advocated by anyone, as far as I am aware, although various theories have made some approach towards it, in so far as they considered tones to be quantitatively different. Many new results flow from this basal reform which considers pitches to be, not qualitatively, but ordinally, different. These new results include a theory of tone as such, which does not limit its accovmt to a reference to the pitch series and to tentative remarks on volume, but shows the stuff and structure common to all tones; a theory of noise, which does not harbour a persistent remainder of doubt, but is convincingly adequate, although the ground of facts is from the nature of the case inexhaustible ; a theory of fusion which meets both PREFACE vii the negative and the positive aspects of all the facts ; the measurement of the real psychical basis of an admittedly immeasurable mode of sensory experience — volume; a theory of interval, which discovers a second field of form and proportion with laws already famiUar in the visual field; or, in general, a purely psychological basis of analysis carried well up to the boundary of musical complexity, upon which a full and sufficient theory of musical construction may in time be raised. No doubt my theoretical constructions must be carried somewhat further before they can be held to have passed fully over into the elements consciously used by productive musicians and appreciative Hsteners. The gap is not a large one and is in great part filled by a psychical field that a theorist of sensory experience dare not rush into — the field of psychical habit and attitude. That field belongs chiefly to the historian and ethnologist. No doubt theory can go somewhat further still than I have gone. But it cannot go very far, for the working musician definitely takes over at a certain point the raw materials of his art from the real psychical processes of hearing, inaccessible in full to observation, and then proceeds to construct from them vast new realms without consulting anything that lies beyond the ken of observation. But I am not concerned about what I have not yet attained. If my theoretical efforts are valid, they will grow easily; if they are unfit to survive, the canker will be found within their body. But I think they are healthy enough to overcome in active hfe whatever weakness may have been born with them. H. J. W. Gi-Asaow, January 1917. CONTENTS PAGE Pbbfaoe V iNTRODUCnON 1 CHAP. I. Auditory sensations and their attributes ... 15 A. Tones 15 B. Noises 34 C. Vowels 41 D. Octave qualities 44 II. The analysis of bi-tonal masses 53 in. Distance and interval 73 IV. The analysis of tonal sequences 86 V. The farther study of tonal masses .... 99 VI. Melody 113 VII.' The formation of scales 123 Vm. Physiological theories of hearing 139 IX. Binaural hearing 175 X. Summary ... 193 XL Short summary . 206 XII. Untechnical account of results 208 Xni. 'Pure' psychology 216 Works Cttbd 220 Appendix 227 kiSEX OF Authors ....... 238 IsDEx OF Subjects 239 INTRODUCTION I. The end and aim of the study of hearing is to explain it. Every- one may be supposed to know what is meant by explanation, so that it is usually considered superfluous to state what that meaning is, especially in the introduction to a scientific book. We expect explana- tion to. follow of itself from a full and correct statement or description of the facts and their connexions. Indeed a thinking mind usually desires no more than this. Not that the thinking mind delights in prosaic formulations. Far from it; it supphes for itself the poetry or the atmosphere ; the mere statement of the facts and their connexions arouses this atmosphere. And by atmosphere we mean the sympathetic surroundings, kindred facts and connexions from other spheres of reahty. Thus explanation seems really to mean the full and correct classification of facts and their connexions, so that they may be grouped by the mind along with already estabUshed sets of facts and connexions of a similar kind. But more than this is usually required by the scientist. He has also to show how his system of facts is connected with those that surround it in the world of reahty. Or if only a part of the events he is interested in can be fully and correctly described, he is required to show to the best of knowledge and beljef what set of facts and connexions, or in a word what set of processes, already observed and famihar in other regions, occupy the unobserved regions of the events he studies. Or he endeavours to clarify his thought of one set of facts etc. by his already clear thought of other sets of facts etc. This efEort of scientific thought is known as theory and in its incipient stages as hypothesis. The study of hearing therefore begins with the statements of the facts of hearing and their connexions. These are wholly and solely matters of experience; they are psychical. For hearing means experiencing. A clear statement of these facts will call up in the mind of the thinking reader similar facts and connexions from other departments of experience, especially from the fellow processes of hearing,— the other senses. And here again these facts will be wholly and solely psychical. Where the facts of hearing cannot be observed or have not yet been successfully observed, the study of hearing will feel impelled to draw w. p. s. I 2 INTRODUCTION upon its knowledge of the other senses and so to form a complete state- ment of the facts of hearing that shall at least be most probably true. Thus far the study of hearing belongs to the science of psychology. II. At the present time even this primary part of the study is full of the keenest disputes. Several reasons may be brought to account for the prevaihng difficulties and doubts. The chief of them is the peculiar complexity of even the simplest auditory experiences, which makes their observation anything but easy. It is not only hard to arrange for the occurrence of an exactly simple sound by the isolation of its physical stimulus, but only a few persons possess the power of making fine and accurate observations on sounds, whether they happen to have educated that power or not. Complexity and obscurity are naturally increased manifold if they suggest the wrong atmosphere of classification. And this seems to have happened in the study of hearing in so far as vision has been held to be the pattern according to which the experiences of hearing might best be arranged. It seems probable that in another sense than vision or in a comparative inductive study of all the senses a better guide to the elucidation of hearing might be found. For we must expect similar parts of experience, in this case the various senses, to work in essentially the same way. Inductive methods are obviously best if the common essential functions of all the senses are to be separated from what is special to each and from what might mislead us, if we take any single sense as the standard and pattern for hearing. It is a famiUar fact that hearing depends upon the work of the ear and of the neural organs attached to it. The study of these is physiology. But the physiology of the ear finds itself very often unable to complete its statement of the facts regarding the working of the ear and its connected organs. It is forced to theorise about the remainder. And it naturally turns for information to the psychology of hearing. If our experiences of hearing are dependent upon the ear, etc., what is more hkely than that the facts of hearing wiU make possible some good inferences regarding the functions of the ear, etc., which carmot be directly observed. But it is obvious that if we are to have good inferences, we must base them upon the best possible psychology. Inferences from one side dare not contradict facts of the other side, but facts of the one side, especially the simpler and clearer facts of experience that are direct objects of observation, form a rule or standard for the theory of the other side. Ultimately we expect to find complete harmony INTEODUCTION 3 between the two sides, the physiology of the ear and the psychology of hearing. And, of course, the more advanced and complete study will tend to lead the way. That is at the present time undoubtedly the psychology of hearing. Let us then see what the psychological study of the other senses may lead us to expect of hearing. The senses may be divided for this purpose into three groups. III. The first group contains the senses that are distributed generally throughout the body, especially over the surface of the skin and the underlying tissues and in various other parts. These are called the cutaneous and visceral senses, and include the senses of pain, touch, cold and warmth. There never has been any doubt about their great psychological similarity. They can be described in almost the same terms, and if we omit unimportant variations of degree and frequency, they can be easily included under one formula. Let us consider the nature of these terms and of the resultant formula. The difference that is immediately evident to us between pain and touch, or between either of these and cold or warmth is called a difference of quality. We do not usually look upon cold and warmth as being so different from one another as they are from pain and touch, but there can be no doubt that once we compare them Avith such differences in mind, we must admit that they are very different from one another and really have nothing in common. We therefore say that warmth and cold also differ in quality, though we may readily allow that they may be more akin to one another than they are to the others. We are for the moment less concerned with their possible kinship than with their obvious differences. This difference is confirmed by the fact that warmth and cold are acknowledged to be physiologically separate senses. That applies in fact to all four — ^touch, pain, cold and warmth. Physiological research has recently discovered that each of these senses seems to be served by more than one set of organs or receptors, as physiologists now call them, in order to avoid the ambiguity of the word 'sense,' which is now used only in reference to experiences. The question thus arises: do these different sets of receptors for one and the same sense give sensations of different quahty. And the answer is : no differences can with certainty or even with any probabiUty be established. Thus we arrive at the highly probable conclusion : each of the cutaneous and visceral senses gives only one quahty of sensation. 1—2 4 INTEODUCTION A second way in which these sensations vary is familiar to every one; that is intensity. We know of course that various things may happen to make a pain more intense; the skin may be heated more, a thorn may be pressed more heavily on the skin, inflammation may increase, etc., but we never fail to recognise in all this that the mere 'feeling' of pain varies in intensity. We never confuse the intensity of the pain with the process going on in the skin, so as to say, for example : 'pain has no intensity; it is only the process on the skin that has intensity.' The same holds for all the other sensations of this first group. IV. It is quite easy to recognise that quality and intensity are attributes of sensation, and these two form the nucleus of probably every Ust of attributes. The constant disputes that have attended the formulation of a Ust of attributes are concerned with the various additions to this nucleus that have from time to time been proposed. Even yet no list has been definitdiy accepted by the majority of psycho- logists, so that the science of psychology is still devoid of any precisely formulated and methodically established foundations. The cause of this lack of agreement is to be sought in no simple confusion of thought, but in a fundamental weakness of method, namely in the assumption that, since the two attributes of quaUty and intensity are acceptable and accepted as they come and appear to our untutored observation, all other attributes must also be taken over from among the details of our sensory experiences as they are found and appear in our search. But this assumption ignores the possibihty that while quahty and intensity may be very simple and are almost never complicated attri- butes, some or all of the others may always be wrapt up in the comphca- tions and modifications which experiences produce upon one another. And that is just what the method we are to follow teaches us to beheve to be true. One Ust of attributes for example would propose to add ' locaUsation ' as a third to its Ust, since all sensations have some kind of locaUsation, although some of them seem to have a much more precise and ready locaUsation than others. Another Ust would exclude 'locaUsation' because the locaUsation of sensations varies and so seems to be due to something else, and not itself to be a primary attribute. Why not suppose it then to be a product of the bunching together of quaUty and intensity, exclude it from the list, and adopt the attribute of extensity instead? The objection to ■tibns fourth attribute is that it INTRODUCTION 5 would force us to classify the extensity of pain and touch with the volume of sounds and would leave us inquiring as to the extensity of smells. Not that such a classification is •wrong, as we shall see, but no good reason was given for classifying things together that seem so different. Some Usts have included feeling-tone, using this peculiar term to express in one word the two variations of feehng-tone — pleasantness and unpleasantness. But most psychologists agree that feehng-tone cannot be an attribute. For why should it have two forms ? Are not these forms more different from one another than are the variations of the intensity of either of them, say pleasant feehng-tone? And if intensity is already in the hst, why is it brought in here a second time with another attribute ? And why does pleasantness not pass gradually into unpleasantness without passing through a stage in which there is no feehng-tone? (Some psychologists have invented an 'indifferent' feehng-tone to fill up the gap.) But if the feehng-tone can be absent altogether, can it still be an attribute? Psychologists seem to have agreed in this case that if a given sensation possesses any attribute, that attribute cannot be made to disappear without the total disappear- ance of the sensation itself, and that therefore feehng-tone cannot be reckoned an attribute of sensation. These reasons do indeed seem to exclude feehng-tone altogether. Let us consider more closely the axiom just stated: a sensation disappears if its attribute disappears. Why? Because whatever thus comes and goes without detriment to the continuance of sensation cannot really be a property of sensation but must be adventitious to it or a product of the comphcation of sensation. Against this it is vain to urge that even the obvious quahty and intensity are intrinsically detached from one another and devoid of inner connexion. For even if we do fail to grasp their inner connexion, we are none the less con- cerned to discover which attributes form 9, constant group in the sense of the axiom. Nor does it really matter that in the course of the briefest observation, lasting a fraction of a second, the attributes are not always aU observed to be present. That may be due to the rapidity of observation and not to the absence of the attributes. We are searching for firm ground upon which to build psychological theory; and if a constant group of attributes occurs, that is of the highest importance for theory, whether very brief duration seems to destroy the constancy of the group or not. The axiom thus becomes in the first place a verbal definition : the 6 INTRODUCTION attributes of a sensation are to be those attributes that on sufficient observation are always found together. But certain discoveries would turn it into much more than this — ^into a real definition. One of these is the constancy of this group of attributes, not merely for one and the same sensation, but for all kinds of sensation. The first axiom would then take on much wider scope and become : if any one sensation has a certain attribute, so has every other. Or: only those attributes are to be held to be the real attributes of sensation that on sufficient observation are found together in all sensations; and there are such. The second discovery is the difEerent psychical status and origin of the inconstant features of sensation that we might feel disposed to call attributes. Thus we should obtain with unimpeachable methods a sure groimd for a purely psychological theory of sensations and aUied experiences. We may therefore proceed, remembering that attributes other than quahty and intensity may not appear in so uniform a guise and that if our determination of attributes is to give us a good founda- tion for a science of sensations, each attribute to be possible must be discoverable in every kind of sensation. V. A third way in which the sensations of the first group vary is their localisation. Pain, touch, cold and warmth are always clearly, and often very distinctly, locahsed. And it is easy in the case of these sensations to show that locaUsation is an experience. When one has toothache or rheumatism, for example, one can hardly ever in any way see where the pain is. Without exploring the skin for tender spots, one feels the pain 'somewhere'; and rheumatism often ffits about in spasms from one place to another, each twinge of pain at once marking itself out from the others by its locaUsation. And pain often seems to be in the wrong place. Cases are famihar in which pain is locahsed in the fingers and toes of a lost hmb. The pain somehow appears as if it were in the lost member. Locahsation is therefore a feature of experience that can be denied as little as can pain itself no matter how 'wrong' it is. But locahsation is not therefore an attribute of sensation. Touch, pain or cold are merely what they are ; we ask for no more enlighten- ment about them when we have them ; we take them just as they are ; we do not need to refer to touch when we name pain ; and pain is not any more truly pain after we compare it with touch or cold, than before ; reference to touch is only a way of bringing the difference of quality logically clearly to mind. Not so with locaUsation: we cannot even INTRODUCTION 7 experience, far less think, the locaUsation without some reference in our experience to the painful part, toes or tooth. If we could completely isolate a sensation, it would have no localisation. But there must be something in the sensation which justifies its getting a locaUsation; if we cannot localise one pain in a finger and another in a toe without some conscious awareness of these parts, by thought, or mental image or the Uke, there must nevertheless be some difEerence inherent in each pain, in virtue of which it can be referred to the correct part. This difEerence would be an attribute, possessed by the sensation without dependence on any other experience. How shall we name it? That depends upon whether the supposed attribute is simple and primary or complex and derived. The schools of psychologists have divided on this alternative and have even received distinctive names, the genetic school holding to derivation and the nativistic school to primacy. The former school urges that localisation may be derived by association from combinations of the quaUtative and intensive differences already admitted. But it has never succeeded in showing that there is in existence a fraction of the variations of quality and intensity which would be required to account for all the variations of locahsation which occur. Nor has it established any convincing theory of the means by which the association of these differences come about. Association merely begs the question. What we need to know is how a particular quahty comes to attach itself by association to a particular intensity, when both are given. If we say : 'Well, aren't they at the same place?' we merely beg the question. For if locaUsation is derived, they have of themselves no place at all; they are merely a quaUty and an intensity. And lastly even if they did hitch on to one another somehow, why should that give rise to a locaUsation? Why not to a feeUng, or an idea? The genetic school is thus defeated at every point, and the field is left to the nativistic theory. It claims no miracle of unfounded associa- tion and transformation. It is true a development must be admitted : the primary attribute develops into locaUsation. But how it does so is a problem which may be left for later study. Looking at the develop- ment backwards in order to determine the nature of the attribute out of which locaUsation develops, nativism claims that the primary attribute has a psychical kinship with locaUsation; it is Uke the latter. Is it justified in doing so? Most assuredly. For we can readily in thought strip from locaUsa- tion its garment of reference, considering only the primary differences 8 INTRODUCTION upon which it rests. We can do so in observation as well. Let the forefinger of each hand be touched. Cease to consider the spots as being in the fingers and consider only their inner differences, staring at them as it were, as one does at times with the letters or sounds of a word, when they lose their meaning and become so oddly absurd. Their differences as mere sounds stand out more clearly than usual. So with the touches : we notice a primary inherent difference between the two which seems best describable as a difference of order. It is the same kind of difference as that between one and two, between first and second. Thus we may conclude; every sensation of any sense of the first group differs from every other in respect of the attribute of order, as we may see from the differences of locahsation that are so obvious in these senses. The careful and correct study of this attribute is of vital importance for the psychology of hearing as of many other departments of experience. VI. A fourth attribute is closely connected with order. But its distinction and its study will for that reason be much easier. When we put a hand into warm or cold water, we have a mass of sensation varying in extent as more or less of the hand is immersed. Now although the extent of the cold or warmth varies, each experience is alike in being extensive. Bach extent of cold or warmth feehng is just as extensive as any other. This extensity is not so closely bound up with localisation, as with primary order; for we should hesitate to say a locahsation was extensive ; space is extensive, so is a finger or a toe ; but a touch's reference to a point of space or to the finger or toe is not extensive ; only the touch is extensive. In other words spatiahty is not implied in extensity any more than it is in mere order. The seeming independence of extensity and locahsation may largely account for the fact that extensity and its temporal counterpart, durance, usually appear even in those fists of attributes that do not contain locaUsation or order. Given extensity, those of the genetic school thought it possible by one means or another to manufacture positions within it out of the natural groupings of differences of quahty and intensity. W. James, who was a nativist as regards extensity, thought that the perception of positions within it resulted from sub- dividing (36, 75). But a moment's consideration shows that orders cannot originate from mere extensity. We could make tactual or visual areas of the same size all over the sensory field; but they would be identical, unless they included differences of order. Even an increase INTEODUCTION 9 of extent is unthinkable apart from inherent difierences of order. Of course difEerences of quality and intensity which accompany ordinal differences may draw the attention powerfully to the latter ; but they cannot create them. Probably James felt this in some vague way. He said (36, 79) : " he who will have thoroughly answered this problem of discrimination, will have laid the keel of psychology." Well, one beam of that keel is a nativistic attitude towards order as well as towards extensity. Two other attributes remain that concern our awareness of time and its differences. They are very clearly akin to the attributes of order and extensity. They may be termed order and durance. To distinguish temporal order from the order upon which locaUsation rests, the latter may be called systemic, as it is the order that appears when a system of receptors yielding one quality is given. But, as the psychology of hearing is in no way seriously affected by the distinction and study of the temporal attributes, important as they are in themselves and for experience in general, we shall omit any further reference to them. Our attention will be confined to the first four attributes — quality, intensity, order, and extensity. The sensations of taste may conveniently be added to those of the first group. They present no new feature of psychological interest unless it be their variation in quaUty. Tastes occur in four quaUties, sweet, sour, bitter and salt. Although we seem to have as good reason of a physiological kind to call them the quahties of independent senses, as in the case of warmth and cold, most people would deem the qualities of taste more akin to one another than those of the cutaneous senses. Unfortunately we have as yet no other means of gauging the kinship of different qualities than our direct introspective impression of their kinship. Thus far at least the rule holds that for every physiologically independent sense only one psychological quality occurs. VII. The senses of the first group detach themselves from the others chiefly because the study of the attributes of sensation in them presents least difficulty and so formulates the problem to be pursued through all the other senses. This clarity and simplicity are doubtless due to the comparative physiological simpHcity of their receptors and to their cognitive functions in deaUng with the objects immediately surroimding the body. They are known to physiologists as the simple exteroceptive senses. A second group of senses is naturally formed by the receptors of the body (known as proprioceptive and interoceptive) that obtain 10 INTRODUCTION for cognition data regarding the states and operations of the body itself and regarding the stimuli that afEeot its inner surface, i.e. the alimentary tract exclusive of the parts near the m^h. But as these states and operations are for the most part controlleS without the aid of cognition, it is not surprising to find that the sensations of this group are in various respects obscure and difficult of study and somewhat complex. But we have every reason to beheve that the formula derived from the first group is perfectly adequate to the second group. This includes the articular, and the muscular (proprioceptive) senses, and the large group of the organic (interoceptive) senses (106, snt, 336f.) (that are stimulated by emptiness or fullness, fiUing or evacuation, of various organs). The articular sense provides data dependent on the relative positions of the jointed Umbs. It is physiologically distinct from the senses of the first group, more particularly from touch. We have no reason to suppose that more than one quahty occurs in this sense. Of the other attributes intensity is the most obscure. This is almost certainly due to a want of variation in the physiological conditions upon which that attribute is dependent, and not to its psychological absence. If a Umb is placed very comfortably and in perfect relaxation, awareness of its relative position gradually disappears entirely ; but it is at once restored by renewed innervation. A uniform intensity would be cognitively irrelevant and therefore introspectively indefinite. But we have still to deal with the datum of position conveyed by this sense. And our thoughts naturally turn to the attribute of order. Could it be the basis of position? To this proposal, we must surely assent. For just as in the case of locaUsation, position includes a system of orders and yet is more than that ; it is position of the limb. To get a notion of the primary underlying attribute we must omit the phrase ' of the limb ' and express the sensory datum as ' order of articular quahty of uniform intensity.' Our awareness of the relation of this order to a certain limb must be gained from it by our somehow collating it with other orders. The same appHes to our awareness of the particular hmb to which the positions apply. In both cases we need presuppose in the sensation nothing but sets of articular sensation of different orders. And the different extents of these sets, e.g. in the contrast of the sensations from a large joint with those from a small one, point us towards the attribute of extensity. The muscular sense provides an interesting variant upon the obscurities of the articular sense. For while it also has only one quality. INTEODUCTION 11 its variability is chiefly intensive and hardly ordinal at all. In fact we are explicitly aware of muscidar sensations as such only when they are considerably intensive and begin to give some awareness of strain or resistance. Intensity may therefore at once be conceded, while we may see ordinal variation behind whatever awareness of localisation of muscular sensation we possess, and extensity in their common varia- tions of bulk or mass, as we pass from large muscles to small muscles. These variations in bulk doubtless imply the presence of sets of ordinal differences in the mass sensation we obtain from each muscle ; and it is probable that varying numbers of the fibres of one and the same muscle are innervated in response to the varying strain or voUtions directed upon the muscle. A variation in the extent of muscular sensation from one and the same muscle would thus be evoked. It is not necessary to suppose, as Myers (85) suggests, that the intensity of muscular sensation varies with these extents; these intensities are surely more Ukely to be dependent upon the strain put upon the fibre in which the receptor for muscular sensation is embedded. But our cognition of strain may take both variations of intensity and of extent from one and the same muscle into account. In organic sensations, amongst which hunger, thirst, repletion, nausea, and many others are included, we find a general obscurity of attributes, but no other serious obstacle to their identification. Their qualities are all rather vague and difficult to distinguish from pressure or mixtures of pressure and pain. Their locaUsation, although by no means precise, is certainly clear enough to warrant the assumption of underlying orders, and they always occur in considerable bulk. We therefore feel entitled to claim that the formula of the attributes derived from the first group, holds also for the second. The special psychological interest of this group of sensations hes in the frequent obscurity and variability of their attributes. But, as we have already suggested, this may properly be ascribed to a want of variation in the physiological correlatives of these attributes, and not to any psychical incapacity of the sensations themselves. If the stimulus to one of these senses is present, it may be sufficiently effective with only one hne of variation. Hunger and thirst need not wander over the body ; their intensity is enough for all purposes. And their physiological vaUdity is further secured by their mass or bulk, dependent as that is upon the distribution of their receptors over the surface or over a representative section of the organ most immediately concerned in the related f imction. So we apparently never get a ' spot ' of articular 12 INTKODUCTION or muscular or organic sensation, as we do with the sensations of the first group, but always an undifferentiated mass or bulk. And yet we may properly infer from the variation of extent which accompanies the gradual immersion of the hand in cold or warm water, that the mass of any sensation of the second group is due to the fusion of many minimal extents of sensation. We also know from the cutaneous senses, and still more clearly from vision, that in area there is no accentuation or discrimination of orders unless within small ranges of that area a rapid variation of intensity (or, in vision, of quahty) is also given. VIII. The remaining senses, hearing, vision and smell, form the third group. Like those of the first group these are exteroceptive senses, sometimes distinguished as 'far' senses or distance-senses (106, 324 f.) from the former, the ' near' senses. Their sensations are much more com- plex and elaborate than those of the other two groups, so that they are often called 'higher,' and the others 'lower' senses. Being 'far' senses they are most important for cognition. Our problem is to express their complexity and to solve their obscurities and difficulties in terms of the simpUcity of the other senses and of the attributive formula estabhshed for them. In vision special difficulties are presented by the attributes of in- tensity and quahty. There seems to be an indefinite, though wholly surveyable variety of quahties in vision, of which the solar spectrum exemplifies a special series. The whole range of these quahtative variations is displayed in the ' colour- figure' (Kg. 1). It is very difficult to find in vision a satisfactory equivalent to the intensity of other senses. For if the range of variations from red to green or from yellow to blue is to be taken as qualitative, so must the range from white to black. And yet there is something common to these series ; in passing from yellow to blue we pass through a series of changes of brightness comparable to the series of changes Fig. 1. R = Red, 0= Orange. Y = YeUow, G= Green, B = Blue, V = Violet, P = Purple, W = White, Gy=Grey, Bk= Black. INTRODUCTION 13 in passing from white to black. How then shall we distinguish brightness and quahty in the two series ? Although it is indeed far from easy to bring vision into full harmony with the other senses, we have hardly reason to believe that such harmony is unattainable. And it is encouraging to find, as we shall, that these particular difficulties of vision do not recur in the sense of sound, if our interpretation of that sense is correct. Successful analysis of hearing would then lend added weight to the probabiUty of vision's conforming to the proposed type. In smell we meet with a sense that has so far baffled all the efforts of physiolo^sts or psychologists. It possesses an amazing variety of quahties which have never even been so surveyed as to appear to be a closed or exhausted system. And they give practically no kind of Emit as to how their complexity might be reduced to the mixture or interaction of a few primary quahties. Of their intensive variations there never has been any doubt. But if they are all localised about the nostrils, as seems probable, they would be devoid of aU variation in order, though not of order altogether, so long as they are locaUsed at all. Of any extensity we have only the vaguest indications. W. James thought vinegar a less extended smell than musk (36, 76). Smell is in fact a most puzzling sense. Of course it is in us in a most degenerate state. But that hardly seems a good reason why it should be difficult for us to give a psychological analysis of it. If the qualities of vision, as all theories of colour vision suggest, promise to allow themselves to be reduced ix> a small number of primary qualities, of which each one or each pair forms a more or less independent sense, it would seem highly probable that the qualities of smeU will some day admit of a similar reduction. In that case some general rule regarding the quahties of sensations, probably that suggested by the sensations of the first group, perhaps with sHght modifications for the cases of kindred quahties, would estabhsh itself. If the ordinal attribute of all the other senses conforms to a general rule, it is hardly Hkely that smell will form an utterly irreconcilable case. A similar remark apphes to the intensity of vision. It is interesting to notice how the different difficulties presented by the various senses thus tend to reduce the probabihty of any one of them proving insuperable. It is my intention in this work to attempt an analysis of the sense of hearing on the fines suggested by this analysis of the first two groups of senses. The analysis promises to be completely successful and thus to add its evidence to the probabihty of the universal apphcabihty 14 INTRODUCTION of the suggested formiila. At the same time we shall find as we pursue the analysis beyond the simplest forms of auditory sensations, that the effort to reduce the least complex forms of auditory experience to tj^es also applicable to the same forms of complexity as they appear in the other senses, adds still more to the probable truth of my formula for the attributes. So far I have of course only been concerned to estab- lish a starting ground for analysis towards the attributive formula. Practically every previous attempt to bring hearing into conformity with the other senses and so to procure extraneous evidence of the successful analysis and arrangement of its facts and their connexions, has followed other hues than those adopted in this work. And (without regard to the present theory) there can be no doubt in the mind of any modern psychologist that every analysis of hearing yet offered leaves so many difficulties unsolved that we must either consider it to be in- adequate to the facts, or our knowledge of the facts to be inadequate to it. Where, as in hearing, the main body of the raw facts of an elementary kind hardly offers serious grounds of dispute, the latter alternative may be deemed improbable. The writer is convinced that his analysis is so much more adequate to the facts as to be preferable to any previous analysis, and as to convince us that the failure of every previous analysis to carry conviction is hardly due to any deficiency in our knowledge of the raw elementary facts of hearing. The writer's analysis thus seems fitted to bring into the elementary psychology of hearing insight and stabihty such as it has never hitherto shown. This result must be of the highest importance to every discipline which is associated with, draws upon, or builds upon, the elementary psychology of hearing, e.g. physical acoustics, phonetics, physiology of hearing, musical practice and aesthetics. CHAPTER I AUDITORY SENSATIONS AND THEIR ATTRIBUTES IX. The whole gamut of the world's sounds falls into two halves which are perfectly obvious in their extremes although there is no clear division between them. Everyone makes a distinction between tones and noises. Tones are smooth, even, and regular in appearance, while noises are rough, uneven, and irregular. A. Tones. A good preliminary survey of tones is given in the series of sounds produced by any musical instrument. When we run over the keyboard of a piano note by note from left to right; the sounds produced differ from one another in a way we usually name collectively as a difference of pitch. The pitch is said to rise as we progress, being low at first and then becoming gradually higher. Even on the piano we notice that the difference between its tones is much less noticeable in the extreme octaves than in those intervening, so that the piano seems to give us the greater part of the whole range of tones, or at least the only part that is of any use for music. But any part, if not the whole of this range of tones, can be produced on many other musical instruments and we can readily recognise the instrument used from the sound of its tone alone. Thus we get a large number of similar series of tones, and if we are to obtain any common survey of the range of tones, we must first settle whether any of these series is the real series of tones and if not what is the primary series. Very few, if any, musical sounds can be produced without the accom- paniment of a certain amount of noise. But apart from that — and we readily learn to neglect it — we usually hear in good musical tones only a unitary sound in which no parts seem to be distinguishable. The tone has of course been made as pure as technique and tuning will allow. But everyone knows that the sounds of the rougher musical instruments often break up into parts which are distinguishable by their pitch. The eUmination of these 'irregularities' of pitch is just what makes the playing of many musical instruments so difficult. 16 AUDITORY SENSATIONS [ch. The trained ear watches for them and learns to manipulate the instru- ment so as to exclude them and to make the tone as pure as possible. But have they even then all been excluded? Perhaps we should find more of them if we made it our special business to analyse the pitch of tones or if we used special instruments to help our hearing. Helmholtz has proved in a variety of ways that the pitch of the tones of musical instruments is nearly always much more complex than it appears to be. In many cases the presence of more than one pitch in a musically simple tone can be heard with the unaided ear, if a careful search is made. And if each probable component of the tone is sounded gently and repeatedly beforehand so as to prepare the attention, the range of such observations can be greatly increased. We seem to hear the prftpare d tone sound ing oninto the tone- to be analysed. Thes e resulta_aift,notj.maginary, as if we re aUy did carry ov er whatjwejexpect into what we actually hear ; for it is often to be notic ed that with the, Sitter methodths^ual temperament of the piano suggests as a probable comgone^a tone^thtrtrdbes not qmte coincide^n pitcTm^JbtfiuDDja- ponent act uai^~ Eeard, but is a h ttle sharpCT "gFBatter than it. More- over these components can be much strengthened with the aid of suitable resonators. And as Helmholtz says, the ear recognises without resonators every component that can be strengthened by them and perceives no component unaffected by the resonator. The results of the psychical analysis of musical sounds are confirmed by physical analysis in various other ways. And the sounds of musical instruments can be roughly imitated on the organ by the combined use of a number of stops which bring together for each tone of the scale a set of sounds whose predominant pitches coincide with the chief component pitches of the imitated sound. The primary cause of the differences between the sounds of different musical instruments is therefore physical. Hardly any musical instrument vibrates so as to evoke a tone of a perfectly simple pitch; it vibrates so as to evoke a sound in which one pitch predominates over others. The component pitches of musical tones are usually membeis of a definite series. Let us call the lowest component pitch of the analysed tone c. The lowest is usually also the loudest or predominant pitch of the tone, from which it receives its name — ^its 'nominal' pitch. The next higher component pitch wiU often be an octave higher, c' ; the third component will often be a fifth hi^er than that, g' ; the fourth <^ ; the fifth e*; the sixth g^; the ei^th c*; the ninth d^; the tenth c*; the twelfth gr*; and so on. The sevraath and eleventh components I] AND THEIR ATTRIBUTES 17 cannot be expressed exactly with the names used in our musical scales. If we played this series of tones on a musical instrument, the physical rates of vibrations which would theoretically correspond to the pre- dominant pitches of each would be respectively, n being the rate of vibration of c: n, 2m, 3w, 4w, 5n, &n, In, 8n, 9w, lOw, lln, 12w, etc. That is to say, they are all simple multiples of n. This can readily be shown with the monochord upon which the presence of components in an apparently simple tone is often demonstrated. The tones of the above series can be obtained by plucking the string when it has been stopped with a fine brush at a half, a third, a quarter, etc., of its length. Or the stopping brush may be apphed to these points of the string after it has been plucked, thus isolating the physical component, if it is present. It is a famiUar fact that the rate of vibration of a string is inversely proportional to its length. We thus obtain a simple rule for the components of the pitch of tones : they are one or any of a series related to the physical rate of vibration or to the pitch of the lowest predominant component in the following manner. Rate of vibration: n, 2n, 3w, in, 5n, 6n, In, 8w, 9w, lOw, 11«, 12»i. Corresponding relative pitch: c, c', g', c^, e^, g^, f, f.... Thus the pitch of any suspected component may easily be calculated. Component pitches may be known as partial pitches, ^h& lowest and predominant partial being the fundamental partial, and the others the upper partials. As confusion is liable to arise if the upper partials be numbered without inclusion of the fundamental, it is usual to include the latter in the numbering as the first partial. Partials are then numbered according to their numeral in the n series above. The even partials are c' (2w), c* {4m), g^ (6w), etc. : the uneven are n (c), 3w {g'), 5n (e^), etc. Helmholtz, whose work on this subject is authoritative, summarised the results of his researches in a few general rules showing the usual components of the pitches of various instruments and the relation between these sets of components and the musical character of the soTmds. These rules are as follows: 1. Tones of simple pitch, hke those of tuning forks with resonance chambers and those of wide stopped organ pipes, sound very soft and pleasant, free from all roughness, but wanting in power, and dull at low pitches. 2. Tones containing the lower partial pitches up to about the sixth in moderate prominence are produced by the pianoforte, open organ pipes, and by the human voice and the French horn in medium strength. w. p. s, 2 18 AUDITORY SENSATIONS [ch. The tones of the latter instruments form the transition to tones with high upper partials. The tones of flutes and of the flute stops of the organ with a low pressure of wind approach to tones of simple pitch. These tones of some six partials are fuller, richer, and more splendid than simple tones, but they are also perfectly sweet and soft so long as the higher upper partials are absent. 3. If only the uneven numbered partials are present, as in narrow stopped organ pipes, pianoforte strings struck in the middle, and clarinets, the tone sounds hoUow, and when a large number of such upper partials are present, nasal. When the fundamental partial predominates, the tone sounds rich ; but it sounds poor when the funda,- mental is not sufficiently superior in strength to the upper partials. 4. When partial pitches higher than the sixth or seventh are very distinct, the tone becomes cutting and rough. The degree of harshness may be very different. When their force is inconsiderable, the higher upper partials do not seriously detract from the musical value of the tones ; on the contrary they are useful in giving character and expres- sion to the music. Such tones are produced by bowed instruments, and most reed pipes, the oboe, bassoon, harmonium and the human voice. The rough braying tones of brass instruments are extremely penetrating and therefore give more the impression of great power than similar tones of a softer blend (29, I79f., 30, ii8f.). X. The character of musical tones by which we recognise from which instrument they have been produced is thus at least a peculiar combination or blend of pitches. This character may therefore well be called the pitch-blend of tones. And the above rules may be further condensed into a single statement: the pitch-blend. of a tone depends upon the group of partial pitches by which it is constituted and their relative strengths. Remarks must be made here on terminology. It is common practice to Rpeak of the lower and upper 'partials' of a tone and of the fundamental partial, without the regular addition of the substantive usually implied — tone. The term 'partial' thus comes to have not an adjective, but a substantive meaning. This practice seems to me to be a happy one. For it will be shown, as we proceed, that the common notion of a 'partial tone' rather anticipates, if it does not also outrun, the warrants of tonal analysis, which yields us primarily only partial pitches. Although this change in terminology is trivial and negligible, so far as concerns the facts of observation in question, it is of the highest importance in so far as it gives a correct leading towards theoretical construction and deduction from these facts. I] AND THEIR ATTRIBUTES 19 Nor has theory been without influence upon the name given to that character of tones of the same nominal pitch which varies with the instrument they come from. The word 'quality' (30, 10, etc.), (however familiar and safe it may be for musical practice) is obviously misleading in psychology; for there is no other case in which that word is used specially to designate a grouping of distinguishable moments, whether these are themselves qualitative or not. The same applies to the word 'colour,' or to the term 'clang-tint' adopted from the Gterman. Besides both these words suggest dangerous analogies with tljie variety and psychical status of the visual colours. Something might be said for lising the word 'clang' alone, which in English is commonly used to designate a speciail kind of pitch-blend, such as that given by cymbals. The French word 'timbre' jalthough increasingly popular in psychological works, is really impossible in English, both in its French and in its Engb'sh pronunciations. The word pitch-blend, on the other hand, has associations already only with mineralogy, which may be considered remote enough to be in- nocuous for psychology. There is no reason why we should not in psychology teach that the object well known in musical talk as the quality of tone shows itself to be psychologically a group or blend of pitches and will therefore be so named within psychology. It may then become a question for musical practice to decide whether it would not be well to adopt 'pitch-blend' in place of its own term because of the aid given by the former towards correct knowledge of the nature and means of producing the variant thus designated (108, 74f.) We have already obtained an answer to our next question : whether tones occur in which only one pitch is distinguishable. Such tones are by no means the philosophical fiction they are sometimes said to be. They do occur, however difficult it may be to arrange at any given moment ^nd for any length of time for the occurrence of a tone of a single and certain pitch. If it be doubted whether such perfectly 'simple' tones occur naturally^ there can at least be no doubt that the series of musical instruments can be arranged so as to present a series of sounds approximating towards simphoity of tone. And special physical devices of 'interference' are familiar which ensure the presenta- tion to the ear of a perfectly uniform and regular aerial vibration and so the hearing of a tone of a single pitch^. Such pure tones can be procured at any height of pitch within a large range of variation and it thus becomes highly probable that every tone, no matter what the height of its pitch may be, can be obtained perfectly simple in pitch. Thus we arrive at the series of tones that is primary to all the parallel 1 Cf. 102, 3ff. Even tuning forks give at least the octave, if not other partials. The octave partial from a fork originates, not in the fork, but In the air as a result of certain physical processes (of. Lindig): "Thus it is practically impossible for simple tones to be produced directly by any source of sound" (p. 4). * Cf. loc. cil. Schaef er adds a second means — subjective abstraction: but, as we shall see, that is really unable to extract a pure tone from a pitch-blend. 2—2 20 AUDITOEY SENSATIONS [ch. series derived from the different musical instruments. No experi- mental means is known by which the pitch of a tone can be modified without any change in the auditory stimulus or in the other charac- teristics of the tone itself. Our series of tones thus seems to be a primary series. And our next problem is to see whether we can derive from this series such attributes as our introductory formula leads us to expect. XT. The whole interest of the psychology of the auditory attributes must obviously centre on pitch. This variant, with whatever is involved in it, is the only important variant in the series of the tones of simple pitch. It is of course open to theory at the outset to identify pitch with any of the variable attributes of sensation. But of these intensity and temporal order are obviously irrelevant, only quaUty and systemic order can be seriously considered. Under which of these two heads does pitch fall? Psychological theorists have been almost imanimous in their pre- ference for the quahtative classification. Pitch is solely, or primarily, a variation of quahty or it includes that within it, whatever else it may be. For a wholly or primarily ordinal classification not a single voice has been raised, so that its prospects might well seem hopeless. The qualitative classification has sometimes been rejected in favour of a quantitative one. " Till the time of Aristotle tones were considered to be essentially not a iroiov, but a iroaov" (111, 136), i.e^-quAntitative, not quahtative. The reason for this Stumpf finds in the Pythagoreans' exclusively mathematical treatment of hearing, which here, as elsewhere, obscured all quahtative differences. But this reason is only good so long as the treatment of tone as quantitative and not at all quahtative is radically wrong. If it is in any way right, the presumption is that the Pythagoreans saw clearly what later theorists have allowed their preconceptions to hide from their view. Mathematical interests might well be the means of drawing the attention to the quantitative aspect of tones. Among modern writers two names may be mentioned. E. Gurney,- writing in 1880 (28, 139), said that differences of pitch are not differences of kind or intensity, but differences of distance and direction, "clearly and indisputably felt as such." But, although this view makes a near approach to certain aspects of the theory developed in the present work, it does not probe down to the fundamental analysis of attributes that we have now under consideration. A quantitative treatment of pitch was urged by K. Dunlap in 1905 (11, 12). I] AND THEIR ATTRIBUTES 21 " Mach was the first and only one to express the idea that tones he next one another in a sensory space, like the colours in the field of vision, only with the difference that the place of any colour is changeable, whilst the place of any tone is unchangeable" (112, 55). In so far as we may identify spatial with ordinal arrangement, we may modify this statement to the effect that Mach was the first and only one to hold that tones include within them an ordinal aspect. But the series of tones varies quahtatively as well, in his opinion. It is therefore only partially ordinal. Although, as I showed in the introductory chapter, a system of locahsations cannot be truly held to be primary in sensation, there can be no doubt that Mach intends the differences he defines to be considered primary. But his view is not devoid of obscurity, as he points out that the differences in question are only analogous to the differences of locahsation found in A^sion, and not really the same (cf. 112, 55, loi, 125, etc.). "A given tone sensation," wrote Mach, "can occur only at a fixed point of this unidimensional space, which must always be fixated if the corresponding sensation is to emerge clearly" (60, i23)i. Mach's work contains many suggestive hints of what I consider to be correct psychological analysis, and I find the use of the term 'fixate' in this representative sentence a happy one. The same apphes to his idea of elements common to all tones, to the ordinal notion of the tonal series, and so on. But these good suggestions were not brought together by Mach in such a way as to convince even himself, not to speak of others. It would therefore be wrong to suppose that Mach had properly discovered these notions in their significance. As they appear in his work they are rather such ghmpses of (what I at least consider to be) the truth as one will find, after any particular psycho- logical field has been cleared up, in countless earher works dealing with that field. K. Dunlap (11, 290) actually considered the classification of pitch as local sign, but rejected it, — "since local signs do not in general vary between two extremes, but rather include a manifold of differences, which do not admit of easy schematization." And, he added, pitch admits of quantitative comparison, while local signs do not. And even Stumpf was impelled to admit, while discussing Mach's views, that the quantitative arrangement of tones is analogous to a spatial arrange- ment. That looks like the inevitable glimpse of a suppressed truth. To me it seems clear that the poor support given to the ordinal ' In the 2nd ed. of 1900, of. p. 180 fi. 22 AUDITOEY SENSATIONS [ch. classification is due entirely to misconceptions regarding the psychical status of space and to inadequate methods of dealing with the attributes. For if an attribute of localisation is admitted in the other senses, and if sounds are also locahsed (no matter whether the binaural basis of their locahsation is famiUar or not), how could any one propose intro- ducing locahsation into hearing a second time? The very idea would be absurd to any one starting out from locahsation. Hence, the com- plete obstruction of progress caused by the prevailing excess of attention to space and locahsation. The methods of deahng with the attributes we have already considered. The nature of the case, on the contrary, really insists upon the ordinal classification so strongly as to put the quahtative alternative out of court, refuted as that clearly is by the sheer chaos of conflicting views to which it has given birth. If it was impossible in our introduc- tory formulation of the probable attributes to adnait localisation or any spatial reference as an attribute of sensation, it would be as absurd as it is unnecessary to drag it into the series of differences given by simple tones. The absence of any spatial difference from that series is no reason for supposing its absence to be due to illusion or to want of psychical variation. For the reason surely presupposed by the absence of psychical variation, namely the absence of the physiological conditions of that variation, has been denied since Helmholtz's day. There is no want of psychical variation at all, but only a want of spatial variation. And this very want confirms our view that spatial indices are not in any sense attributes of sensation. For the ordinal classifica- tion the presence of a system of orders is enough. The question then is : is the series of simple tones, the series of tones in each of which only a single pitch can be distinguished, a system of orders? In 1883 Stumpf wrote with regard to the classification of pitch as quahty: "from, the psychological point of view it is so obviously correct as to need no defence" (111, 136). I feel inchned to write the same about the ordinal classification. For both statements seem able to claim justification directly from the phenomena before the mind in the tonal series. Taking this phenomenal presence to be the chief concern one might feel inchned to say: "the mere name is a matter of moonshine ; what's the difference so long as you and I are pleased with our views? We shall never agree." This would indeed be the case if the proposed terms were nothing but names for the special phenomena of tone, as is primarily true of the name 'pitch.' But they are not so. They involve more complex operations of thought than do the simplest I] AND THEIR ATTRIBUTES 23 types of classification. Here we have to take many phenomenal objects together to consider their hkenesseSj differences, connexions and changes, and then to express our ideas about them on the basis of this large survey. The fact that the object of study is primarily phenomenal is now irrelevant. It is being treated largely as if it were real, as if its nature were largely beyond, or independent of, its presence before, or direct contact with, our thought. But it is always possible for thought to return from the realistic attitude armed with a reformed and classified conceptual vision or disposition, and taking the phenomenal attitude, referring itself most directly to the phenomenal object, to say: now, are not these tonal differences ordinal, and not quahtative? and to receive a positive answer with complete assurance. After all, that is what Stumpf meant when he said that "from the psychological point of view " his classification is " so obviously correct as to need no defence." He is wrong only in impljring that there is but one psychological point of view in this case, and that any psychological point of view can dispense in such a case with a defence on the systematic Unes I have just indicated. XII. With this in mind we may now review the arguments in favour of the ordinal classification: 1. Phenomenal evidence, (a) Direct. My first and last argument is then : the series of pitches is ordinal ; it is unidimensional ; and in it every tone occupies one and only one place, which can be determined to a very high degree of accuracy. If we take the tonal series in a number of discrete pitches, e.g. the tones of the chromatic scale, and apart from differences of volume to be considered later, the series can better be described conceptually in terms of order, as 'this one,' 'that one,' 'the next one,' and so on, than in quahtative terms 'this sort of one, 'that sort of one,' 'the other sort of one.' This is confirmed by the use of already estabHshed ordinal series, — e.g. place upon a set of real or imaginary Unes, or the series, a, b, c, d, etc., — for the naming of pitches. If pitches were really quahtative, we might have expected to find them named after the objects that utter the different, pitches, e.g. the names of birds and animals, as colours and smells are named after flowers, etc. If reply be made that the spectral colours are now named with numbers, it should be noticed that these numbers apply only indirectly through the fines of the spectrum to the colours themselves and that this conceptual system has not yet been apphed to the colours of the colour body, and probably never will be, in spite of the obvious 24 AUDITORY SENSATIONS [ch. advantages which would accrue therefrom. In short, the series is ordinal. If we take the series in a continuous form, as given in Stern's piston bottles, or the like, it forms an ordinal continuity. (6) Indirect. Those who adopt the quaUtative classification admit the presence in what we ordinarily call pitch of features whose common designations ultimately imply ordinal differences. Stumpf, who has already been quoted in this connexion, says the power of spatial symbohsm among tones is extraordinary. The conception of the tonal series as a 'rising' series "seems to the present day musical mind to be so directly given in the nature of tones, and so obvious, that the expression 'tone-quality' in place of 'pitch' is liable to convey no meaning" (111, 190). Whatever may accoxmt for the 'rising' aspect of the series, it seems clear that a rising series implies an ordinal basis, just as much as does localisation. Of course it obscures the solution of the problem to look for really spatial relations within the 'rising' aspect. 2. Evidence from discrimination. The threshold of discrimination for simultaneous pitches lies, in the middle of the musical scale (from G to e', or from 90 to 1200 vibrations per second), between ten and twenty vibrations per second of difference (103, 9it.). Above these low limits of difference it is always possible to distinguish and to separate the component pitches of a sound. This is true even in the case of pitch-blends, where our ordinary knowledge of the nature of the stimulus does not lead us to expect to find component pitches. It will therefore hold d fortiori in the case of chords, where we do expect to find components. This independence and self-maintenance of original components is not found in vision, and vision is the only sense that gives us an easily changeable and unmistakable system of quahtative differences. Analogy between the mixture of pitches in a blend and the mixture of visual qualities can therefore hardly be held to support the quaUtative classification of pitch. But if the ordinal classification of pitch is adopted, the ease of pitch analysis receives a ready explanation. This evidence of discrimination is seconded by the evidence of memory known as absolute hearing or absolute ear. This exceeds by far the finest memory for shades of colour. 3. Systematic evidence, (a) Noises and vowels. The other classes of sounds yet to be studied — noises and vowels — admit of explanation with the help of the ordinal attribute in a simple way that is not open upon the quaUtative Une. The latter is compeUed to assume the presence i] AND THEIR ATTRIBUTES 25 of a new set of qualities in noises or in vowels or in both, or to attempt to establish relations of primacy and derivation amongst these sets of qualities. But any such relations must remain entirely hypothetical, because there is no analogy to them in any other sense, not even in vision. They therefore fail to carry conviction even within the quaUta- tive camp. (6) The systemic attributes. Under the name pitch we commonly include not merely an ordinal series, but a series of changing volumes (which we shall study later). In the other senses we have reviewed, volumes and extents or masses imply the conjunction of extensity with ordinal differences. "We should therefore expect the volumes of the pitch series to involve ordinal, not quahtative, differences, which in no other case combine with extensity to give volume. (c) The formula of attributes. The classification of pitch as order seems indicated by the mere apphcation of our introductory formula of attributes to the series of simple tones. The presumption regarding qualities is that hearing will contain only one or a few discrete miscible quaUties. As the latter have hardly been claimed to be present till within recent years, they are at least not strikingly obvious. One quaUty is patent, viz. the quahty of hearing as such, that distinguishes itself from taste or vision. For a parallel to order, our formula does not lead us to look for any form of locahsation, as we have already tentatively declared that to be more than the content of any single sensation. Why not then try to fit pitch upon order ? Then the volume of tones would j^eld us extensity, and all would be well. Pitch as order would thus fully confirm the formula of attributes for all but the crazy sense of smell, which after all could be accommodated in spite of its obscurity. Hearing, as generally analysed to-day, is the only sense that offers any serious diflGlculty in respect of the ordinal attribute. If it were accommodated, the sense of smell might pass unchallenged. (d) The integrations of sensory experience. The ordinal classification receives great and decisive support from a systematic study of the integrations of the several attributes in the various senses. If distance, direction, motion, etc., result in many senses from the integration of the attribute presumed to be inherent in differences of localisation, — an attribute which very many of those who adopt the qualitative classification of pitch consider to be at least non- quahtative, — and if in sound we discover processes closely akin to distance, direction, motion, etc., viz. distance and interval, direction, and melody, etc., which are admittedly dependent upon variations in pitch; then it follows that 26 AUDITORY SENSATIONS [oh. the attribute inherent in differences of pitch is at least non-quaHtative, and is presiunably identical with that inherent in differences of localisa- tion, — ^in our case order. By a parallel study of the senses we should thus be called upon to regard pitch as essentially ordinal ; and conversely the classification of pitch under order offers a key to the proper arrange- ment and interpretation of the complex phenomena of hearing. 4. Psychophysical evidence, (a) Anatomical. These psychological pleas are supported by two psychophysical ones. The first is ana- tomical. The afferent nerves attached to the receptor of hearing are spread over it in a thin Une of varying curvature. No matter how this hne of receptors functions, whether transversely or longitudinally, or in both ways, we shoidd expect it to have some parallel representation in our elementary experiences of hearing ; and the series of simple pitches is the only parallel to be found. Thus for our ordinal series in hearing we should have an ordinal series in the body. (6) Genetic. If that plea is granted, we can then readily understand how the series of tonal experiences has been developed, without trans- gressing the bounds of well accepted biological principles. We caii see the finger of nature shaping the marvellous organ of hearing towards perfection, and need not vainly inquire how that organ came to be alUed with a series of qualities with which it has no inner kinship and which it could not have produced of itself. In short the series of simple pitches itself and the whole theory of hearing asserts and demands the ordinal classification of pitch. I have here anticipated much evidence that can only be adequately dealt with separately as we proceed. But it is well to puU the whole mass together aroimd the crucial point of the analysis. We have thus completely disposed of the quahtative classification in so far as that may be appKed in any simple manner to the whole series of tones of simple pitch. Other forms of the quahtative classification which have been offered in recent years, wiU be dealt with as the sets of facts on which they are founded, come within our view. XIII. But the ordinal nature of pitch is only part of its whole content. It is a 'rising' series besides. And if we have identified the attributes of intensity and order, we have not yet definitely settled how we are to accommodate the attributes of extensity and quality. Of these quaUty already has a place, as we have indicated under 3 (c) above ; it is mere hearing as such ; there must be at least one quality in all hearing. The only question which now arises is : are there several 1] AND THEIR ATTRIBUTES 27 or many forms of quality in hearing? Can any changing feature of the tonal series other than its order be properly classified as quaUtative? I shall now discuss this question in relation to the 'rising' variation of the tonal series, leaving other aspects of the tonal series till later. My formula of the attributes calls for the attribute of extensity in conjunction with order. Thus we obtain the problem: is the 'rising' variation better classified as quaUtative or as extensive? Of course extensity alone, being a non-variable attribute as far as our attributive formula has revealed it, could not explain a variable series; but it could do so in the complex variable form in which we have discovered it, especially in the second group of senses, viz. in extents, volumes, or masses. The following considerations favour the extensive attribute: 1. Phenomenal evidence — direct. My first and last argument is as before: the series of tones of simple pitch contains a variation of volume or mass. Low tones are great and massive and all-pervasive; high tones are sharp, thin and circumscribed; as tones rise in pitch, their volume shrinks gradually, pulling itself together as it were, till it is finally almost too small and thin to be noticed. This is true whether the series is presented continuously or in discrete pitches. And it is confirmed by the conceptual terms commonly used to indicate these variations of tones without regard to their specific pitches. In EngUsh we speak of high and low tones, of sharp and flat tones, where neither specific pitch-place or pitch-blend is referred to. Stiunpf, who collated the terms in use in a number of languages, said: "not all tongues use spatial expressions for differences of pitch, and those that do so, not in a thoroughly uniform manner. But yet expressions analogous to the modern ones are the most general, and are to be found alongside even where others predominate in technical usage " (111, 192). Nevertheless at the time of writing the first volume of his work on hearing, Stumpf was of the opinion that deep tones only seem to be more extensive; they are not phenomenally so, they are not 'given' so (111, 210). So he was forced to trace the spatial associations and the spatial apprehension of tone to such differences as commonly accompany differences of pitch, especially to differences in their duration, in their spread throughout the organism during hearing, in their feehng- character, and in their strength. The association is thus essential, not accidental (111, 223). We may well agree that Stumpf has thus shown how the association between tones and spatial relations, especially those of height and depth, has arisen. Tones are not spatially higher 28 AUDITORY SENSATIONS [ch. or lower; that is obvious to everyone in spite of the use of these terms ; the terms are clearly used in a figurative sense. Nor are they spatially massive, or thin, or flat. We transfer all these terms to them because, being terms of practical hfe, they are more famihar and demonstrable to everyone, and so afEord a ready means of describing tonal differences. The basis for a transference of terms by association is thus given. But we surely do not mean by these terms that high tones are either more continuous than low ones (we cannot use the term 'smoother' as Stumpf did (111, 203ff.) because that is really meaningless, unless tones have a volume which can be smooth) ; or are more intensive for the same power of stimulus ; or come from smaller instruments ; or pene- trate less throughout the body; or take longer to hear and recognise and to fade out. None of all these differences corresponds directly to what is imphed in the spatial references that have to be explained. Surely we mean that tones do differ phenomenally in their extent or volume. And although Stumpf did not expUcitly reject the reasons given for his earher view, yet in his second volume he definitely accepted the primacy of a variable 'extension' in tones of different pitch^. A word of caution must be repeated here even at the risk of boredom to those who understand. By order and extent or volume I do not mean spatial order and spatial mass or volume. If the analysis I offer is correct, I am sure the failure of previous psychologists to reach it is due, apart from lack of method in the study of the attributes common to all the senses, first and foremost to the fixed idea that there can be no order and continuity unless they be spatiaP. ^ 112, 66. A. Lehmann (50, 119) also adopts this view and argues from the relation between extent of sensation and extent of stimulated surface in other senses, e.g. touch, to a variable extent of stimulated nerve fibres for tones of different volume. K. Dnnlap (11, 291) may also be mentioned- "Differences in pitch are directly comparable to differences in planar or linear extent, and the physiological condition of differences in pitch accordingly is probably diffeience in number of nerve-endings stimulsted." ' The obstruction of the spatial preconception is specially evident with Stumpf, who indeed comes within an ace of seeing it as such (112, S4ff.). From his nativistic principles he ascribes a special difference (pq) to the experiences of either ear, but although th^e differences become associated with the spatial localisation of the ears as parts of the body and are involved in the functions of binaural locaUsation, they need not, Stumpf says, themselves be localisational in the ordinary sense, but only distinctive of the experiences of either ear. Having thus properly for the moment got beneath the localisa- tional process as it were, and having suspected the presence of a 'we know not what ' he proceeds to ask "whether amongst the tones of one and the same ear differences of the kind pq are to be found," and replies : "not a trace." But in this answer he has already il, AND THEIR ATTRIBUTES 29 The order of numbers is not spatial nor is the order of times, although the most modern of philosophers Bergson is so obsessed with the idea of space, that he would have us beheve the intellect itself to be similarly aflBicted and to have crushed the pulsing soul into its cast-iron moulds. No, there are many kinds of order, and space is only one of them. Why should there not also be many kinds of continuity — of however many dimensions — and tonal volume or any sensory extent be only one species thereof? Then we need only inquire why tonal volume comes to be named with terms borrowed from spatial volume and why the pitches of these volumes are said to rise. The answer, as Stumpf has shown, is clear enough. 2. Evidence from discrimination. As we shall see in more detail later on, we can discriminate tones in respect of their volume as well as in respect of pitch. But our judgment is much finer in the latter case. Wherever only small differences are given, we rely wholly on the change in pitch (123, 320f.). Where the pitches offered he so far apart as to fail to exhibit the musical relations usually borne by tones of the same octave, their difference of volume becomes more obvious. It is not possible to estabhsh the difference of discriminabihty of pitches and volumes by direct experimentation with the tones of the musical range, because no independent variation of pitches and volumes is possible. But in the extremes of the tonal series, in very high and very low tones, at the hmits of the musical range and beyond them, pitches become much less discriminable ; they seem then to be less discriminable than the accompanying volumes (112, 57 and 123, 3i7fE.). 3. Systematic evidence, (a) The presence of volumes is not quite so distinctly traceable in the case of noises and vowels; because the series formed by these sounds are not so perfect. But there can be no doubt that noises and vowels differ amongst themselves in respect of volume. In fact differences of voliune are more characteristic of noises than are differences of pitch. (6) The impUcation of the attribute of extensity by the attribute of order and (c) by the attributive formula need only be mentioned. (d) The special systematic plea for the admission of variations of volume, and so for the attribute of extensity, is given in sotmd by riaen from his momentary plunge back to the surface of the localisation barrier. Apart from the obvious results of mere association given in the expressions 'higher' and 'lower.' he says, " we notice no spatial togetherness of tones and none such has been maintained even to most recent times." And thereupon he admits the primacy of 'extension ' amongst the tones of one ear, denied in his first volume! 30 AUDITORY SENSATIONS [ch. the phenomena of fusion that are so characteristic of this sense. These seem to be exphcable only by reference to coincidences of volumes. 4. Of the psychophysical arguments the anatomical (a) is not so evident and demands the careful examination and discussion of the rather uncertain observations that have been made. But it seems clear on various grounds that the occiirrence of even a tone of simple pitch requires the excitation of a short length of the series of neural termina- tions in the cochlea. The exact amount of this length is not quite evident either on mathematical physical grounds or on the basis of experimental injury of the cochlea. The genetic argument (6) is here the same as before, in so far as with the assumption of volumes the mystery of the quahtative view is removed. XIV. The series of tones of simple pitch then fully satisfies the requirements of our introductory attributive formula and the sense of hearing thus far conforms to the probable sensational type, making this type therefore still more probable. It thus becomes probable that the same scheme of allocation of the attributes will apply to the other groups of sounds that do not fall within the series of tones of simple pitch. The next problem now is whether these tones are both actually, and really or theoretically, the simplest auditory sensations. XV. It must be at once clear that as tones form a series of decreasing volumes, they may well be the simplest auditory experiences we ever actually obtain, but they cannot be the theoretically elementary sensa- tion. They may be the molecule of sensation as it were, but not the atom. That must be true of every tone except possibly the tone of smallest volume, viz. the tone of the highest pitch at the upper limit of the range of hearing. The rate of physical vibration required to pro- duce this last tone varies for different persons and decreases somewhat in advanced years. It may be set approximately at some 20,000 vibrations^. But it is not certain whether the soimds produced by higher rates of vibration are merely the noises which accompany the production of these rates of vibration or are themselves soimds (noises) directly produced by these rates of vibration in place of the previous tones. An answer to this question is not very important. In dealing with the senses of the first group we have seen how the * V. e.g. 123, 327. On the limits of the pitch range compare K. L. Schaefer (102. 7ff.). The lowest limit may be put at an average of 16-20 vibrations. Below that the occoTTence of tone is uncertain. I] AND THEIE ATTEIBUTES 31 variation of the extent or area of sensation is dependent upon the area of skin affected by the stimulus and thus upon the number of minimal spots of sensation of different order evoked at once. The variation of mass or volume seen in the senses of the second group thereby received a ready explanation. It seems inevitable that we should extend this explanation to the volume of hearing^. Each tonal volume would then be a mass of — shall we say — ^undifferentiated tonal orders. But surely there is a differentiation of tonal orders within the mass. Is not the pitch of a tone obviously the intensively predominating order of the volume of that tone? Of course great difl&culties arise as soon as we attempt to discover in each case whether more than one order is involved in the predominance we find, and if so how many. Such questions are almost as hard to settle as it is to say whether the spot of cutaneous sensation is the absolute minimum or not. But certain indications are of great service. Stumpf says : " I will not conceal that simple mild tones sometimes seem to possess a pecuhar elasticity in the matter of pitch. When the pitch of the tone of a tuning-fork is to be reproduced upon the vioUn, it can happen that the player oscillates within a quarter tone. When a bottle is blown whose tone lies in the small octave, and the corresponding tone is sought on the piano, the former can sound now Uke /, now Uke /# ; it seems Uke that one of the two tones which is just being played" (112, lu). The pre- dominating part of the tone is here obviously far from punctate. Probably the predominance of tone becomes sharper and clearer as the pitch of tone rises, being broad and gradual hke the rise of a grassy knoU in the low tones, and becoming sharp and pointed like a pyramid in the higher regions. The addition of upper partials to a tone helps to reduce its smooth predominance. How this comes about we shall consider later. For the present I shall neglect details which are subordinate to the general solution of the problem and speak of the pitch of a tone as the predominant order in that tone. We can go still further in our intellectual apprehension of tones. Tones of simple pitch, e.g. those of a weU-played tuning-fork seem to everyone wonderfully smooth and regular, beautifully rounded off systems (of elementary tonal sensations). Leaving out the aesthetic reaction we may say : tones are regular systems of ordinally different, elementary sounds, in which 'one' element predominates. And there is no sign of any want of balance and no asymmetry in any of these systems, and no difference between the different tones in respect of 1 This extension is made as such by A. Lehmann (50, 120, 131); of. below, p. 145. 32 AUDITOEY SENSATIONS [ch. the system they consist of. There could therefore be no reason for denying the supposition that the predominating pitch of each tone is central to the whole system. In disciLssing the question whether there are perfectly simple tones Stumpf (112, 272) referred (without indication of the source) to a theory by Hostinsky which has some affinity with the theory just stated. But its foundation seems to be rather physiological than psychological; an apparently simple tone is held to consist of as many tones as there are neighbouring resonant fibres. And in so far as its founda- tion is psychological, it rests upon indirect evidence, not upon direct : because so- called simple tones form a series which would be impossible were they really simple sensations. Stumpf said of this theory and of Mach's : "In neither case is an attempt made to establish the presence of these elements, it is a case of mere hypothesis. These must justify themselves by their purpose, by the theoretical need they serve." My theory does that as well as being a direct expression of the phenomenal nature of sounds. In discussing the same problem of simple tones at another point (112, 111&.) Stumpf brought forward as evidence against the view above referred to (Hostinsky) that in a simple tone none of the hypothetical neighbouring 'tones' are actually to be heard by the analytic attention. This expectation appears from my point of view as a misunderstanding of the problem. We cannot expect any but thp predominating order to appear as such in the tone. But Stumpf came very near to my view in the passage quoted in the text about the pitch of simple mild tones. Only he rejected this clear indication by saying that even "if the tone really in the sensation changed its pitch within a semitone according to the momentary direction of att/ention, this would anyhow be quite different from hearing, apart from a middle tone, simultaneously others differing by a quarter tone upwards or downwards. Elasticity is not extension." The theory of Hostinsky may be compared with that of Lehmann (50) referred to above (p. 28, note 1), and sketched below (pp. 145, 159 f.). Lehmann's theory is a little better than Hostinsky's in so far as the former recognises the psychical reality of volume and gives the extensive parallel in touch. C. S. Myers and others have made less confident suggestions regarding the volume of tones. But none of them has really developed the matter beyond a first suggestion. XVI. But all these predominating pitches form a continuous series of orders. And this series has two ends, all the pitches higher than the lowest lying on one side of the pitch of the lowest tone. The ends of the series are therefore not only distinguished by their difference of order, but also by the different size of the volumes in which they regularly occur. Thus we get a perfectly adequate basis for the associa- tions which are incorporated in the terms by which we name tones, viz. high — ^low, sharp — ^flat, rising — falKng. And all the psychological characteristics of the tonal series thus far encountered have been explained. I] AND THEIE ATTRIBUTES 33 We may finally return to the pitch-blends which, as attentive analysis shows, constitute the differences in the musical sounds of the same nominal pitch as they are produced from different instruments. Apart from special efforts of analysis these sounds seem to be quite simple. Neither their predominant pitch nor their volume seems to vary with the instrument from which they come. They vary only in the pitches which dan be distinguished within these volumes. In all of them only one pitch predominates — the fundamental pitch of the tone. But whereas in tones of simple pitch no other pitches are distinguishable, the whole tone being a regular system of sounds; in pitch-blends several other auditory orders of varying dominance appear, all being much weaker than the fundamental, but varying irregularly in dominance amongst each other. None of them leaves the imity of the whole unchangeable volume so as to fall outside of it and away from it, and so to constitute a really separate tone. It is for this reason that the pitch-blends of good musical tones are ordinarily noticeable only as a variant upon the corresponding tone of simple pitch (i.e. the pitch of the fundamental). That is after all the predominant pitch. The other pitches which dominate relatively within the whole only serve to give a slightly different character to the system of sounds that constitutes the musical tone. This system has not the simple regularity of the tone of simple pitch. Its system has a shghtly different character; it becomes asymmetrical, receiving a greater basis of dominance at one or other part of its volume. If the proportion given by the dominance of the fundamental is removed by its weakening, the chief character of the tone disappears, and the tone is ' poor.' The more the fundamental and the partials near it dominate, the richer will the tone be. When the higher partials become more prominent, the tone must take on a high or bright character, because its higher parts become more noticeable. Of course, as Helmholtz indicated in his summary rules, other factors than just the spread of the component pitches over the whole volume help to determine the character of a pitch-blend, e.g. fusions, harmonies, etc. The nature of pitch-blend may be otherwise expressed in relation to the tone of simple pitch. This pure tone we may for the moment consider as the ideal of music — a fictitious philosophical ideal. All good musical pitch-blends are approximations towards this ideal. In them all the upper partials are so subordinated to the fundamental that they cannot be distinguished with the unaided ear, imless a systematic, analytic, search is made for them, or unless the hearer has acquired w. p. s. 3 34 AUDITOEY SENSATIONS [ch. an unusual disposition for attending to them. An effort is obviously made by the musician to make the tones of his instrument as balanced and regular as possible, like those of pure tones. But he does not go beyond a certain point in striving towards this ideal. Once the easy analysis of partials is obviated, he is within a sphere where the advantage of keeping away from the ideal greatly exceeds the advantage of approaching it further. For a great variety of approximations to the ideal is thereby attained. These together make up a practical ideal which is close to, but not identical with the theoretical ideal with which each one separately is related. Other evidence in favour of this interpretation of the relation of pitch-blend to the psychological constitution of the tone of simple pitch will be brought forward (below, p. 71 ff.) when we are free to consider the phenomena attendant on the simultaneous presentation of the stimuh which separately would give tones of simple pitch. Having completed our preUminary study of tones we may now pass on to the other famiUar group of sounds — noises — with the presimiption that the results we have obtained from the study of tones will suffice to explain the peculiar phenomena of noises, if that explanation is not also derivable directly from these phenomena themselves. XVII. B. Noises. " The nature of the difference between musical tones and noises," said Helmholtz in the opening hues of his work (29, 14; 30, 9), "can generally be determined by attentive observation without artificial assistance. We perceive that generally a noise is accompanied by a rapid alternation of different kinds of sensations of sound. Think for example of the rattling of a carriage upon paving stones, the splashing or breaking of a waterfall or the waves of the sea, the rustUng of leaves in a wood. In all these cases we have rapid, irregular, but distinctly perceptible alternations of various kinds of sounds, which shoot forth spasmodically. In the howhng of the wind the alternation is slow, the sound slowly and gradually rises in pitch and then falls again. It is also more or less possible to separate restlessly alternating sounds in the greater number of other noises. ...On the other hand a musical tone strikes the ear as a perfectly undisturbed uniform sound which remains imaltered as long as it exists, and it presents no alternation of various kinds of constituents. To this then corresponds a simple regular kind of sensation, whereas in a noise many various sensations of musical tone are irregularly mixed up and as it were tumbled about in confusion." Later on, by ioference from i] AND THEIR ATTEIBUTES 35 his physical and physiological theories, he showed how steady transition "between noises without any determinate pitch, and compound tones with a determinate pitch may be produced." "This actually takes place," he said, " and herein hes the proof on which Herr S. Exner has properly laid weight, that such noises must be perceived by those parts of the ear which act in distinguishing pitch^." Now we are not at present concerned with the identity of the organs of hearing. In fact we must ask by what right any such proof is thus established. Does not the proof he in the psychical identity of being that is shown to hnk tones and noises in spite of their great differences ? Noises are essentially the same things as tones ; only tones are regular and of determinate pitch, while noises are irregular and of more or less indeterminate pitch. Not that they have no pitch at all. It merely changes rapidly, or there are many pitches at once, so that none predomi- nates so as to determine which pitch the mass of sound shall be held to have. Obviously then, one and the same organ, especially such a complex series of organs as those of the basilar membrane, may produce both tones and noises. It has only to act regularly in the one case and irregularly in the other. Variants that are essentially the same in nature, even although this nature be psychical, can be produced by the same organ. The argument then is primarily psychical. The critical point of Helmholtz's exposition is therefore the postula- tion of indeterminateness of pitch and the assumption of its existence in noise. No exception can be taken to the methods by which Helmholtz estabhshed the existence of indeterminate pitches. But one might well ask whether Helmholtz's psychological system will accommodate such a thing and how. It certainly does not exclude it; for he took over the chief terms ^ describing tones in a rather free untechnical sense and without comment from ordinary speech. And so he was free to extend the notions conveyed by them whenever he saw reason to do so. This is of course a merit in his method. And impatient of psychological subtleties, one might feel incUned to say: Helmholtz did well not to bother over purely subjective matters, which in any case cannot be decided, being a mere matter of personal fancy. But such a view would ignore the primacy of the psychical problem, which has just been 1 30, 151 from H.'s fourth edition. Earlier he did not agree with Exner. ' In the first pages of his work Helmholtz spoke of the force, pitch and Khmgfarbe (which Ellis translates as 'quality') of musical tones. Later on (29, 232) he spoke of the different quality of a tone sensation according to its pitch, thus using quality in the technical attributive sense. 3—2 36 AUDITOEY SENSATIONS [ch. indicated, and will be set out more fully as we proceed, and would try to improve on neutrality by adopting an unfriendly attitude. If Helmholtz's progress was not obstructed by any wrong preliminary classification of pitch, it was not aided by a right preliminary view. And it never got the help of this right view, although in certain respects the physical and physiological ground which Helmholtz prepared for the reception of psychological analysis would have suited it admirably. Popular psychology, Uke popular dietary, has good reason to be highly successful. But it is just as likely to be incapable of improvement. And Helmholtz was forced to justify whatever psychological assump- tions he made or adopted, as soon as they seemed questionable. Thus in his study of pitch-blends he began by assuming in an apparently very harmless way, that "the ear... does not hear merely thai one musical tone... but it hears besides a whole series of higher tones, which we call the... upper partial tones" (29, 37; 30, 22). Had Hehnholtz kept strictly to the facts, he should have written here partial jntches instead of partial tones, just as I have done. But he seems again to have taken over the common usage without questioning its psycho- logical vahdity and without even carefully scrutinising its exactness as a description. Had he limited himself to partial pitches his judgment upon the discussion raised by Ohm and Seebeck^ and his psychological reflections upon the analysis of tones (29, i02fi.; 30, 62fi.) would un- doubtedly have been very different. He would not have thought the difEerence between the analysis of mixed colours and pitch-blends was due to greater instrumental practice in the latter case. The use of the innocent looking word 'tone' instead of 'pitch' led him to expect of tonal analysis by mere attention what it can never do. It can never make tones more separate than they originally are when given to it. An immense amount of obstruction has been created by this little word tone so innocently and so plausibly inserted instead of the certain fact of pitch. Stumpf's discussion of the psychological nature of noises, for example, was profoundly affected by it. At many points he used the word tone where obviously only pitch was meant. Thus when a * 29, 100 ff.; 30, 58 ff.: "The dispute turns upon whether in all cases upper partials can be perceived analytically in their individual existence; that is, whether the ear when unaided by resonators or other physical auxiliaries, which themselves alter the mass of musical sound heard by the observer, can by mere direction and intensity of attention distinguish whether, and if so in what force, the octave, the twelfth, etc. of the funda- mental exists in the given musical sound." I] AND THEIR ATTRIBUTES 37 single stick is thrown on the ground, it was said, no 'tone' is heard; but it is readily recognised if a series of sticks of the appropriate sizes are thrown on the ground. A moment's recollection of this famiUar experiment suffices to show that in the serial part of it only pitches are heard and recognised, not tones, unless we mean by tones merely recognisable pitches. So too when a tone was said to be heard from a brook. And Stumpfs test of the absence of 'tone' from subjective (ear) noises seems to have been: how nearly their pitch could be estimated (112, 500ff.). With regard to theories, Stumpfs chief objection to the view that noises consist of many simultaneous tones httle different in pitch, was that the resultant sound does not, or would not, lose " its tonal character and its analysabiUty" (112, 504). This double concept seems somewhat strange. And of noises, in so far as they are said to be a very rapid succession of very many tones of different pitch, he said, we should still hear the "rapid change of tone." But what constitutes 'tone' when the steadiness of pitch is gone? Is it the mere change of volume? And do not noises have volumes as well? Do not noises show among themselves all or any of those kinds of differences which accompany change of pitch in tones? Having rejected these two theories Stumpf then favoured the view that noises are tones of a definite, not necessarily changing, pitch, and are distinguished from tones in the usual sense either by their being momentary or by their being a rapid succession of intermittent momentary tone impressions. Here he seems to have got rid of the tonal phenomenon, while still retaining (as very brief) real tones in the noise. But even this theoretical device did not quite convince him. He added : " One set of so-called noises are intermittent tones of the highest or lowest region (growing, hissing, etc.). Here it is by no means impossible with increased attention to recognise the tones as such. Still I would not maintain even in these cases, that soma scrap of pure noise is not left over^." And then some noises, as he said, are surely constant, e.g. ear noises. Noises might therefore still be special sensations not reducible to tones, but Uke enough to them to justify their reference to the same organ as a different quahty of hearing from tones. With all this, he thought, some noises might still be more like low than high tones or conversely. Thus a ' 112, 509. Stumpf wrote as late as 1914: ''That all noises contain admixtures of tone, I should now no longer maintain" (123, 341). At the same place he spoke of noises with a limited zone of pitch, which yet do not include 'real tones,' or do not therefore become 'tones.' 38 ATIDITOEY SENSATIONS [ch. good meaning would still rest in the oft used expression : "a Mffererux of pitch, but no tone, was noticeable'':" Clearly the idea by which the whole of Stumpf's discussion is ulti- mately governed, is this : it is almost impossible to assent to any sort of reduction of noises to tones, because although noises may have some sort of pitch, recognisable under the most favourable circumstances, and some sort of volimie, and any or all of the other variable features of tones, and so be classifiable with them as sensations of hearing, yet they simplv are not tonal. Stumpf does not, as far as I am aware, anywhere give any proper analjrtic justification of this important impHcation of the tonal concept. Perhaps the clearness of pitch, or the unity of many clear attributes in tone might be referred to. But so long as noises have pitch and the other attributes at all, I cannot see how clearness could be held to divide the sensations of hearing into two such peculiar groups. The nearest approach to a justification I can find, is in Helmholtz's opening words quoted above, especially where he said: "in a noise many various sensations of musical tone are irregularly, mixed up and as it were tumbled about in confusion" (29, 14; 30, 8). It is evident therefore that however much is common to noise and tone, something stiU remains pecuhar to each, so that noises are not any more really tones because pitches can be recognised in them with special efforts, than tones are reaUy noises because under special circumstances their pitch is often hard to recognise*. This outcome of Stumpf's discussion greatly confirms us in our theory of tone as a regular, probably symmetrical, system of sounds. Not the pitch, nor the volume, and still less any other attribute or feature is the essential mark of tone, but the place of the predominant pitch in the whole volume, which thus forms a regular system of sounds. That is the 'tonal' character. In order to explain noises then we do not need to attempt to reduce them to tones. We need only trace the presence of pitch and volume in them as far as we can, and by the evidence of the trend of their variations up to the point where * 112, 610, expressed also by Hensen, v. 112, 600. ' Max Meyer discussed Stumpf s treatment of noise (76) and wrestled with the obstruc- tion hidden in it that I have expounded, without being able to overcome it. He thought that if a chord were not analysed at all, it would appear as a noise (p. 238). He defined noise as a series (rapid changes) of tone sensation under conditions which make a deter- minate judgment regarding the existence of pitch impossible. The conclusion is near enough to my own to pass unchallenged, but Meyer did not get at the root of Stnmpfs difficulty. So Stumpf might well reply r your opinion differs from mine, bnt you do not convince me I am wrong. I] AND THEIR ATTRIBUTES 39 our observational analysis fails, make it highly probable that noises whose pitch and volume cannot be determined are required to complete the scope of the theory of the psychical constitution of auditory experience. XVIII. Tones are regular systems of sotmds. Irregular systems should occur, as should also all degrees of irregularity from the tonal ideal to complete chaos. Irregularity is produced by displacement of the predominant order from its usual position to any other condition. A priori this disturbance could arise in various ways. Simple displace- ment of the point of predominance seems normally to be more or less excluded by the nature of the physical stimulus. Certain abnormal observations will be mentioned later (p. 50 (6)) which seem to represent this case. The large class of pitch-blends forms for our ear a minimal departure from the ideal regularity, by which they seem rather to gain than to lose in interest. Apart from their adherence to certain more or less regular patterns already described, their 'tonahty' will form the object of later consideration. The same appUes to all those groupings of pitches which give more or less harmonious impressions. The ' harmony ' is introspectively closely akin to the regularity of tone and is probably of the same nature as it is. Far from being anjHihing strange in sound, as so many have held, it would then be only a development or comphca- tion of the very thing 'tone' itself. We have at this point no d priori method for following out the grades of harmony, but we know how to decrease this effect and make it pass towards noise. Apart from these special cases irregularity may be got by continuity and rapidity of displacement; but this gives no decisive degree of irregularity, because the volume varies with the displacement of the pitch, and so the regularity of system — ^the tonal character — still remains in spite of the constant change of pitch. When soimds are very brief or momentary, there are two possibihties : either no pre- dominant order is produced in the time allowed, but only a mass of orders in which no sort of system of intensive differences can be detected ; or system and dominance do occur, but the sensory mass lasts too short a time for the characteristics of the experience to become psychically effective towards cognition, i.e. to be observed. Of the two hypotheses the former seems to be more generally accepted. At least two identical, or very similar vibrations are required if 'tone' is to be heard and recognised (1, I97ff.). And yet it does not appear how a maximum or a system is to be produced by the second vibration if it is not 40 AUDITORY SENSATIONS [ch. already present in the first. The second is either just another first, or if the effect of the first lasts on, its hypothetical irregularity will hardly form good ground for the attainment of regularity. But it is possible that two vibrations are required only because one periodic vibration cannot be physically given or defined unless two are given. In other words the first cannot be finished off perfectly, so as to give the required balance in the sensation, unless the second is at least begun. The second or in general the last would, then, tail out irregularly. Irregularity may also be attained by increase in the ordinal scope of the 'spot' predominant in the volume. This would mean a decrease in the definition of the predominance, and seems attainable by a rapid and irregular oscillation of the rate of vibration round about an average. Jaensch (34) claims to produce noises in this way. Of course a pitch could still be ascribed to these noises with or without the help of com- parison, as in the experiment with the dropped sticks. A detectable pitch need not be strictly, any more than it is popularly, considered to be incompatible with noise. A sound might also contain two or more indefinite pitches. The greater the number, the more irregular would the whole sound be. Following these methods of multiplication and blurring of pitches, we should attain to sequences so irregular as to produce sounds in which no trace of predominance or of system could be detected. These would be 'pure' noises. A pure noise would then be a mass of sounds in which dominance is so irregular as to be undetectable or in which no dominance occurs. The latter state in a constant form should be a possible result of various affections of the inner ear. That there are tones without noise would mean that there are systems of sound in which no trace of irregularity is to be found. That there are noises without tone would mean that there are systems of sounds in which no regularity and no approach to regularity can be detected. We should thus have surveyed all the evidence that can be brought to bear upon the problem of noise. It is primarily psychological — ^the whole trend of the variations of sound away from the perfect system and predominance of pitch that constitutes tone towards multiphcation and irregularity of pitch imtil such degrees are attained as completely bafiEle analysis. Parallel to this runs the evidence of the comphcation of the physical stimulus. These hues of evidence in conjunction with our theory of tone exclude any need for a separate being and origin in noise. The former are commonly held by those who accept the reduction of noises to tones; whereas the peculiarity of the 'tonal' I] AM) THEIE ATTEIBUTES 41 experience over and above its pitcli, although this pecuharity has been explicitly handled by none, is the sole reason of a psychological kind for the separation of sounds into a special class i. Any sort of physio- logical argument is, of course, irrelevant, for it could never affect the fact that tones and noises are both auditory, any more than the theory of the independence of rods and cones in the retina affects the psycho- logy of vision. Hence with the resolution of the tonal difficulty, all may agree that tones and noises vary from one another, not in any attribute, but in the nature of the mass of auditory ' atoms' of which they consist. Of the presence of volume in noise there can be as httle doubt as of the presence of pitch in noise. And what doubt there is, may be resolved in the same way, if not more easily ; for in this respect we do not need to analyse the sound, but only to compare it with others as a whole. XIX. C. Vowels. It is a familiar fact that in the utterance of the various vowels u, o, a, e, i, etc., the mouth is brought into different positions. The occurrence of vowels is obviously dependent upon these positions. How these positions actually determine the character of the soimd produced in speaking is far from clear and has been greatly disputed. There are two chief views. One asserts that in the act of speaking, the cavity formed by the mouth acts as a resonator reinforcing certain partial pitches of the sound produced by the vocal chords. A special type of pitch-blend is then produced which we name by the utterance of it. The chief difficulty of this view hes in the supposition that the same sets of partials required for the various vowels can be present in aU the voice-tones of the range within which these vowels can actually be produced. The other theory asserts that in the act of speaking the cavity formed by the mouth is blown by the air passing through it or past it through the nose, as a bottle is made to produce a tone by air blowing over the mouth of it. This view gets over the difficulty of the first; identical partials or sets of partials may now be admitted, no matter what may be the pitch of the voice-tones, the tones produced by the vibration of the vocal chords. Neither of these theories would offer any new material for the special ^ K. L. Schaefer (102, 17) says that, while many believe that there is in noiae, apart from a more or less large number of tones, a specifically noisy element, he believes that noises like clangs (pitch-blends) are " nothing but a sum of tones, although a sum whose composition usually deviates radically in various respects from that of clangs, which rests on musical principles, and whose complete physical and physiological analysis into parts is very very much more difficult than it is for clangs." That is a tortuous way of admitting that the complete analysis of noises into tones is not possible. 42 AUDITORY SENSATIONS [CH. study of this work, concerned as it is with the principles and outline of the psychology of hearing, not with such detail questions, which more properly fall within the scope of physics or phonetics. But in recent years a new turn has been given to the study of vowels, by which they would become of primary importance. This new turn was made possible by two things. In the first place the partials of the chief vowels show a tendency to occupy positions an octave apart from one another. This appears in Helmholtz's table to some extent (o = b'^, a = 6^", E = h^') (29, iTi; 30, no) and it was extended and generaUsed for the five vowels u, o, a, e, i, by R. Konig, whose set of tuning forks for the demonstration of the vowel pitches is well known, being tuned to h, h', 6^ 6*, 6* (or rather S'', 225 vibrations). In the second place the difficulties so obviously felt in the elementary psychology of hearing, as that had grown up on the fundamental decision to look upon pitch as qualitative (with or without any reference to volume, which in any case was set aside as quasi-spatial), seemed to call for some new venture in theory. But for that, there would indeed have been room for the improvement and extension of Konig's results, but not for any re- interpretation of them. It was Kohler (42) who propounded the view that the vowels are the sole and primary qualities of hearing. According to him the series of vowels he strictly in octaves over one another, their vibration frequencies being multiples of 264, which would give the pitch of c for an ' o' of 220 vibrations. Between one c and the next higher the 'quality' of the tone changes gradually into another radically different quahty, in the same way as the quality of visual sensation changes from red to yellow. And we have popular 'absolute' names for these quaUties, as for red, sweet, etc. None of these things holds for the alternative treatment of pitches as quahties. These argu- ments of Kohler' s helped to weaken the old qualitative position of pitch. The new view may be expressed diagrammaticaUy in relation to Hering's colour system thus (123, 325; 42, lie): 1^^5.264=0' {s)mon Fig. 2. (After Stumpf.) Here, in the example chosen, the rising of the tone from c^ to c* would bring a gradual departure from the o (toe) quahty through the a of all to the a of father, just as red passes through orange to yeUow (cf . Fig. 1). I] AND THEIE ATTRIBUTES 43 So for the others also. It is hard to say if any given isolated tone is the pure o or not, but after a little practice it is very easy to indicate the point in the tonal series where the u valency just disappears and a new quahty, that of o, appears. These special points are not just peculiarities of the mouth or of any one language, but absolute turning points. The average values got by Kohler by the 'easier' method appear in the following table (42, 130, 137) : Table I. Obar. m u a e S. — 261 2x261 4x263 8x261 G. — 264 2x261 4x264 8x262 K. Jx263 262 2x258 4x262 8x264 M. i X 264 266 2x264 4x264 8x262 The value of the mean variation is never greater (and often much less) than a quarter-tone, so that the optimal positions of the vowels in the scale of pitches seem to be very precise. The octave law was extended by Kohler to include M on the lower side and S and other sounds on the higher side. One fundamental defect appears immediately in this theory. Either it is wrong or the meaning of the term vowel is very difEerent from its usual one. Stumpf claims that the sound of a powerful male voice, contains some thirty partial pitches that can be objectively verified by the resonance of tuning-forks. Vowels spoken by such a voice would never be pure tones. If, then, the vowel-tone (pitch) is always present, it can be only a partial-' tone' of the whole. As the range of a mobile voice is Umited to some two and a half octaves, it should be able to produce clearly only as many of the vowels, or these con- siderably 'coloured' by the vowel quaUties of the higher partials. The theory thus loses its relation to vowels in the ordinary sense and retains only its relation to octave differences, which we shall discuss later. Apart from that however, the theory has received at Stumpf's hand (123, 324fE.) very damaging criticisms. Apparently the difficulty of recognising pure vowel quahties when they are given in isolation is much greater than Kohler would have it appear. And for the specific nature of the vowel quahties the introspective evidence claimed does not exist (cf. 137, los.). Z7 and o are not as different as red and yellow, even although they are recognisable as types, hke the sounds of the viohn or the piano. Stumpf fails too to recognise any great similarity between o, a, e, and the pitches indicated for them. Such internal 44 AIIDITOEY SENSATIONS [ch. difficulties of fact, interpretation, and system make KoMer's attempt to give vowels a central importance for hearing futile. And so we may- leave it, without reference to its external consistency with the wider facts of hearing^. An attempt has recently been made to make the vowels supply a felt want in the psychology of noise. On the basis of the method abeady referred to, of making the vibrating frequency of a sound vary round an average, Jaensch (34, 35) propounds the view that the vowels, far from being the quahties of tone, which they at most only resemble, are the specific and serial quahties of an older sense of noise. Thus a new hne of analogy with vision is introduced. The pitch series is like that of the series of colours because the stimulus of each is a defi- nite rate of vibration ; the noise series resembles the series of neutral brightnesses from black to white. Apart from all the difficulties that arise when we try to work out this analogy, Jaensch's experiments by no means warrant the theory built upon them. The only difference in the stimulus for tones, vowels, and noises shown in his experiments is an increase in the mean variation of the individual values of the vibration frequency about one and the same average. That alone would suggest the question whether the introspective difference between the three is not a decrease in definite- ness of pitch. Introspection affirms this. Thus Stumpf says noises have a pitch, but it is not so well defined, and always fills a certain stretch of the hne of vibratory frequencies; that, he says, probably explains their phenomenal character (123, 341). It is hard to see from Jaensch's experiments why vowels should be held to be the qualities of noises rather than of tones, for they are said to resemble tones, but not noises; or why we should not assume the existence of three senses in hearing for tones, vowels, and noises, severally. But taken at their face value Jaensch's results offer a very acceptable confirmation of our view already developed that tones and noises are both complex auditory experiences, auditory molecules, as it were. We may there- fore place vowels between them, so that the study of vowels serves to extend and confirm our previous studies. XX. D. Octave quulities. Another direction remains in which new material has been sought for the elucidation of the elementary ^ An excellent account of the work on vowels in relation to the assumption of Kobler's vooality as a, quality of tones is to be found in 51. For a good example of the rather uncritical determination of that theory see there, p. 746, where it is held that vooality can be determined by a partial that is inaudible when presented alone. I] AND THEIR ATTRIBUTES 45 psychology of hearing. Although the great series of auditory qualities taken from the whole range of pitches, as expounded e.g. by Stumpf in 1883, appealed with satisfaction to the great majority of psycho- logists, doubts kept recurring to a few. The chief trouble of all these quahties is the great number of them, which does not seem reducible to a few, hke those of vision. Of course the smallest number of qualities to which any single series of differences is reducible is two, those that occupy the extreme ends of the series. This merely possible view was actually propounded by Mach (60, 122; 2nd ed. 1900, 181). He compares the pitch series with the series of colour differences leading from red to yellow. A comparison with the black to white series would be equally effective of course. Mach's proposal received hardly any support. The chief difficulty was that a leading physiological theory hke that of Helmholtz offered no warrant for it whatever. F. Brentano^ alone adopted the suggestion made by Mach, but with considerable modifications for its improvement. He reduced the number of quahties of the pitch series by recognising as primary only those differences included within the range of an octave. The differences given by the repetition of octaves were explained by the assumption of a brightness component of sound increasing from one octave to the next above. A view very similar to this was stated summarily by W. McDougall (153, 73) as early as 1899. He suggested that " all the elementary quahties (of tone) are contained in a single octave, which might be hkened to the complete colour series, and that the differences of pitch that distinguish the same quahties in different octaves... are of the same order as differences of extensity or voluminous- ness in the case of visual, tactual or temperature sensation, and are due to differences in the number of sensory neurones excited." The strength of this theory hes in the powerful appeal it makes to one of the special interests of hearing — the similarity or at least equiva- lence of the tones of successive octaves. The interests of theory were bound to be influenced by this pecuUar phenomenon so long as the elementary psychology of hearing was not perfectly satisfactory and no convincing explanation of the phenomenon had been given which would make a transference of it into the elements impossible. The latter explanation, as we shall see later, was never given, so that the trans- ference was at all times possible. The only serious objection to the transference was the immanent incoherence of the theory itself, which, 1 8, 101 ff. For a much earlier report of Brentano's views by his pupil Stumpf, v. 112, 199. 46 AUDITORY SENSATIONS [ch. although satisfactory as far as the octave is concerned^ is unable to do justice to the other degrees of similarity and musical relations within the octave and to the perfectly relative nature of the octave itself. These and other objections led Stumpf (112, 196£E.) in 1890 to reject Brentano's proposals entirely. Stumpf was satisfied in accepting the octave relationship and the other facts of fusion as ultimate facts of hearing, preferring his already discussed theory of the elements to any remodelling of them on these hnes. But the poverty of Stumpf's elementary psychology of hearing as a source of fruitful explanations^ led him comparatively soon to the acceptance of Brentano's distinctions. He gives himself the date 1902 as that of his conviction, 1912 as that of pubhc admission^- And his reasons are : " not the analogies with the colour sense, which are a nice illustration, but no proof, but reasons taken from the experiences of hearing itself, especially the common deviation by one and even two octaves in the determination of absolute pitch, the ease of the octave transposition, which is often done quite unwittingly, and the harmonic equivalence of the octave iu chords (inversion of intervals and chords)" (123, 311). AU these reasons were present to his mind in 1890, but were rejected as being mere consequences of fusion. Recently a series of papers has been published by Geza Revesz in which both broad and special foundations are sought for this distinc- tion between octave-quahty and height-brightness. It is hardly possible to use the word pitch to represent the latter concept, because so much of the ordinary meaning of the word pitch has been absorbed by the term quahty, and it is not easy to say just what remains over. But the remainder includes at least the rising aspect; probably almost what Stumpf thought of under 'volume,' when he distinguished it from pitch- quality in 1890 (112, 203f.). In order to distinguish this new attribute from quahty, it is well to use the word 'height' for it, in order to avoid clashing with the meaning of the word pitch, as it is used both in ordinary speech and in psychology. After all pitch is what we begin to theorise about. So if we cannot take over the ordinary meaning of that term into our psychology, let us use another word. In referring to Stumpf's quahty of 1883-90 I shall say pitch-quahty, in referring to his recent quahty, I shall say octave-quahty. Let us now consider closely the arguments for and against octave-quahty. ' Cf. 123, 314, where Stumpf admits this as the "Undurohfiihrbarkeit der psyoho- logischen Konstniktion." ' Cf. 121, 334, note, where the question was still left open. I] AND THEIR ATTRIBUTES 47 1. Direct. Trained and untrained observers have long noticed a great similarityj if not identity, between tones an octave apart, whether they are of the ordinary musical kind or are perfectly simple in pitch (cf. 123, 309 uote, 312 ; 94, 248). There is no need to urge the point. It may be granted at once. There is something which hnks a tone to its octave very closely. But the question is : what is it and what is its psychological status ? The mere fact of similarity or whatever else one may call it, indicates neither identity, nor simpUcity, nor psychological status. In this connexion it is important to notice a recent remark of Stumpf 's in which he says: "Statements about the fundamental properties of our sensations should in general be based only upon the observations of those who by long practice are at home in the province referred to, and have also tested their single observations time and again" (123, 307). That is true as it stands for any province of facts. But if it means that no one is to have a voice in the discussion of the primary attributes who has not an absolute ear and the practice in experimental observation of sovmds that Stumpf and his important co-operators Abraham, V. Hombostel, and others have, it is plainly absurd. For, as I have already shown, judgment upon the psychological status of an attribute is a matter of wide theory ; it has, therefore, no relation to the fineness of discrimination of that attribute or to the abihty to name its varieties. This is surely the only sort of explanation that can be given to the fact that so highly quahfied an observer as Stumpf himself could in the course of time uphold diametrically opposed sides on one and the same question, without having any new weight of argument on the side now taken and without being able to demoUsh any of his previous objections to it {v. 112, 196-204). Stumpf himself offers an objection of a direct kind; if octaves are identical quaUties, we should expect the identity of c' and c* to be as easily recognisable as that of a stronger and a weaker c'. This sound objection may indeed be pushed farther. If c' and c^ are the same except for their accompanying brightness or height, we should expect the octave partials of a musical tone or octaves in a chord to be indistinguishable, as Stumpf himself very properly said in 1890 (112, 200), and, if anything at all, only differences of brightness or height to be left in it. But the opposite seems rather the case : the pitches c' and c* are distinguishable, but in a perfectly steady tone the brightness or volume components of each are no longer traceable (137, 36f.). Where such close fusion of tone is attained as in pitch-blends, we should at least expect identicals to lose their distinction (cf. 112, 200). 48 AUDITOEY SENSATIONS [oh. Of course, if we are determined to keep octave qualities at any cost, we may plan the matter out in various ways. We may say that, besides octave quaUty, tones have both pitch and volume, or that what distin- guishes c' and c* in a pitch-blend is their pitch or height alone. But the latter view would imply that the 'similarity' of c' and loc. n] THE! ANALYSIS OF BI-TONAL MASSES 67 impossibility of getting a large number of units of different size on to one measuring unit. So for our present purposes we may simply take over the ratios from present knowledge without further inquiry into their origin and apply them to psychical volumes. For the interval of the fourth the 'lower extreme' and the 'pre- dominant' points of the upper tone would he respectively one-fourth of the whole length of the lower tone towards the lower extreme order of it away from its predominant order, and one-eighth away on the other side^- Similar values^ for the other grades of fusion and the intervals included under each by Stumpf and Kemp {v. below, Table VI, p. 104) are given in Table III. The hne separating the columns ' Lower extreme' and 'Pitch-order' may be taken as representing the predomi- nant point of the lower tone. The values in each column then give the distance, in terms of the volume of the lower tpne, between the latter's predominance and the lower extreme order of the higher tone and between cit. p. 181 f. If my method and results are correct, I have now measured real Tolume by superposition. I have not however measured the mode or Gentait 'volume,' which must remain in its psychical essence, as it has always been, a mere magnitude. For a clear assertion of this — ^that volume is an immeasurable magnitude like intensity — see Stumpf ( 1 ] 2, 68). Those who hold that magnitudes are somehow measurable — on the distance method, for example — ^may pot now turn upon me and say: you have just succeeded in measuring a magnitude. For I have not measured volume by the distance ritual, nor have I measured it as a njagnitude; but 1 have measured its real basis by an inferential process, by superposition, the same inferential process by which real physical • distances are measured. As I said in my paper, some day the real psycliical basis of intensity — if it has one — may be measured in this way. But that will not be a measure- ment of intensive magnitudes. Are those who stiE differ really ready to accept the ultra- sensationaUsm implied in the denial of my argument ? I doubt it very much. If volume and interval are really in psychical essence something more than their psychical basis of auditory atoms, they are essentially immeasurable, although they are magnitudes. But their real psychical basis is measurable. I do not deny the possibility of psychical measurement. But J do deny the possibility of measuring psychical magnitudes as such. In strict logic it is the 'as such' that matters. If any one cares to omit it and to substitute for the magnitude in question some other in order to be able to say he has measured that magnitude, I suppose he will. But in that case all discussion is at an end- 1 The predominant point of the fouith therefore divides the distance (in the strict sense) between two pitches an octave apart into two equal parts. Generally, the half- distance pitch is got by subtracting the inverted ratio of the two tones from ijnity and dividing by four. If the resulting fraption is found in the fourth column (a;) pf Table III, the required tone is ^ven by tjie interval of the first column ; otherwise by simple calcula- tion, e.g. c:a=i; i.e. l-f; =f-r4 = iV; i-^- m or e; e divides the distance c-a into two equal parts. On distance compare below, p. 75 ff. 2 Cf. 137, 34. There by mistake the values for the natural seventh (4 : 7) have been given instead of those of the tritoiie (3? : 45). 5—2 68 THE ANALYSIS OF BI-TONAL MASSES [CH. the predominances of the two tones ('pitch-order'). The values for the intervals beyond the octave are set alongside. In these, both the special defining points of the upper tone faU above the point of predominance of the lower tone. It is important to notice that the lower extreme point of the upper tone falls in the intervals beyond the octave exactly where the predominance of the upper tone fell within the octave. Table III. (In terms of the lower' s volume) Interval From the pitch-order of the lower to the hlgher's : Interval lower extreme pitch-order {X) lower extreme 1:2 i i 0-^0 1:4 6 2:3 m i ■ i a) i Oh-5 1:3 4 3:4 (*) * * (i) * 0^-4 3:8 III 4:5 (A) t\ A (A) A 0-1- III 2:5 3 5:6 (A) A A (A) A O-fS 5:12 VI 3:5 (A) tV A (A) A Oh- VI 3:10 6 5:8 (A) A A (A) A 0-1-6 5:16 T 32:45 m) u M (1*) M + T 32:90 n 8:9 (A) A A (A) A 0-i-n 4:9 7 9:16 (*4) A A (A) A O-t-7 9:32 2 15:16 (A) a A (M) A Oh-2 15:32 VII 8:15 iU) A A (A) A 0-1- VII 4:15 AH these arrangements are obviously devoid of the coincidence or balance that is so obvious in the first two — ^the octave and fifth. AU of them are irregular, but to some extent this irregularity gets worse as we proceed, while one of the two defining points of the upper tone comes closer to the predominance of the lower. All these fusions mil therefore naturally belong to a lower grade than the fifth, whether our conceptual estimation of their balance is as graded as our auditory apprehension of them or not. We shall consider these details more closely in another connexion. For the present our deduction agrees II] THE ANALYSIS OF BI-TONAL MASSES 69 sufficiently with the introspective and indirect determinations of fusion expounded above. Our study of the tonal mass produced from two different sources of sound thus teaches us that in almost all cases pitches can be found in it that correspond exactly to the pitches of the original sounds pro- duced separately. In this sense analysis is perfect. In the case of slow beats, as we have seen, the original pitches cannot be recovered. But of the other attributes of tone neither volume nor intensity can be recovered in their original forms by analysis. Fusion, we have concluded, is to be defined as the coincidence and inseparability of volumes which remains over even after pitch has been perfectly analysed. The merging of intensities in one another is proved in general by the familiar fact that a sound can be heard in the silence at an intensity far lower than that required when another sound is being made^. One sound seems to drown the other to some extent. This is again a sign that sounds coincide at least to some extent in their 'stuff' as it were^. Some of the 'atoms' or 'spots' of soimd of which they are composed, are identical, as we have seen in the study of fusion. XXXIV. Let us illustrate these things with a diagram. Let the two primary soimds have the pitches c' and c*. The diagram of c' alone would be as in Fig. 4. The line Vh-Vl represents the total volume of the tone, VI being the lower extreme and Vh the higher. P represents the predominant order that constitutes the pitch, while the perpendicular Pi represents the relative intensity in which that predominant order is present. The relative intensities of the other orders that make up the whole volume are found by perpendiculars parallel to Pi. But the diagram does not claim to represent these differences truly, but only to indicate them in principle. Their exact relations are a matter for difficult special research. The diagram for c^ looks in principle exactly like that for c'. In actual fact the rise of its intensities may be somewhat different especially about the point of predominance. Only the dimensions of the whole are half as large. 1 Cf. 112, 220 f., 420. Weber's law alone would suffice to explain this if a coincidence of volumes is admitted. 2 A remark by W. James (36, 84) is of , interest in this connexion: "At most, the high tone is felt as a thin, bright streak on a broader, darker background." Ct. however. Stumpf s denial of all the implications of this suggestion (112, 58). Cf. with this, Dunlap (1 1, 291) : " the higher note is contained in the lower note both psychologically and physio- logically, just as if a short streak of light were superimposed on a long one." ro THE ANALYSIS OF BI-TONAL MASSES [CH. In the combined diagfftin of the soimd heatd when both sources of sound are played at once, the lower half is identical with that of c'. A second point of predominance appears in the upper half. But what exactly are the intensive relations of the curve of the upper half we cannot say, except that they must be somewhat greater than are those of c' for the simple reason that a second point of predominance appears. A similar diagram is given in Fig. 5 for c' + g'. There the curve of g' overlaps two-thirds of that of c'. A point of predominance is not created at the end point of coincidence W. Evidently more is required for predominance than just an increase of intensity. But this increase is not so good as absent or ineffective, any more than it is at the extremes of a tone of simple pitch. It is elective in so far as it marks out the balance in the whole tonal mass that is characteristic of the fifth and that makes the fifth so like a perfectly balanced tone of simple pitch that it is often taken for one. The common difference tone c" will in this case bifiiig out a third point of predominance at Yl, the rest of the volume of that lOw tone stretching away out as fat as the whole length Vlr-Vh on the left of VI. II] THE ANALYSIS OF BI-TONAL MASSES 71 Certain other phenomena are now readily deducible from our diagram. In low tones and in simple mild tones the curve of predominance may be much rounded or flattened^, so that the pitch seema indefinite and to a certain extent displaceable at will. Such rounding may be produced by the interaction of sounds. Or by the same influence the addition of a second sound may seem to pull the pitch of the first sound towards the new pitch (112, 396ff.). But this shift of pitch is evidently only due to the rounding of the predominance. For the true pitch or the real point of predominance is always to be got by attentive observation. The theory represented by our diagram provides a basis for a detailed study of the relation of sounds of difierent intensity to analysis. Stumpf's conclusion "that the higher tone must possess a greater excess of intensity if it is to cover the lower one than conversely*" seems to be deducible from our diagram. For the volume of the lower tone always encompasses completely the volume of the higher, which can then only suppress the former if its predominant part lies near that of the lower tone and if its intensity is great enough to swamp the predominance of the lower tone. The lower tone on the other hand has the advantage of being the only clearly defined volume, which can be flooded by higher tones from one side only. If the higher tone is more than an octave above the lower, the latter can be flooded only on the upper side. Any sort of mutual flooding of volumes will be much less likely with considerable differences of pitch (cf. 112, 229). In fact the intensive relations within the tonal mass from two sources of sound might yield a fairly good statement of the relative volume curve of a single tone, if one tone were plotted out in terms of the minimal audible intensity of all other simultaneous tones within an octave on either side^. XXXV. Concluding our consideration of the mass of sound evoked by two sources of sound at once we may say that the whole nature of the mass is continuous with that of a tone of simple pitch. In fact * Cf. 112, 114. The greater threshold for discrimination of simultaneous pitches in the lower regions also' suggests this. v. 103, 94 f. 2 112, 228. On this subject in general of. 102, IS ff. The tones considered must be of comparable intensity. Of course a weakish low tone is obliterated altogether by a loud or shrill high tone, from which the ear (i.e. the attention) cannot free itself. Sohaefer and Guttmann (103, 87 ff.) have shown that in the middle parts of the scale of pitches two pitches are not discriminated until they are some 10-20 vibrations apart. • Other details on this subject wiU be found in 112, § 21. We cannot pursue it further now. It must at present suffice to have indicated the chief lines of theoiy iii this region. 72 THE ANALYSIS OF BI-TONAL MASSES [ch. the theory of the mass of two sources is just a modification of that of the 'pure' tone. Both are masses of elementary auditory 'atoms.' Both are regular and balanced and therefore admit of the predicate 'tonal' appUed specifically to the sound of simple pitch. No complete analysis is possible except of the aspect of pitch, where predominance is equivalent to discreteness. These pitches are not tones, in spite of the prevailing tendency to identify the two notions. But in view of the fact that not only the whole mass of sound but also the parts sur- rounding each pitch are so regular and balanced as to admit of the pre- dicate 'tonal,' it is permissible to speak of the analysis of the whole mass as an analysis into tones. Yet we must not forget that tones cannot be separated completely from one another. The separation is much less good than is that of two neighbouring patches of colour; for the contrast effects on the area surroimding a patch of colour do not really belong to that patch as a unit. Every tonal mass, in fact every sound, is then a unity except in so far as it can be analysed (cf. 112, 77). Unity and fusion are primary and ineradicable, analysis is only possible so far as the discreteness of pitches and other favourable circumstances, such as the movement of tones in a mass, permit. There is then after all no new problem in the unity of pitch-blends, or of fusions of the octave, fifth, fourth, etc. Their problem is the one problem of tone in the strict sense. And all analysis is a consequence of the predominance of pitch. But analysis goes only so far as it is induced by favouring circumstances or is pushed by will. And then of course aU attitudes or habits of attention are in a sense artificial. Attention falls back from them under ordinary circumstances into an easy posture which corresponds roughly to the mass-nature of the experiences in question. And in the case of sound that mass-nature is fusion with a central predominance of the funda- mental pitch (cf. 112, 384ff.). We thus obtain a central point of view for the most varied series of phenomena and many standing conflicts of theory are aUayed. The comparative smoothness or roughness of pitch-blends depends upon their approximation to the balance of the ideal pure tone. So does the grade of fusion of two tones. We cannot, therefore, explain con- sonance by agreement or disagreement of partials; but we can reduce the similarity of both to the same basis in principle. The same holds for any other adventitious accompaniment of tones which might be invoked as the basis of fusion, e.g. difference-tones. Any smoothness or fusion-like character which may appear in groups of difference tones II] THE ANALYSIS OF BI-TONAL MASSES 73 must be due to the approach made by the mass to the balance and smoothness of a pure tone. But each group of pitches (volumic pre- dominances) has its own balance. We do not need to appeal to one group for the smoothness or roughness that we cannot find theoretically in another group. Here we touch again on the principle of nativjsm. Theory shows that we do not need these transferences of properties between experiences, just as principle urges us to decline to accept such an irrational and imcontrollable process. Even the roughness of beats is but a further instance of departure from the ideal smoothness of the pure tone. Beats undoubtedly do sound rough, if at all frequent ; but they could be so theoretically rough as they seem to be to everyone since Helmholtz, only because the basis of the resonance theory passes over in the region of beats insensibly into a series of small volumes, blending and oscillating with one another, even although no volume was expUcitly attributed to the tonal element in that theory. We involumtarily think of a parallel Uke visual flicker — and that is a volumic parallel. A non-volumic theory of hearing has no right, strictly, even to the rationality of the roughness of beats. AU the theories, then, hke all the facts, lead to, and ground in, the volumic differences of tones and the ordinal differences that are implicit in them and are made explicit in pitch. CHAPTER III DISTANCE AND INTERVAL XXXVI. The further psychology of hearing must naturally build upon the results of the study of simple sounds. Whatever success we may have in carrying on our early conclusions will help to confirm their validity. Then the psychology of hearing will be continuous in its principles and explanations. We have already seen how the psycho- logy of the attributes provides a basis for the adequate study of fusion and tonal complexes. We shall now see how the same basis suflS.ces for the study of distance and interval that in the ordinary psychology of sound appear so mysterious and peculiar. In the senses of vision and touch, distance is known to everyone. We can feel the distance between two points or the thickness of a book 74 DISTANCE AND INTERVAL [ch. very acciirately with the thumb and first finger. We can compare the distances between two pairs of stars or between two pairs of dots on a page with great exactness, quite apart from any measurement of these distances. Of course a small and somewhat variable error is made, but that does not detract from oiir ability to compare distances very rapidly and efficiently. A great deal of our ordinary practical work involving appreciation of distances can be done quite well without any measurement. Although we are aU thus familiar with distances, using them and speaking of them constantly, psychologists have been of very different opinions regarding their psychical nature. At present many incline to regard distances as special parts or kinds of experiences, something other than the attributes of elementary sensations, an addition to them which supervenes only when two or more elementary sensations are given. Apart from theoretical objections, a moment's consideration wiU make this seem very probable. The distance between two visual points is quite another thing than the two points themselves. One might reply that between the points there is a continuous stretch of visual sensation, since the whole field of vision is always full of sensa- tion. But that mere fact would be no reason for our selecting the stretch of sensation between the two points as of special interest, except as a stretch or distance. Besides, when we speak of distance, we do not mean more or less the sensation lying between two points. What interest is there in that vague sensation, undistinguished as it is from its surroundings ? What we mean is the specific experience of distance. All this is no explanation, but only an attempt to point out what is meant. Moreover if we refer to the sense of touch we find that we have distances without any continuous background of sensation. If two points are touched at once some inches apart on the forearm, we feel a distance between them. We can also feel distances with the tip of the finger or tongue. When we reflect on the relation between distance and the attributes in these senses, we find that distance is dependent on the occurrence of two sensations of different order. Thus from senses in which the facts are easily distinguished we get a pattern by which to test aU the senses. And this procedure leads to a general rule in the same terms, which is supported by another rule saying that distance is only found in those senses in which suitable variation of the attribute of order occurs. There is no distinct trace of distance in the olfactory and Ill] DISTANCE AND INTERVAL 75 muscular senses. In those senses in which only unrelieved masses of sensation occur, especially the organic, and also cold and warmth, no distances can emerge properly. In articular sense distances are very important. Here We see further confirmation for our classification of the variations of position in this sense as primarily ordinal (cf. 133, 172fi.; 135, 250fi.). Such agreement amongst the senses must lead us to expect to find in hearing an experience similar to distance, founded upon differences in the attribute of pitch; and its presence is admitted by a number of psychologists. XXXVII. The amounts or sizes of the distances between pairs of pitches can surely be compared with one another, Uke visual distances, especially if we compare the distance between one pitch and two others above or below it, e.g. c-e with o-f. Anyone who is not quite unmusical, wiU notice an increase in the distance between these pitches. However, even for these more or less obvioUs cases doubts may be raised. Stumpf, for example, is inclined to think that then our judgment is based on the difference of pitch of the upper tones and not directly upon the difference of distance between the pairs of pitches (111, 248). But he is quite sure of our ability to judge very small tonal distances, smaller than a semitone (111, 252f.). That much is at least comforting and forms a beginning. But in different parts of the musical range of pitches, e.g. c-e and g-a, it is much more difficult to compare distances. In fact we seem sure only when we make rough comparisons. These again hardly do more than assure us that distances exist throughout the range of pitches and are comparable as to size. It is said that the distances given by one and the same ratio of pitches are not the same in different parts of the tonal range. According to Stumpf, who admits the difficulty of judging^, they increase from the depths up to about the third accented octave. In the lower part of the great octave ^-c) the interval of a third (4 : 5) is just recognisable as a distance ; i.e. it is a minimal distance. The fifth then also seems hardly greater than a third in the middle musical range (112, 4D3f.). In the upper ranges of pitch a similar contraction of distances over against one and the same ratio of vibration is also to be observed. ^ df. hie criticisms (113) of Lol-enz's attempts (68, 26 ff.). At p^ 465 StUmpf speaks of judgnlents of tonal distance as a field "in which cleat? results are Clearly out of the ques- tion." On p. 459 he gives a tew approximations of his own: the middle between c' and c' is about d"; between c' and j" about 6'; between c' and d^ about g'%. 76 DISTANCE AND INTERVAL [ch. These things hold whether the sounds that form the interval are given simultaneously or successively (112, 406). XXXVIII. Compared with the ease with which we handle distance in the other senses, especially vision, such helplessness in hearing seems mysterious. We might have expected to be able to say with great precision whether the distances between two pairs of pitches were equal or not, no matter how far apart these pairs of pitches were, just as we do with visual distances, looking first at the one distance and then at the other. But, as it is, unless the suggestion had been given to us in various ways^ from the other senses, we might hardly have thought of looking for distance in hearing at all, were it not that it seems so to contract in the extremes of the pitch range. But we must be careful not to be misled by the constancy of physical ratios. How do we know that one ratio offers the same possibility of distance in the various parts of the range of pitches? Surely it gives no such guarantee, any more than the number of inches between two points on the skin gives an index of the comparative size of the distance we shall feel between them. On the theory of pitches above developed this parallel is quite true and exact. The only proper basis for judgments of distance is distance itself. And we can only ask for a reason why we apparently judge auditory distances so badly. One obvious reason is that every one finds it so much easier to state the relations between pitches in forms of interval (111, 249fi.). For interval is not only a constant ratio of vibrations, it is also a constant experience. All the ease that we should expect to find in our dealings with tonal distances, we actually find in our work with intervals. We trace their equalities in the same and in different parts of the musical range very easily. We can do with intervals what is hardly possible even with visual distances, at least with such accuracy; we recognise them in isolation without comparison. We must therefore examine intervals carefully in the hope that we may then throw some light upon the obscurity of tonal distance. ^ I refer not only to my own rule of the relation between orders and distances, which is too new and revolutionary yet to have been accepted by others, but also to the curious classification of all differences between sensations, whether qualitative or not, as distances (of. HI, 122 f.; 125, xxiv tt). This classification is of no direct importance for our discussion, but it seems dangerously confusing. There may be a degree of 'distance' between red and orange that can be stated in figures, but there is no distance in the ordinary ordinal sense of that word. Ill] DISTANCE AND INTERVAL 77 XXXIX. Interval is ptysically defined as the special feature of auditory experience that is common to all tonal complexes evoked by two sources of sound vibrating in a certain fixed ratio at any part of the musical range of pitches. It is not identifiable with degree of fusion, for, as we have seen, intervals like thirds and sixths, or seconds and sevenths, are not easily distinguishable amongst one another in terms of degree of fusion, although they are readily distinguishable as intervals. Fusion considers only the degree of disruption or disorder in the whole mass, the ease with which it could justify the judgment that it is evoked from two sources of sound, with or without simultaneous attention to the discriminabihty of the pitches it contains. Interval applies to the whole experience, not before or after the analysis of its pitches, but both before and after, at all times, as a characteristic whole especially in reference to the setting which the whole mass provides for the primary pitches it contains. That whole setting must be a characteristic thing. And on our theory of fusion it is so. We may at once state interval as the outUne of the whole mass of sound which, within the musical range, is charac- teristic of any ratio of vibration, especially such ratios as are already characterised by fusions, e.g. octave, fifth, fourth, or by any other features such as we shall indicate later. In the case of fusion the character of the whole mass of sound given by any ratio of vibrations was set into special relations to its unitariness or balance. Or, in practical terms, to the ease with which it could be correctly judged to have been evoked by two sources of sound. The discrimination of pitches in fusion is only a secondary matter or a parallel indication of its unitariness. The fusion proper is there equally before and after discrimination of the pitches in the sound. In interval, not the unitariness, but the characteristic volumic outline — the ordinal incidence of its variations in intensity and predominance throughout its whole extent — ^is set into relation to the pitches and their difference. It is a fine point of analysis to say whether this difference includes or excludes the distance between the pitches. Which do we actually mean when we hear an interval? Do we mean that these two pitches stand in a characteristic mass and so fall into one class with all other pitches which also occur in a mass of the same character; or does the reference hold for the distance between the pitches? Distance might perhaps be less involved than are the pitches, in so far as we do really ignore all the differences 6i distance that occur in the same intervals. 78 DISTANCE AND INTERVAL [ch. Pitches are not ignored ; for, being predominant in tonal masses, they are in all practical senses, and in some others too, the chief object of interest in sounds. Probably, in interval as such, neither distance nor pitches are the chief concern, but only the interval, the characteristic volumic outhne as such, while the pitches it contains stand for other reasons well forward for attention. Of course the accompanying distance may be ignored as of no special interest, but it cannot be suppressed. XL. However fusion and interval are distinguished from one another in careful reflection, there is no doubt that they are really very closely connected with one another. In a sense they are merely diverse aspects of the same thing — complex tonal mass — ^although interval owing to its special interest or 'intent' naturally carries us further in distinguishing and recognising these masses. Probably both are in varying degrees responsible for the standardisation of the range of pitches. Fusion perhaps leads the way in so far as owing to its influence octave parallels pass in unreflecting minds as identicals. But wherever the reflecting and observing consciousness sets to work with sounds, interval must take the first place. Then the characteristic mass volume of the octave, so perfectly balanced, so easily repeatable, and so Uttle affected by redupUcation, must inevitably give an in- eradicable 'set' to the whole attitude of observation towards the range of pitches. No other interval is so little affected by reduplication. If, for example, two fifths are given at once, the second is not a perfect repetition of the first: for neither of the two new points (the extreme of the top volume and its predominance) coincide with any of the points of the first fifth; they merely upset its balance a httle. But in the octave the lower extreme point of each new volume always coincides with the predominance of the lower one (v. Fig. 6). The fifth might therefore be used as a unit, but it would not be able to maintain itself as a unit in spite of simultaneous duphcation as the octave does. This holds, in fact, for any other interval than the octave. The incompatibility of two identical intervals, e.g. c, c, g^ is, therefore, by no means mysterious, c-e and e-^jf separately are good thirds, but together they create an unbalance, otherwise that of c-g$ separately. Of course, theories that can show no inherent connexion between tones of different pitch, cannot account for this — to them, erratic — behaviour of two-tone fusions in triads^. ' Cf. Biemaim's ciitioism of Stumpfs theoiy of consonance (69, 419 ff.). in] DISTANCE AND INTERVAL 79 The law of the octave is then the supreme law in the reflective use of the range of pitches, or, in the more usual term, in music. If any other intervals are used, their presence and pecuUarities must be entirely subordinated to the Umits imposed by the octave. Under these circumstances it is only natural that the standpoint of the octave should in the course of time work itself ineradicably into our dispositions of observation. For after we have run through all the intervals that we admit between any two pitches an octave apart, the same intervals recur in the next octave without any change of standpoint. But it is clear that if we wish to change our octave basis, for example from c to d, it cannot be done without intermediation. For the volumic Two Octaves Two Tijfhs Fig. 6. relations of c as an octave and of many of the other intervals admitted as compatible with the octave c, are not coincident or continuous with those of d. We can now understand why octave relationships come to be so prominent^ in the pitch range that they could be thought to be based upon identity of quality (cf. above, p. 44 ff.). And we can see too why we must take special means to confuse and suppress our ready disposition towards the octave attitude, if we are to appreciate fully the original continuous pitch differences of all tones. We need to play chromatic passages rapidly in order to baffle the octave disposition. We can hardly exaggerate the strength and readiness of that disposition in modern minds that are famiUarised with it from the first contact with music or even with the most accessible musical instruments such as piano or organ, apart from the natural inevitability of the phenomenon 1 Of all intervals the octave is far the most easily recognised, v. 117, 168. 80 DISTANCE AND INTERVAL [CH. itseK. The whole range of pitches is made manageable by it. Pitches are standardised then in relation to their volumic outhne. We might represent the whole musical range by a convenient diagram^ (^^S- '^)- The lines vf, v', v^, v^, etc. represent half the volumic proportions of the tones of the pitches c", c', (?, etc. The letters p", p', p^, etc. stand, there- fore, at the points of predominance, or at the pitch-points, of these volumes. On the other side of p" from ifi there would be as much volume again as is indicated by the bottom line of the diagram. The diagram represents the relations of the tones of the musical scale truly. but it does not necessarily indicate correctly the real relations which may subsist between the pitches as orders. We do not yet know for example what relation the number of orders in the lower stretch of volume of c' bears to the number in the lower stretch of c*. This 1 The only diagram of pitch-volume I have met with in psychological literature is Titohener's (126, 94). "This attribute [of size or diffusionl" he says, "runs, in general, parallel with the attribute of pitch; but at the ends of the scale it changes more quickly, in the middle region more slowly, than pitch, so that deep tones appear very large and diffuse, and high tones very small and concentrated, while the intermediate tones seem aU to be more or less of the same size." I am not aware that this view has received wider acceptance. m] DISTANCE AND INTERVAL 81 problem can only be settled bjr a direct exauaiiiation of the dis- criminability of pitches. XLI. Such an examination shows that the differential threshold of pitch does not conform to Weber's law, as was naturally expected on the hypothesis of the qualitative classification of pitch, or as would be expected also upon a quantitative interpretation of pitch, such as that suggested by K. Dunlap. We do not need to multiply the rate of vibration of a given tone by a fixed ratio in order to get the tone just distinguishable from the first in respect of pitch. The increment is very nearly an absolutely constant quantity. It increases absolutely (not relatively) slowly with the rise of pitch within the musical range, as the following figures show : 3^ (ca. 70 vbs.) + 0-4 vb. ; e" (125 vbs.) + 0-4 ; c' (250 vbs.) + 0-4 vb. ; a' (417 vbs.) + 0-65 vb. ; a^ (834 vbs.) + 0-9 vb. ; «» (1668 vbs.) + M vbs. ; / (2965 vbs.) + 7 to 10 vbs. ; c^ (4000 vbs.) + 10 to 40 or more vbs. ; after this point the threshold soon rises enormously (cf. betow. Table V, p. 96). These values are averages taken from' tests made by Stiicker (110, 396f.) upon some 30 professional musicians, 14 of whom belonged to the K. K. Hofoper in Vienna. In reckoning the values I have omitted a few abnormally high values (poor discriminations). Meyer (67, 358) working with the special training in observation possessed by Stumpf, got for an increase of 0-35 vibration per second on the tones of 100, 20O, 400, 600, 1200 vibrations respectively 71, 83, 80, 84, 67 per cent, correct judgments. These sets of results show great similarity. They are the only apparently reUable and imiform results I have been able to find. Determinations of the threshold of pitch differences made by unmusical and untrained ears are allnost worthless; they are erratic and irregular and unless got by strietly incognitive methods, probably illusory. A grouping of them is given by Vance (129, 133) : for 64 vbs. some 3 vibrations give the threshold ; for 128, ca. 1-5 vb. ; for 256 ea. 1-5 vb. ; for 512, ca. 2 vb. ; for 1024, ca. 3 vb. ; for 2048, ca. 6 vb. Practice has of course a marked effect upon them, as F. 0. Smith has shown (107). The simultaneous threshold for pitches shows somewhat the same differences of relation to pitch (102, 25). Absolutely it is much bigger, some 10-20 vibrations in the middle range. No doubt the size of this increment at any pitch depends upon the spread of excitation round the point of predominance and upon the displacement consequently necessary if a noticeable shift of the centre of predominance is to be got. And as the volume of a tone w. p. s. 6 82 DISTANCE AND INTERVAL [ch. is the smaller the higher it is, we may suppose that its volumic outline is proportionately smaller all over, as it rises. This would mean a greater pointedness of predominance in direct proportion to the rise of pitch or in inverse proportion to increase in the number of vibrations per second. Apart from slight differences due to rounding, and apart from the desire to extend the range of predominance, which gives a tone a certain elasticity of pitch as we have seen above (p. 31), the displacement required for discrimination under the best conditions of attention might therefore well be about the same fraction of a vibration per second. This analysis however would not imply that the numbers of orders which compose a volume are anything but proportionate to the magnitude of that volume. Indeed how could we expect it to be other- wise? The very notion of volumes impUes a reference to the original multitude of orders constituting it. Although this inference receives no support from matters of fusion, which rest upon the coincidence of volumes, nevertheless it seems natural that apart from the measures of coincidence volumes should really decrease evenly with rise of pitch, as they seem to do, especially since the predominance of pitch seems to be central to each volume. We can only assume that both volume and distance are proportional in magnitude to that number, apart from special conditions and processes of standardisation which may supervene. Thus we should conclude that distances are not only not equal in octaves, but rather about halved in size for any interval with each octave upwards^ (cf. infra, p. 90 f., 162 ff.). XLII. The impression of an increase of distance with a rise of octave must therefore be due to the increase in the number of dis- criminable steps that comes with rise of pitch. That would make the former apparent increase illusory. We need not hesitate to accept this conclusion, for all judgments of distance within the musical range are, as Stumpf admits, exceedingly difficult and doubtful. Theoretical leading might very well reverse them entirely. ' Cf. Ill, 62; 113, 459 f., where Stumpf writes: "Experiences and ideas of other kinds, such as... especially the smaller (apparent and real) extension of higher tones and associations connected therewith, drive us further still: the tone realm seems to grow smaller and smaller upwards. A melody repeated in the octave higher, with retention of the distance relations, appears with regard to the absolute size of the steps like a reduced copy of the original. In real music, in the whole context of music, this illusion is in fact the dominating one," etc. The argument of our text would make it out to be no illusion, but reality. Cf. similar expression in 120, 83. Ill] DISTANCE AND INTERVAL 83 XLIII. But even if auditory distances do thus originally differ that would not preclude their being later standardised by means of interval. We find in the sense of touch great original differences between the distances given by stimulation of points objectively equally far apart on different parts of the skin, tongue, finger tips, palm of hand, etc. But these come to be standardised approximately to the proper amounts of real distance, although the tongue hardly acquires this correction. So too the original differences amongst the tonal distances of the octave and other intervals would become equivalent in the octave standard. Their original differences would not thereby be annulled. They could be recovered by suitable devices and brought before our observation as such. But this standardisation would at least soften for us the crude original differences of distance, and explain the unobtrusive nature of tonal distance over against interval. We should therefore conclude that the essence of interval is proportion of form ; for the variations of distance would then make no difference, since all the elements of the form of any interval would change equally with any raising or lowering of the interval. Nevertheless, as was noted above, the pitches that stand in an interval must not be lost from sight; the interval is referred to the pitches or rather to the 'tones' to which the pitches belong. In a recent pubhcation (120, 85) Stumpf notes that "Messrs Abraham and V. Hornbostel in the last few years have made long series of experiments upon distance judgments in tones... the famiharity with our intervals being as far as possible set aside or made harmless by the cir- cumstances of the experiments. They found that it is really possible to equate small tonal distances with some certainty at different absolute pitches ; and the distances thus judged equal showed the same relations of vibrations, not the same differences as one might think." It must be evident that Abraham and v. Hornbostel, in the face of the immense, if not insuperable, difficulty of uncovering the primary distances of pitches, have simply wandered unwittingly into mere interval, which is of course a matter of relations or proportions of volume. In fact their getting equal relations of vibrations is good indirect evidence that they did not judge on the basis of distance at all, reaUy. For on any count the probability of pure distance judgments giving equal ratios is not great, surely. I have given what seems on my theory to be the proper method of dividing a true tonal distance into two equal parts above in a note on p. 67. The middle pitch between two c's is an /, between c and a an e, between c and g an el? (cf. 14). But when we compare 6—2 84 DISTANCE AND INTERVAL [ch. ttiro theoretically equal distances — e.g. o-f and f-c' or o-g and g-g' — we do not readily detect amidst the sounds anything identical. It may be there for all that. We must remember that we simply cannot strip tones and tonal sequences of their volumes so as to reduce them to mere points, between which distances might be traced. No matter what we do, the basis of proportion, i.e. interval, and from habit the attitude towards proportion, are both there ; and they may well make the proper abstraction of distances impossible. Perhaps somebody will succeed with the abstraction some day. XLIV. The octave and the other intervals are also of great im- portance in the naming of pitches. Relative nomenclature is obviously based entirely upon these relationships. And so in fact is every absolute nomenclature or absolute 'ear.' For absolute ear there is required, besides, an absolute point of reference in auditory orders themselves. In those who have a more or less perfect absolute ear, we may suppose that every auditory order is an absolute individual, which can under favourable circumstances be recognised as such apart from all names. We can all do this with some accuracy, if we are allowed to find at once the given tone on a musical instrument and to practise doing so in order to overcome certain common sources of error, such as octave trans- position, confusion of repeated trials without renewal of the given tone, etc. We then get a tone in the instrument which Ues within a fairly small variable error of the given tone. The same result comes of a similar test of our ability to localise a point touched on the skin. But whereas we all learn to name the point touched on the skin with an equal degree of correctness, only very few learn to name tones correctly, without the help of some familiar instrument, voice or piano, etc. It is difficult to explain this difference between the two senses. Probably the chief source of the difference is the extraordinary emphasis laid by music upon the relationships of pitches. Musical instruments constantly vary in the absolute pitch to which they are tuned. In fact a standardisation of pitch is only required for certain special reasons; e.g. on the piano for the convenience of those who have a very precise and rigid absolute ear, and who would be ' thrown out' if they had to play a work a quarter or half tone lower than usual; or for the convenience of those who are singing to piano accompani- ment up to the Umits of their voice range, and so on. Absolute ear is a great help in musical practice, but it is quite dispensable. In fact it seems to be rather difficidt to acquire even when there is a certain Ill] DISTANCE AND INTERVAL 85 tendency towards it (cf. 43, 26if.) and easily lost, if it is acquired with any difficulty. In some favoured persons it is acquired early and more or less unwittingly and never lost. Perhaps these persons have some 1 special refinement of hearing, such as a much greater delicacy of volumic j outUne and especially of predominance, than have others. Or perhaps ; a highly favoured auditory disposition gives them the power to maintain i their absoluteness of ear in spite of the universality of musical relativity. ' In that case we should all naturally possess absolute ear and then proceed to lose it or to lose the power to convert it into absolute nomen- clature. In dealing with noises we all seem to retain a good deal of v it, even the most unmusical of us, who recognise voices, noises, etc. I But then noises are irregular sounds in which many orders emerge irregularly, rather than only one or a few all the time. And noises are not subject to the same relativity as music is. There is evidence also that young children may be brought to show absolute ear, if the snares of relational changes are avoided. And they can learn it the more easily, the less they have ialready had to do with music (32). They are taught tones in association with certain letters, and can repro- < duce the tones absolutely when given the letters^. Dogs perhaps \ also have a good measure of absolute ear. The whole problem is subtle and compUcated. But it is not of any primary importance in the psychology of hearing. We are now in a position to see how unimportant Stumpf's evidence in favour of octave qualities is. Errors of one or two octaves in absolute ear and the ease of octave transpositions are inevitable results of the relativity which the octave brings. They show that even absolute ear is in part subdued to musical relativity, seizing as of 'first importance only the placing of the pitch in its place in the octave, and of minor importance accuracy in specification of the exact octave. All these arguments, as well as the harmonic equivalence of the octave, are obvious consequences of the relativity of the octave and its importance for all music. * Cf. D. Katz (38), and the tone- word method ot training by Karl Eitz mentioned there, a method which uses names for tones very much like those of our tonic sol-fa system. CHAPTER IV THE ANALYSIS OF TONAL SEQUENCES XLV. The chief differences between the study of tonal masses and sequences is that the latter is relieved of the problem of fusion. A gradual transition from mass to sequence is of course possible, if one of the two primary sounds is intoned and stopped a little before the other and the interval between the incidence of the two is gradually increased, until the second occurs distinctly after the first. It may be noted then that the phenomena of adventitious pitches and fusion become less and less readily noticeable. But they do not disappear as soon as the two sounds are heard as a succession. With strong tones of deeper pitch there is even then an instant of coincidence in the brief gap between the sounds^. But this rapidly disappears and the two sounds are then heard as a succession without any overlapping. Each tone is then easily distinguishable from the other in every respect, intensity, pitch and volume. An exact comparison of intensities is not easy unless the tones are of closely neighbouring pitch ; but some estimation of their relative intensities is possible in spite of considerable . difierences of pitch (111, 348). Various reasons account for this. In the first place it is diflS.cult to measure the physical basis of the intensity of tones of rather difierent pitch and to equalise them. In the second place it is impossible to bring intensities to any sort of psychical over- lapping, especially as the pitches to which they are in this case attached are supposed to He some way apart in order. Other reasons might be brought forward from physiological sources, but these do not concern us now. Comparison of volumes is not affected at all by the succession of the sounds. Of course the same sort of judgments cannot be expected from successive sounds as from those that completely overlap. In the latter case the characteristic coincidences that constitute fusion emerge of themselves and hardly even require the help of attention for their observation. But it is evident that the same relations will exist in succession as in simultaneity. If the first tone is followed by its octave, the ordinal incidence of predominance of the former will be ^ Cf. 112. 89; low tones are damped less easily. CH.iv] THE ANALYSIS OF TONAL SEQUENCES 87 coincident with that of the lower extreme of the latter. The transition to the second tone will therefore be prepared by the first. In fact a basis of transition passes from the first to the second. The only change is the movement of pitch to the middle of the upper half of the first tone and the cessation of the lower half. Similarly in the case of the fifth the reception of the second tone is prepared by the first, in so far as the characteristic points of the second lie at equal distances on either side of the point of predominance of the first tone. But the fifth cannot give the same degree of identity as the octave; for none of the chief points of the first tone are identical with any of the second, excepting of course the upper extreme point, wHch is common to all tones. It is clear then that relations exist between successive tones that will inevitably standardise the range of successive pitches in exactly the same way as the range of simultaneous pitches is standardised by the volumic outline of tonal masses. Apart from the natural steps given by such cases as octave and fifth, any step will give relations that may be applied universally throughout the whole range of volumes and so create an 'interval.' And that interval may be learnt and remembered^. Experiments on the purity of intervals showed that successive intervals can be adjusted as finely as and, especially in thirds and octaves, much better than simultaneous intervals. For the third 70 cases of correct judgments were got for 2*18 and 5 vibrations respec- tively for succession and simultaneity. For the octave 90 correct judgments for 0-46 and S-l vibrations respectively (146, 366f. ; 115, 55). We shall consider the formation of scales later. So far we can see that there is not the least problem or anomaly in the parallelism of the relations estabhshed in tonal masses and sequences^. * CSE. M. Meyer (75, 207-214) on Quarter-tone music, where some experimental evidence is brought to show that when intervals previously strange, grow familiar, they are expected and anticipated, and then become more pleasing. ' Of covirse this is quite without prejudice to their difference. In sequences melodic values {v. later) stand forth prominently. A major second is not a melodic discord; but it is the same interval in sequence as it is in the mass; and in sequence it is as devoid of balanced relationship to the tone preceding it as the two are in the mass; and so on. There is no sense in running the statement of an aspect to death by generalising it in opposition to all other statements of aspects. We must see the facts and their theoretical basis in their fullest breadth; of. 121, 329, where F. Krueger is quoted as having opined correctly that no one listening to the scale would ever speak of it as a series of dissonances (45, 246.). That is to encourage us to beUeve that successive tones get their relations from common partials! Krueger himself believes that "the transference of the notion of consonance to tona.1 sequences would never have taken place, or would be unintelligible unless, chiefly from reasons drawn from physics and musical theory, we classed the mass 88 THE ANALYSIS OF TONAL SEQUENCES [ch. For Stumpf and others there is an insoluble problem in the parallehsm, because, not having any psychological basis for the fusion of simultaneous tones, he can have none for the parallel relations of successive tones ; and as the latter do not fuse, successive tones seem to call for another cause than the physiolo^cal one supposed to underlie fusion. Stumpf accepts as the most probable basis of the consonance of successive tones their relationship through common partials, a principle adopted from Helmholtz but disproved and rejected by Stumpf as an explanation of fusion. Stumpf's reasons in the latter case are: (1) consonance and dissonance can be got in absence of aU partials; (2) any appeal to memory is illusory, fusion will not arise out of habit any more than the locomotive will rim from custom, when the stoker forgets to coal up. These two reasons hold equally for successive tones. Stumpf's appeal on behalf of successive consonance " seems rather inconsistent and helpless (115, 55ff.; 112,195; 121, 328f.). His own feeUng for this inconsistency leads him to give the relations of successive tones another name — " Verwandtschaft" — although it is clear from the facts and his discussion that something identical with consonance is implied thereby, in spite of Krueger's sage remark. There is an identical aspect, as I have pointed out, and also a difference, but it is not a case of all difference and no identity. The only problem of these parallel relations is how they are brought into connexion in cognition. For in spite of the connexions just ex- pounded, the two series are separated by obvious differences. The direction of observation in each is different. In the tonal mass the coincidence of volumes is actually present ; it cannot but be felt. And it can be specially observed as soon as comparison of tonal masses favours its effect upon attention; it will then be more distinct, but not any more fused (i.e. any more like the perfectly balanced 'pure' tone) than before (of. below, p. 99 ff.). The attention in this case does not establish the relation; it only favours its effectiveness. In the tonal sequence on the contrary, the relation is hardly actual until attention has been directed upon it. But that direction of the at- tention must be easily provoked even by the mere sequence of the tones, in the case of octaves at least; and with greater difi&culty in the other cases. The transition from simultaneity to succession, which can easily be produced on many musical instruments, must help of two tones and the sequence of the 'same' tones under one name (fifth, semitone, the same interval, etc.)" (ibid.). Inverted commas and extraneous reasons go well together, but thej do not suit the good intrinsic sense of the 'transference' referred to. IV] THE ANALYSIS OF TONAL SEQUENCES 89 to encourage the attention to see the identity of relations upon which successive and simultaneous intervals are founded, especially in persons highly disposed to auditory observation. XL VI. But many are not so disposed. They feel the coincidence and balance in tonal masses, as we have seen in the study of fusion. But they do not readily learn to recognise and name the difierent forms of balance and stiU less the different volumic outlines that are devoid of much balance — ^the dissonances. The parallel relations of tonal sequences they find as hard to learn ; and of course they do not spontaneously grasp the connexion between the two series; they have to learn each independently. Their incapacity to trace the connexion spontaneously is no evidence that masses and sequences are devoid of common relations; it proves only that the difierences between the two are great enough to obstruct the view of these common relations for those who do not observe and learn sounds readily. This incapacity of unmusical people has been proved experimentally by V. Maltzew (61, 192). There is even a difference in the memory dispositions for ascending and descending sequences. The judgment of descending intervals is found to be much harder, even by persons who have had considerable musical education and practice. Longer time is spent in recognising a descending interval than an ascending one {ihid.). The peculiarity is probably based upon the habitual attitude towards tone masses already noticed whereby in a constant mass of sound the whole takes the pitch of the lowest component, even when that is not also the strongest (cf. 112, 384ff.). The point of observation is naturally central to the whole volume, unless it is drawn by special circumstances to one or other side or induced to spread itself over some extent. Stumpf pointed out other indications of this habitual standpoint. We judge of the tonal series from below Upwards. Rising makes the impression of tonal recession, falling that of approach. We begin a scale involuntarily from below, not from above, and we end it below again^. When a major chord is given successively or simultaneously, we take the lowest, not the highest, tone as tonic; we consider the major, not the minor, third as the first interval (111, 149). Also, when an interval is mistaken for unison, the tone heard in the majority of cases is the lower of the two (117, 166). This habit is doubtiess much strengthened by the octave standardisations of music. * M. Meyeir (75, 204 ff.), who shows some experim^tal evidence for the preferfenoe for a descending interval as the last of a series of intervals. 90 THE ANALYSIS OF TONAL SEQUENCES [ch. The octave standard is given not by the enunciation of a single tone, but of an octave along with one or more of its characteristic subordinate intervals, and that octave, seen from below, is of course implicit in the subsequent music in so far as all successive intervals are such as are compatible with it, or if not directly so, are introduced with sufficient preparation, or with little or no preparation for some artistic purpose or effect. The difficulty of identifying ascending and descending intervals might well be compared with the difficulty of identifying upright and inverted visual patterns. Towards visual pattern we also learn to adopt a standpoint. We must, of course, take as examples, figures that we are accustomed to see only in one position, e.g. figures and letters: 08 ^ fnopngdapm e- ^ n W 'lA ^ ^^°V^ & 5 ^ fUnodpuB^s Inversion makes Inversion changes each Inverted words the lower halves of these four letters are hard appear smaller into another to read V. Maltzew found no true evidence to show that the relations between successive and simultaneous intervals are estabUshed by our converting the sequence into a mass in representation, as Stumpf recently maintained (121, 328f .). In cases of doubt or difficulty this procedure is hardly ever even attempted by observers and is of little use even then (61, i90ff.). It is indeed difficult to see how an observer should proceed in order to sum the volumes, of two successive tones in order to get the simul- taneous volumic outhne^. The coincidence of characteristic orders is enough for the tracing of identity when masses and sequences are given in sufficient proximity. But we could hardly expect two successive volumes to be summed or even identity to be readily traced through common points, if only the sequence was given and the mass had to be imagined therefrom. Distance is just as unable to explain the con- struction of intervals in the case of sequence as in that of masses. And it would show no difference between ascending and descending intervals. V. Maltzew's own view (61, 196) is that interval is based not on any graded difference, but on something that is peculiar or typical for each interval. That is of course quite certain from the nature of the facts. And it is confirmed by the fact that when learners begin to take notice of this characteristic difference between intervals, they make great ' Cf. with this the difficulty of summing two sine curves at sight, e.g. the sine curves of sounds in ratio of 4:6. IV] THE ANALYSIS OP TONAL SEQUENCES 91 progress in learning them (61, 210). She proposes to call this peculiar 'content' of the experience 'step or passage experience.' But she is as unable to give this experience any definite psychological expression as Stumpf was in the case of his degrees of fusion. It is confusing to refer to the absolute way in which colours are named and recognised. That comparison and the use of the word 'qualitative' suggest that interval is a kind of psychological quaUty. But it would be absurd to look for qualities at this stage of the complications of hearing. Let us take other analogies. A word can be recognised and named without comparison with other words. So can a visual pattern or figure such as a circle, a cross, a square, etc. And these are after all the same thing psychically as words, which are only visual patterns. Not only that, but on our reading of it, interval is really and truly a matter of form, of volumic outline. And it is known now in the sense of vision that we have a very fine sense of proportion of forms (9, I38ff.). This sense of proportion in sound would give proper expression to our abihty to sing any interval or melody on any given pitch. The basis or scale of proportions is then given in the volume of the starting tone. And distance would in no way obstruct the proportions, for it would itself be proportional to the volume of the starting tone (cf. above, p. 82)^. But for that volume we could no more define the proportions of a melody or interval by giving the lower tone than we could define the proportions of a square by fixing a point as the beginning of an outhne drawing of it. Such reflections lead us to see that there is no more difficulty in there being unequal distances in one and the same interval in difierent parts of the musical range than there is in difEerences of volumes in the tones of that interval. It is all a matter of proportion. So long as all the proportions are maintained, interval remains identical, if interval is a matter of proportions. No doubt the basis of judgment in interval is the experience of 'passage.' That expression is quite compatible with the terms of our analysis, v. Maltzew's failure to get beyond this expression is ^ Distance would then be of essential importance in the recognition of interval only when the intervals investigated were confined to a relatively smaJl range of pitches. Then judgment on the basis of distance would lead to the confusion of neighbouring intervals, as happens so frequently when intervals are given in very short duration. Consonance (fusion) then loses its eSect upon recognition; but distance seems to be less affected by reduction of dilation. THo doubt this is due to the fact that distance is only » part of a tonal mass, the part that stands in the more frequent 'focus' of the attention upon the ordinal field of tone, whereas the apprehension of fusion, as we have shown, requires the apprehension essentially of the whole of a tonal mass. Cf. 117, 169 ff. 92 THE ANALYSIS OF TONAL SEQUENCES [ch. due to the lack of any true psychological method of analysis. We cannot expect observers untrained in psychological method to piish their direct analysis as far as it can be carried by theory. For theory is analysis guided by all available facts. And it must be able even in dealing with experiences to go further than any direct analysis could. Theory pierces in experience, as in any other realm of existence, into the real structure of experience. That structure cannot be said to be non-existent because it is not observed in all respects in ordinary analysis as it is properly held to be in theoretical analysis or 'in reality.' For experiences can only be described by being taken into cognition. And it is not evident that our highly practised direct methods of cogni- tion should be suffi J tone at : is not ilsuaUy > tone at : judgment breaks down at : 1. d* e*-6't> 6* + Modified from v. Maltzew (61, 2l»). In the last colimm 0^=0 means that c^ and tones above it up to o^ were usually said to be of the pitch a. When we seek an explanation of this pecuhar phenomenon, we can expect no help from the theory of octave qualities^ much as that seems to account for the existence of nominal pitches in general. On my showing, nominal pitches are the result of the standardisation of the whole range by the octave interval. Thus the judgments showing distortion of pitch revert to the same basis, whether the observer be endowed with absolute ear or not. For an explanation we must suppose that, in the extremes of the musical range, tonal volume (probably for some physiological reason) begins to be a httle more extensive than it should be (cf . p. 63, note 2, above). This is doubtless due to the sraallness of the volmnes in that region and to the difficulty of getting IV] THE ANALYSIS OF TONAL SEQUENCES 97 the sensitive surface in the ear to respond in extents decreasing regularly according to the decrease in wave length of aerial vibration^. Thus the volume giving the octave, where the lower extreme of the upper tone would coincide with the predominance of the lower, would arrive a little too late, in relation to the usual ratios of vibrations. In other words : the volume heard from a certain number of physical vibrations would be a little greater than it should be, were the relation between volume and rate of vibration still unaffected. So in order to get a higher volume whose lower hmit shoidd be at the point of pre- dominance of a lower volume (the upper Hmits of the two volumes being necessarily identical, of course), we should have to use a rate of vibration more than double that required to evoke the lower tone; instead of 2x vibrations, it might be 2x + y vibrations per second. But 2x vibrations are physically, nominally, shall we say, of pitch c, whereas 2x + y vibrations are, similarly, of pitch d ; but the latter, not the former, is heard as the octave of the c of a; vibrations. My theory of interval, whether simultaneous or successive, shows that this process of standardisation does not involve any explicitly ratio- cinative process. The observer 'sees' directly that the tone given by 2x vibrations does not touch off the points related to the points touched off by x vibrations in the regular proportion known as the octave ; but that it touches off points more or less nearly related in the proportion known as the major or minor seventh. He therefore hears the new higher tone flat, or calls it b or 6l7. In the same way, in the lower regions of tone, we may suppose that the ear ceases to offer sufficient room or a proper basis for the great volume required, and that the areas under stimulation are somewhat cramped. Thus the volume evoked by x vibrations per second would be too smaU; its centre of predominance would lie a little nearer the upper end of the tonal-pitch series than it otherwise would. So in order to get a volume whose central predominance shoxild he at the end of the volume evoked by a 2a! rate of vibration, we should have to use 1 Of. Abraham and Briihl (1, 197), where it is shown that while two vibrations suffice for the production of tones whose pitch lies below j*, tones are heard with three vibrations up to b*, with four up to d^, with five up to P% with ten up to cfi, and with twenty even beyond that. At the lower extremes there is some sign of a similar change, but it is much less distinct. The lowest limit for two vibrations is 0^, for four vibrations B^. In V. Maltzew's case it is a matter of volumes, in Abraham and Briihl's of the definition of the predominant order in the volume as well. These values given by Abraham as observer are similar to the values given by observer 4 in the table from v. Maltzew's paper (p. 96). W. p s 7 98 THE ANALYSIS OF TONAL SEQUENCES [ch. iv a rate of vibration lower than x, say x — y. That is to say the x — y rate would be called, say, c, while the x rate would be called, by simple inspection of the proportion of the ordinal position of the evoked volume to the ordinal position of the volume evoked by 2x, say d. That is, very low tones would be heard a Uttle sharp. But as the centre of predominance of the lowest tones lies still far away from, though of all tones nearest to, the low end of the pitch-order series (physically — the apex of the cochlea), we may well allow that much of such distortion of low pitches need hardly be expected. It is possible, as Stumpf suggests (123, 320), that, within the range of smaller errors of a semitone or less, repeated work with these extreme pitches should lead to quite correct estimation of the pitch. The physiological process would then, as Stumpf says, gradually accommodate itself properly to the physical stimulus. But it is clear that this process of adaptation wiU only go a certain length and that we cannot expect it to appear where all judgment of pitch breaks down. Thus we see that the musical range of pitch is the whole range within which the octave standard remains vahd, while stiU (approxi- mately) maintaining its consistency with the ratios of the aerial vibration. Beyond this range the volumes of tones evidently become quite inconsistent with the demands of the octave standardisation. They do not conform in any manner, not even if we seek out the required proportions of volumes without regard to the physical ratios of vibra- tions. No doubt the balance and symmetry of volimies then largely disappear. This need not, however, imply that in these extreme regions no differences of pitch-order are observable. These orders may well change without there being any proper basis for their standardisation to mTisical nominal pitches. If we were to construct a diagram of the relation of change of volume to increase in the number of physical vibrations, we should have to reduce the relation somewhat for very low tones, to keep it constant throughout the definitely musical range, and to increase it gradually towards the upper limit of hearing, stopping it as indefinable soon after the musical range had been passed. The rest of the range of hearing is the range of mere audibiUty. CHAPTER V THE FURTHER STUDY OP TONAL MASSES XL VIII. We are now in a position to consider criticisms, restric- tions and extensions applied to Stumpf s treatment of fusion. It is quite evident on my theory that fusion introduces into tonal masses a great deal of that regularity of system and balance which in its greatest perfection constitutes the pure tone. A fused mass approximates more or less to the unity of the single tone. This approach to unity may therefore legitimately be taken either as an index towards, or as a definition of, fusion (40, 143), if it is understood that all definitions of fusion are to avoid stating the exact basis and essence of fusion, as in fact all the definitions of Stumpf and others do. At the same time Stumpf is quite right in looking upon fusion as an. "unalterable pectiliarity of sensory material" (112, 128). And it would follow therefrom, as Kemp says, that " every change that a degree of fusion suffers, is only apparent; it is only the apprehension of the fusion that changes" (40, 144). It is not apparent how by means of attention any change could possibly be produced in the volumic coin- cidence of two tones. Of course no one could deny in face of the great progress made since 1890 in our knowledge of the influence of attitudes upon observation^, that by suitable instruction an observer may be more rapidly and singlemindedly directed upon the specific phenomena of fusion, the pecuKarity of sensory material to use Stumpf's phrase. Similarly he can be led to abstract fusion from any other phenomena of tonal masses, e.g. from their pleasantness or from their harmonic affinities. That sort of abstraction was not impossible for Stumpf even in 1890. If we attend to the whole impression, it is more efEective upon our observation; if we attend to the discriminable parts — the pitches — they determine our statements most. If we are practised we can discriminate parts sooner, if we are fatigued we cannot discrimi- nate them so fast, because when special attitudes are opposed by fatigue, the habitual or natural attitude is the easiest (cf. above, p. 72). ' My Beitrdge zu einer Theorie des Denkens (131) was the first decisive contribution to a study of the influence of the 'instruction' on a, mental process Cf. my abstract of this book (132). 7—2 100 THE FURTHER STUDY OF TONAL MASSES [ch. But it is hard to see by what right greater abstraction can be read as greater fusion. If it is, we should surely have to find some other instru- ment than abstraction if we are to get through to the original fusional differences of tonal masses. Our aim in the direct study of sensation is to bring our knowledge into conformity with sensation through the medium of observation. I fail to see that Kemp (40, 146) has shown any superiority in Kiilpe's method of dealing with fusion over Stumpf's. Kemp says it is a fact that the impression of fusion is changed by many circumstances. But practice and fatigue, which have just been mentioned, clearly do not affect fusion, but only the analysis of pitches, or perhaps better the analysis of pitches as against the appre- hension of the total impression, including the pitches, and without their discrimination. But neither of these things is really and properly fusion. It is" no departure from actuahty towards the 'ideal' to say that the phenomenon of fusion does not primarily include the dis- criminabiUty of pitches, but is present in equal degrees whether the pitches be discriminated or not. That is just the sort of thing that is justified by our later knowledge of attitudes of observation. Only one group of facts might perhaps be brought under Kemp's statement, viz. the influence of intensity upon fusion (40, 159). Kemp accepts Stumpf s law of the independence of fusion from the intensity of the components only for absolute intensities within a middle region. For relative intensities he says it holds only for Stumpf's fusion, not for Kiilpe's, which is concerned only with "the impression, the experience of fusion." According to Kiilpe the characteristic feature of fusion is the increased difficulty of analysis that is due merely to the simultaneity of the sensations (40, 145). The single components retire in fusion in favour of the mass impression. This conception of fusion would seem to give the analysis of pitches much more importance than they obtain in Stumpf's conception. Surely there is here a failure to appreciate the merit and justice of the abstractive analysis of fusion from the discrimination of pitches, claimed by Stumpf. When the resonance box of one of two sounding forks is closed, the unitariness of the mass-effect and the imperfection of the analysis are both very much increased. Naturally; because the predominance and whole strength of one tone have grown very much less and it is notoriously more difficult to pick out a weak tone in a mass than a strong one especially if the higher is the weaker tone (40, leo). For the lower tone readily swamps the volume of the higher. The nearer we get to V] THE FURTHER STUDY OF TONAL MASSES 101 the intensive proportions of a single tone by great weakening of the upper tone of a pair, the nearer we get to the perfect unity of the 'tone.' But that does not mean that so long as the volumic coincidences characteristic of any one fusion suffice to give the mass a noticeable character, the fusion of the mass is changed by the weakening of the proportions due to one of the tones. Fusions do not aU run into com- plete 'purity of tone' by continuous degrees proportional to the weakening of the intensity "of one of the pair. It is the merit of Stumpf 's second law of fusion that it abstracts fusion from the peculiar and different difficulties of analysis of pitches that accrue when in a tonal mass either the higher or the lower component is. gradually weakened. Stumpf's point is: so long as the characteristic features of a fusion can be seized, so long that fusion is one and the same, no matter how different the relative intensities or the pitch-regions are. And my analysis bears out this position very nicely. The matter may be summed up as follows : if we take fusion strictly as approximation to the regularity and balance of the 'pure' tone, then differences of intensity would produce shght differences in fusion ; but these differences are very slight so long as both tones are readily audible; greater differences in approximation to the 'pure' tone are determined by the volumic coincidences of the fusing tones; so long as the two tones are distinguishable, these must provide a constant basis of departure from the pure tone; such differences are noticeable without any violence of abstraction and are far more important than are the minor differences of approximation to the pure tone produced by intensity ; these minor differences would, however, gain in importance as the difference between grades of fusion decreases, i.e. below the grade of the fourth ; here they come into competition with other influences, e.g. pitch-blends which also form a shght departure from the perfect tone^. And Kemp admits two things that bring his position very near to that of Stumpf. 1. "Two masses of different fusional degree differ 1 On the influence of intensity of. Faist (19, 125 f.). A slight difference towards indiscriminability is produced for his young unmusical subjects by pianissimo-strength except for the octave. His results show that the higher tone is more easily swamped by the lower than vice versa, as I have already deduced (above, p. 71). As Faist shows, it is difScult to say whether this influence of intensity upon analysis also holds validly for fusion. Faist's results are based only upon analysis (i.e. are there two tones — pitches — in the mass, or only one?). Cf. 114, 288 f. Of pitch-blends Faist finds that they increase the high fusions and decrease the low ones, i.e. they simply further the prevailing tendency as we should expect of them. Stumpf reports Faist wrongly (114, 292). 102 THE FURTHER STUDY OF TONAL MASSES [ch. with respect to their fusion not only by the fact that in the one case analysis can be carried out more completely than in the other. The essential point is that in the two cases a quahtatively different fusion is experienced" (40, 149)1. This idea of quahtative difference is pushed still further by Meinong and Witasek (64, 199) to the impossibility of comparing grades of fusion at all. Obviously on my theory such an extreme view is tenable : the ' form ' of any one fusion is as incomparable with that of another as are the forms of square and circle. But never- theless grades of fusion are certainly comparable according to the degree to which they approximate to the perfect symmetry of the pure tone, i.e. according to their degree of fusion. 2. The differences of fusion produced by special attention to fusion are small, and if they are kept constant they need not swamp the differences given by different ratios of vibrations (40, 153). In view of the fact that it is difficult enough to get great constancy of order in the lowest degree of fusion among different observers^, these minor differences must be so small as to be negligible. And if attention to fusion increases fusion as much as attention to a partial increases the subjective strength of that partial, we must remember that the latter is a very debatable matter and might be decided differently according as pitch is classified as quality or as order. When I attend to a visual point without moving to fixate it, it does not grow more intense. When I attend to an auditory order, need it therefore grow more intense? Do I not merely give it the same subjective benefit of attention as I give to the visual point? My attention passes over to its order, so to speak. With pitches as quahties there is no basis for any movement of the attention, so that the apparent effect of attention must be ascribed to change of intensity. For the pitch quality itseK cannot be supposed to change or come into being with the attention. Nor is there any other variable character to explain the prominence given ' As Stumpf remarks (114, 298): "how is the octave recognised if not by its fusion? And don't we recognise the octave even when the one tone is weaker 1 Don't we recognise an octave with the same certainty as usual, so long as the weaker tone is still anything like clearly recognisable." This is true for the octave; it is primarily recognisable as octave only by its fusion; but it is clearly not true for intervals of the third grade of fusion (fourth and others); they are so individual and so easily recognisable because they are intervals, i.e. because of their volumic proportions not by their volumic coin- cidences or approximation to the balance of the pure tone. And then we must ask : are the fifth and octave not also intervals in this sense and recognisable as such? Cf. Stumpf 's remarks on the fusion of the double octave (114, 894). ^ Cf. Kemp's own results (40, 186 f.). V] THE FURTHER STUDY OF TONAL MASSES 103 by attention. Hence it is supposed that the attention somehow brings about an increase of intensity^. The concept of fusion adopted by Kemp is that of "a phenomenon be experienced in its peculiarity only by attentive observation of . mass impression" (40, 153). Unitariness and difficulty of analysis are only secondary marks of fusion. I cannot see any advance in this formulation beyond Stumpf's position. One might indeed see a trifling retreat, for whereas Stumpf does try to indicate his sense of the psychical presence of fusional grades, although he fails to express it, Kemp merely refers us to the phenomenon in experience itself. There is no empirical formulation here to contrast with Stumpf's ideal fusions; there is no formulation at all. It is for these reasons that it was said above that no essential advance had been made beyond Stumpf's residts^. And no wonder ! Neither Stumpf nor any of his successors found any psychological method of getting beyond a gesture towards the special phenomenon to a psychologically formulated concept of fusion, such as has been given in this work. Kemp's experiments were carried out with certain improvements of method. The method of the comparison of pairs was used; by pair is understood in this case the mass sound consisting of two 'tones' of a certain interval. These pairs were given successively and compared in point of fusion without regard to the interval formed or the pitches of the tones or any other feature of the tonal masses. Thus the advan- tages of the incognitive method seem to be obtained. Its special service in this case is the avoidance of any witting transference of judgments ^ I may say in this connexion that I think wanderings of the attention, in any proper sense of the word 'wandering,' are only possible within a sensory field, i.e. with the system which all the ordinal variations of a single sense create. The terms 'focal' and 'marginal' can also be properly applied only to the distribution of attention in such a field; these terms are generalised from a special case of sensory field, viz. the visual field with its specialisation of a most sensitive area. In the auditory field (Le. of pitches) there is no 'focus' at aJl, but if the attention is directed upon one pitch, it is thereby diverted from another; and the diversion is in general the greater, the greater the distance between the two pitches. When differences of quality or, if it is possible, of intensity, occur apart from ordinal differences, we can speak of attending or not attendmg, but not of any wandering of the attention. Wanderings of the attention are probably possible in the temporal field, but not with any such ease and precision as in the systemic field. ' Cf. 40, 179, where Kemp supposes that there is really no difference between his method of observing fusion and Stumpf's — namely to face and judge the mass impression. Also p. 189 f., where he shows a willingness to suggest that any differences between the results of different observers on fusion is due to the variant observers not having observed fusion only, but something else as well. Is not that like an admission of the ideal constancy of fusion attributed to Stumpf? Cf. 114, 290. 104 THE FUETHER STUDY OF TONAL MASSES [oh. from one experiment to another through the medium of the names of intervals. The use of the method presupposes such an indisposition to reproduce the names of intervals as persons devoid of absolute ear or special musical famiUarity with intervals possess. But there is no reason to suppose that a comparison of the fusions of intervals is made any less objective by their recognition (40, 175). Stumpf's judg- ments were certainly given under full knowledge of the pitches of the tones used. For a person possessed of an ear Uke Sttmipf 's no incognitive method and no instruction could keep the intervals out of cognition. In any case fusion is not primarily, but only secondarily a matter of difficulty of analysis of pitches and an abstraction of volumic coin- cidences, i.e. fusion, is possible both with and without analysis of pitches. Thus, as Kemp says (40, 179), we may fairly conclude that there was no difference between Stumpf's method of judging fusion and Kemp's. Only the experimental methods differed. And the agreement of results (cf. 40, 188 f.) confirms this view. If we take as an observer's minimum the placing of octave, fifth, and fourth in the first three grades of fusion (i.e. expecting their discrimina- tion of fusion to reach at least one step into the group of minor grades) we find the following series from the papers of various experimenters : Table VI. Observers No. Intervals in order of fusion Faist (19, 104) III 6 VI 3 T 7 II 2 vn Ki^ {ibid.) III II! 3 VI T 6 7 vn 2 Meinong (64, 193, 198) VI ni 3 6 T 7 n vn = 2 Pear— Wa. (89, 66) ... III 6 3= VI — II 7 VII 2 Pear— We. {ibid.) ... 6 111= VI 7! — 3 VII II 2 Kemp (40, 186 f.) ... 5 III 3 VI 6 T II 7 2 VII Kemp exceptions {ib.) i VI III — — 7 II — — Most frequent of all 10 — III 3 VI 6 T II 7 2 VII Stumpf (112, 135) ... 1 Srds and 6ths All the others Of Pear's observers one was the most musical, the other the least so. Two of the other three observers did not even get the fifth into the second place. One of Pear's observers and Witasek put the fourth down from the third place to amongst or below the thirds and sixths. Meinong and Witasek doubt whether any definite order can be got out of the qualitative differences of fusion (64, 199). V] THE FURTHEE STUDY OF TONAL MASSES 105 In the face of the agreement between the results just shown such a conclusion is of course unjustifiably sceptical, but it is surely true, as I pointed out above (p. 58), that these grades of fusion are not nearly so distinct from one another as are the octave and fifth from one another and from the rest. There is undoubtedly in the fourth and those below it a greater departure from the unitariness and balance of the pure tone. The fourth may be supposed to be better balanced on the lower side of the upper predominance {v. Table III, p. 68), as that extent of volume is divided into two parts by the lower extreme of the upper tone. Of the next four intervals we migh,t readily assent to the minor sixth's coming last as the proportions are there complicated by the introduction of halves (or 6 — 2 : 3 — 5). The other three might well be on a level ; it is especially difficult to see what there is to choose between (III) 2—3 : 1—4 and (VI) 4—1 : 2—3. The intervals of the last group give still more comphcated propor- tions. But we need not follow out the parallel between conceptual and auditory proportions further. That parallel should certainly hold for the first few distinct degrees, but the differences perceived by the ear, subtle as these are^ and visible only through considerable statistics, need not be evident in conception. Or in other words they are as evident there as we could well expect. These proportions may also as Kemp's observations show present other features for hearing than, their mere approach to the unity and balance of the pure tolie. One of these Kemp calls the sensuous com- patibihty of the tones : an imdisturbed concurrence, kinship, friendship, between the tones. This is said to be least for the major seventh, which was therefore used as a model for observation. Of this one observer says: "the two components strive with one another, I can't grasp them together ; when I try to bring them together, one of them always eludes me." Another observer is brought to notice the phenome- non by being instructed to direct his attention more " upon the single components of the whole mass." When he had observed it, this observer said: "it seems to me to run parallel to the fusion, as if it were always the same thing that is observed, only from another standpoint." The 1 Cf. Stumpf (114, 286 f.): "So much at any rate is certain, that intervals like II and VII fuse considerably less than III. That the sevens (4 : 7 and 5 : 7) lie between, and can in themselves as well be termed consonances as dissonances, does not prevent us from distinguishing at least the dissonance group from the thirds group, even if we go exclusively by the differentia of fusion. But a specific opposition, such as is intended in the distinction of dissonances and consonances, is never in this world to be deduced from fusion alone. Other differentiae must co-operate hereto" (p. 287). 106 THE FURTHER STUDY OF TONAL MASSES [ch. phenomenon was detected by one observer, whereupon its study was undertaken by Kemp, another observed it "relatively easy," a third (the second quoted above) experienced considerable difficulty with it, three observers failed to carry on through their observations what was brought home to them with the model of the major seventh. This introspective uncertainty is borne out by the results of four observers, which only show two clear steps of diiference: the fifth, major sixth and third as against the fourth, minor sixth and third. The proportions of the parts of the tonal volumes for these masses (v. Table III, p. 68) are: 2—1:1—2; 4—1:2—3; 2—3:1—4; and 2—2:1—3; 6—2 :3— 5; 2—4:1—5. It is possible that in these proportions, which in the first set are of opposite direction in the whole volume, a basis for a difference of har- mony of proportions might be found as in pictorial art, where distances and proportions are often brought to balance round a centre. It is significant that in Kemp's results the series of grades for the pleasantness of intervals is almost identical with the series for sensuous compatibility (40, 202). But neither of the series is very distinct in any one observer or regular as between observers. It may therefore be supposed that sensuous compatibility means balance of proportions. That is much the same as basis for movement of interest, basis of pleasantness, as is the case in pictorial balance as well. The fifth and fourth are not very pleasing intervals and these are said by observers to be of 'empty fusion' while other intervals are of 'full fusion' (40, 190, 211). These terms point to the fullness and variety of a mass as against the formality and want of variety of a close approach to imity and balance. But where introspective observations furnish only indistinct series, we need hardly look for more than a probable basis in the conceptual statement of the volumic proportions of tones. When we come to 'harmonic compatibility' (40, 201) we go beyond what is given in a mere isolated interval as such. Of course, something must be given in an interval that prompts its musical dissolution; but that may not be grounded in the interval merely as interval or mass, but in that interval as one of a special set, whereby each interval of the set has acquired special relations in virtue of being in that set, not in its own virtue as a mass or interval. That is confirmed by the word Kemp used as a lead in the instruction of his observers: "judge of the finahty of the chord." It is absurd to suggest that a chord is more or less closed off or final, all by itself ; not-being-closed-ofE points away to its completion, points through the paths of memory. V] THE FUETHEE STUDY OF TONAL MASSES 107 "Two tone masses of high fusion give the impression of greater fusion than three tone masses of high fusion (e.g. eg and ceg) ; similarly two tone masses of low fusion appear better fused than three tone masses of low fusion (e.g. ch and cgh); but two tone masses of quite low fusion appear less fused than three tone masses of high fusion" (e.g. cb and ceg) (40, 206). These rules follow directly from my theory of fusion as approximation to the balanced mass of a single tone. Other things being equal, a greater number of tones means less fusion; and yet some three tone masses can be much better balanced than some badly balanced two tone masses. It is also obvious that one badly fiising interval must make a whole tonal mass of low fusion (40, 209). If the quaUtative differences of intervals disappear largely in three tone masses, these becoming much more similar {ibid.), that means that in three tone masses the essence of fusion — its approach to the unity and balance of the pure tone — becomes more apparent for most observers than the defining volumic points. But these are still quite obvious to those observers who have a very highly trained ear; they can recognise them at once apart from their fusional degree altogether. Kiilpe's law (46, 298), according to which, if a three tone mass con- sists of intervals of equal fusion (e.g. ceg and c^g) the greater fusion of the lowest lying interval (ce) determines the greater fusion of the whole, is deducible from the fact that the usual region of observation of a tonal mass is its centre, i.e. its fundamental component (cf. above, p. 65). Parts near this centre will then be more effective on the whole, than more outlying parts. This law was verified by Pear (89, 59, 87) and Kemp (40, 207f.). Stumpfs third law of fusion (112, 136) maintains that, by the addition of a third or further tone, the fusional degree of two given tones is in no way affected ; although the greater number of tones makes analysis more difficult. Kemp's examination of this law confirms it (40, 235) in so far as the two tone mass is abstracted from the three tone mass more or less completely. The abstraction is hardly possible, as we should expect, when the third tone Ues between the two which are the object of abstraction. This fact confirms my classification of pitch as order; for if pitches were quahties, their mere resemblance would be no sufficient reason for any such hindrance to abstraction, especially as two such tonal quahties may be of closer 'resemblance' than either of them is to one which Ues between them. Moreover the abstraction, again as we should expect, is easy when the third pitch lies far to one side of the other two. It is better when the third tone 108 THE FUETHER STUDY OF TONAL MASSES [oh. lies below than when it lies above them (40, 236). There is no reason to suppose that we cannot confine our attention to one part of a tonal mass. The volumic balance of that part must of course be to some extent adversely affected by any third tone, especially if any of the defining points of the latter enter into its main centre. But in so far as the main centre of comparison remains relatively free and retains at least approximately the volumic outline it would have if the third tone were absent, we may well speak of an equality of fusion of the two tone mass and of that mass abstracted out of the three tone mass. But it must be clear that this abstraction is not the same sort of process as the suppression of the illusion in the Miiller-Lyer figure by a special attitude of attention. When the process of abstraction does not succeed, the fusion of the three tone mass will, of coiirse, be judged on the unity and balance of the whole mass, not on that of a part. Then Kiilpe's rules (46, 294) are said to hold: when to one interval others of lower or higher fusional degree are added, the impression of fusion (i.e. the total fusion) is lowered or raised. Thus for example when e is added to eg, two lower fusions — the thirds — are added to a high fusion — the fifth. When c is added to eg, the fifth and third should improve the minor third eg. When g is added to ce, the fifth should heighten, and the minor third lower, its fusion. These compari- sons and results all seem justifiable on the basis of volumes, provided that in speaking of the fusion of the three tone mass we do not consider the number of predominances as detracting from the total fusion more than the relation of the parts in the whole creates a new balance or fusion. There is, perhaps, in this kind of balance a shght change from the balance that approximates to the perfect symmetry of a pure tone and that almost conceals the two predominances ; it is a balance in spite of predominances^. That is so, however, only in so far as we consider the presence of distinguished predominances inconsistent with the notion of fusion. Stumpf does not think it is. He says: "what I call fusion can in itself be perceptible only when the fusing tones are distinguished from one another" (121, 33o). If that is so, I fail like others to see why fusion cannot be present amidst three pitches as well as amidst two. Surely the sensory stuff of three or more tones can fuse more or less, as well as the sensory stuff of two. Stumpf's attitude to this (and to some other aspects of fusion) seems * Cf. Stumpf (114, 290 f.): "Try it; we will only find that increase of unoleamess that is produced by the addition of each new simultaneous tone, and that spreads itself equally over all the tones involved." V] THE FURTHEK STUDY OF TONAL MASSES 109 more obstinate than reasonable, so long as he offers no theory of the real basis of fusion and of the real connexion it estabhshes between the stuff of tones, without prejudice to the independence of their pitches. Much can be saiifor Stumpf's view, as the preceding paragraph shows. The question then is not one in which any once stated view can be rigidly maintained against all others and forced through. We must allow a Uttle here, a Httle there, and a great deal according to the attitude of observation taken towards a tonal mass. That is surely one of the things most obviously required in the theoretical treatment of music (cf. 121, 328). We need by no means hesitate to admit that even fusions can apparently be affected by the momentary attitude of apprehension, in so far as this leads the attention to take greater or less note of a well balanced or ill balanced region of a tonal mass. The theory of the volumic coincidences and proportions of tones thus seems able to provide an adequate basis for all the chief phenomena of fusion, whether that is taken with reference to the whole of a tonal mass or to any special part of it, in so far as any part can exist in the whole without being seriously affected by the third component and can be considered separately. The doctrines of Stumpf and Kiilpe are therefore supplementary, not contradictory, and are both compatible with this theory of the basis of fusion. Thus far we have considered fusion and interval only within the limits of the octave. About the conditions beyond the octave there is difference of opinion and imcertainty. Stumpf (112, I39f.) asserts as a law of fusion that beyond the octave the same degrees of fusion return ; the ninths have the same fusion as the seconds, the tenths as the thirds, the double and triple octave as the octave. One must not be misled, he says, by the greater ease of analysis. In Table III (above, p. 68) I have shown that in the double octave the lower defining point of the upper volume always falls exactly at the point where within the octave the predominance of the upper tone fell. For the observing (musical) mind that has already standardised intervals, this coincidence shotild be enough to establish a very close connexion between an interval and its extension beyond the octave. But it is clear that the approxima- tion to unity of a tonal mass cannot be the same in the two cases. To some extent the reduction of the upper volume will mean a greater unity, for more of the whole lower tone is free of irregularity; but it is clear that the predominance of the upper tone will stand out more clearly, being farther from the other, while the upper volume does not reach to the lower predominance. Apart from the relations between no THE FURTHER STUDY OF TONAL MASSES [ch. intervals within and beyond the octave just referred to, the connexion between the two is just the one case where one might safely appeal to memory (or even to the relationships established by partial tones). For intervals, as we have shown, are standardised throughout the musical range and learnt as individuals. It is to the relations established by memory and by habitual attitudes towards a famihar system of tones and their more familiar relations to one another that we must refer the basis of harmony, a basis that is certainly the result of a long analytic process, now become exphcit in our music. This basis, that of the fundamental triads ceg^ and ce'^g, Stumpf finds so different from consonance, which rests upon fusion, that he invented a new name for it — concordance. But he seems, wrong in relating the divergence of consonance and concordance to a supposed divergence between two tone masses and three tone masses. There is no such divergence, if we maintain the same attitude. As long as we retain our interest in the approximation of a tonal mass to the balance of the pure tone, so long can we talk of its fusion, whether it include within itself two or three or more pitches. But when we change our attitude and consider the capacity of a three tone mass to point through our memory to a definite system of tones (forming a scale or key), then we have left considerations of fusion out of account. At the same time, however, we have not thereby suppressed the existential basis of fusion. Both attitudes can be combined or brought to compromise and both undoubtedly do afEect many of the tonal combinations we admit at any time. Moreover, it may hardly be possible for two tones to define any tonal system of ours, whether the attitude be turned upon it or not. But that is no extra reason why we should make a gulf between two tone masses and three tone masses in the matter of fusion. Stumpf, with his sharp distinction between consonance and concordance, seems to have pointed to a true cleavage of interests in the tonal basis of music, but he seems at the same time to have confused the issue and to have wrongly referred it to the difference between two tone masses and three tone masses in the matter of their fusion, whereas the issue really refers to the difEerence between that approximation to balance in a tonal mass (whether of two or three or more tones) which is called fusion, and that relation of the tones or intervals in a tonal mass which makes them capable of defining for us a large system of tones and tonal relationships and so of giving a specific set to our musical anticipations for a time. This latter relation is probably the more important for any explicit musical V] THE FURTHER STUDY OF TONAL MASSES 111 consciousness, but the former — ^fusion — undoubtedly maintains its force in a rather — shall we say, subconscious, afEective, way ; powerful and decisive in its effect upon the pleasantness of music, but yet less explicitly before the analytic eye of the musician than the systematic relations of tones (which, as we shall see, have developed out of these more primitive relations). Of. 121. Table VII. Stumpf (112, 139) 0" 6» 42 etc. Ellis (30, 191) — III 5" Paist (19, 104) 0" 5" IIP Kemp (ifiirf.) 52 0« III* Faist's scholars (19, 108) 5» 0" IIP Witasek (64, 191) 5a 0' IIP Meinong (final, 64, 198) 5« 0» IIP „ (preliminary, 64, 193) 0^ 5a IIP Probable order 5" 0=^ IIP Table VIII. Meinong' s final series (64, 198) Volumio proportions Mean variation of all four Mean variation of three 52 3 : 1—1—1 i 02 4:2—1—1 1 0-4 III^ 5:1—2—2 li 0-4 VP 10:4^3—3 2i 0-4 32 12:2—5—5 3 1-3 6= 16:6—5—5 4 0-4 42 8:2—3—3 2 04 fp2 45:13—16—16 11 1-3 7» 32:1^—9-9 8 2-2 VIP 15:7—4—4 3i 1-3 IP 9:1 1 1 4i 1-3 2^ 32:2—15—15 8 5-7 ■ In Table VII I have grouped together the observations of various writers on the highest grades of fusion beyond the octave. I have 112 THE FURTHEE STUDY OF TONAL MASSES [ch. neglected the relation of these steps to the first grades within the octave. The most important fact to notice is the number of times that the twelfth (the fifth over the octave) is placed first and before the double octave. Meinong's preliminary determination must, of course, yield to his final result, got by better method. The change seen in Meinong weakens the strength of the case for Faist's observation. Thus we may oppose the majority of these observers to the law stated by Stumpf that the series of fusions beyond the octave simply repeats the series within the octave. Now consider the volumic proportions shown in Table VIII. There the first figure in the second column gives the proportion from the lower limiting order of the lower (the whole) volume to its predominance (its middle point); the other three figures represent proportionately and respectively the stretch of volume separating the mid-predominance (the pitch of the lower tone) from the lower Hmiting order of the upper tone, the stretch from this latter point to the predominance of the upper tone, and then the other half of the upper tone. Is it not remarkable that the latter three stretches in the fifth should be all equal? It seems to me that that might fairly be read as a kind of regularity and balance, which as such exceeds the regularity shown by the double octave : 4 : 2 — 1 — 1. I refer to the rest of the Table for what it is worth. As we concluded regarding the fusions within the octave, it is not easy to establish conceptually a series of degrees of balance which wiU clearly be parallel to those estabHshed by the ear, even if we accept Meinong's series as generally valid. But it would perhaps appear probable that Stumpfs law of the identity of fusions within and beyond the octave rests upon the musical relationships of the corresponding tones rather than upon their fusion strictly. These relationships would be estabUshed through the medium of the absolute- nesses of interval. At any rate, if my theoretical determinations regarding the volumic basis of fusions be accepted in general and in particular, that would seem to be the most probable explanation of the sweeping decisiveness of Stumpfs statement. At least, we may think so imtil Stumpf can show an acceptable full theoretical basis for the facts and his laws of fusion. It is doubtless very difficult to apply to the intervals beyond the octave during their observation exactly the same concept of fusion as to the much less easily analysed intervals within the octave, except in the case of the fifth and octave. Meinong's series clearly approxi- mates otherwise to Stumpfs law, except for the place of the fourth. V] THE FURTHER STUDY OF TONAL MASSES 113 Meinong may also have been led chiefly by considerations derived from musical knowledge. But with these indications we may leave the matter for further advancement by observation (114, I.). CHAPTER VI MELODY XLIX. Melody is one of the most characteristic features of music, and the study of it follows naturally upon that of fusion and of interval. It might even be urged that the study of melody should come first, since it appears first in the development of music and its apprehension is earlier and easier in the individual. A common answer to this obj ection is that the tuning of the tones of the scales even of primitive music involves a reference to consonance, as scales are formed with the help of the chief fusions. In spite of the motive for this answer that Kes in the prevailing inability to find any such basis for the consonances of successive tones as is found for those of simultaneous tones in the fusion of tonal masses, it seems likely that the plea is a vahd one. Scales are no doubt largely moulded by the chief consonances. But the consonance of fusion need not be their only source. As we have seen the consonance of sequence is by no means lacking and must speedily become CArident to an attentive observer, no matter whence the first call to consonances came. Besides the answer under discussion impUes that some rudiment of a scale precedes all melody. This idea is natural enough; for all our melodies are completely subject to our scale systems. But, as we shall see, the impUcation is by no means necessary. It is good to take our notions from the facts before us ; but, as we have repeatedly urged, in the study of hearing we may make great errors uidess we find a proper method of approaching and analysing the facts. Whatever may be the case in any effort deserving the name of music, it is quite absurd to think that no series of tones of different pitch could be formed without reference to the consonances of tonal masses. All the birds must get on without this help, as probably do young children too. If it be asked in reply how we know that the tonal sequences of birds make melodies for them or for us, we can only answer with another question : how do you know they don't ? Evidently w. p. s. 8 114 MELODY [OH. again we must first seek for some giound on which to build a notion of melody, which we may then seek in the facts of tonal sequences. I need offer no apology for seeking a method in our knowledge of the other senses; tor every appeal thus far made to them has been confirmed. Whatever else they are, melodies are certainly series of tones of different pitch, whereby of course repetitions of the same pitch are not excluded. Pitch we have classified as order, so that we may read the former as the latter and ask what special features accrue to the experiences of the other senses when a series of sensations of different orders is presented. We come then upon motion. Or starting primarily from the other senses and knowing there that motion is a characteristic feature of sensory series in which order varies, we may ask if the psychology of motion as we know it in the other senses can be confirmed in hearing. Hearing itself also directly suggests the connexion between melody and motion. "All melodies," Helmholtz says, "are motions within extremes of pitch. The incorporeal material of tones is much more adapted for following the musician's intention in the most delicate and pliant manner for every species of motion than any corporeal material, however Ught. Graceful rapidity, grave procession, quiet advance, wild leaping, aU these different characters of motion and a thousand others in the most varied combinations and degrees, can be represented by successions of tones^. " Helmholtz had no theoretical reason to see motions in melodies, so that his words are quite sufficient evidence of the motional suggestiveness of melodies, if any such evidence is required. Any one can hear it for himself. Or rather it is so obvious that it cannot be overlooked. Motion is very familiar in the sense of vision and less so in touch and articular sense. Apart from special experimental study it is in these senses readily defined as involving progressive differences in the two ordinal attributes, the systemic and the temporal. It is not so readily diagnosed as a distinct addition to experience over and above the progressive differences it involves. This diagnosis is very difficult in so far as the habitual attitude towards the simpler phenomena of sense is that of analytic cognition, such as we apply to the dynamic study of moving bodies or to the mathematics of 'moving' points. But it is evident to everyone from his own experience that we can notice motion without attending to the points through which the motion 1 29, 397; 30, 250. Cf. E. Gumey (28, 103) : "If one thing is suggested by any other, physical movement is continually suggested by melody." VI] MELODY 115 passes, and without knowing how long the motion takes to pass over a given distance. We do not need to hnow the systemic and temporal orders involved; they need only be there to constitute the motion, and then we can compare the speeds of two motions, simultaneous or successive, in different parts of the field of vision without any sort of analysis of our experiences. The motions simply are different ; that is, they are experiences of one class, of different variety, and are compared as we might compare two shades of a colour. Motions are prominent only in those senses in which the attribute of order varies distinctly, i.e. only in vision, touch, and articular sense (133, 157 ft. ; 134, 26ff. ; 135, 251). L. If with this much of motion in mind, we turn to the sense of hearing for its confirmation, we cannot but feel rather disappointed. We can make a tone ghde up and down continuously over a tonal distance. We can easily see the resemblance between this process and motion ; we call it a gUding, or a motion of tone ; and we can easily trace its speed ; we now do so often in the tone of a motor, as its speed of revolution varies. But this kind of motion plays practically no part in music. Of coiurse its absence may again be due to the systems of intervals adopted in the various musical scales. But it would be surprising if music went out of its way to avoid such an elementary mode of sensory experience as motion — the only other one than distance included in the single system of a single sense. Besides we should then have got no whit nearer to an analysis of melody. If we hope for a visual clue to melody, we must therefore look around in vision again. And for what? Why, for a parallel to a kind of connexion that accrues when a series of sensations of fixed, but discrete, orders is given ; for a motion that comes without any real motion, but only from a succession of orders. And a motion originating thus is famihar to everyone in cinematographic projections. There each picture of the series is shown at rest on the screen for an instant to be followed by the next after an interval in which nothing falls on the screen. As everyone knows the objects shown in the picture appear to move continuously. Certain phenomena of the greatest interest for hearing appear when the rate of succession of the pictures is gradually reduced. Let us simplify the experiment and suppose that instead of an object only a moving point is shown, for no object is required. In the old fashioned 'wheel of life,' the forerunner of the modern cinematograph, a picture 8—2 116 MELODY [CH. was sometimes given of a juggler throwing a ball up into the air and catching it again. Or in modem theatres we sometimes see a picture of an aeroplane arriving; nothing appears at first but a moving spot on the white backgroimd of the screen. Suppose five successive stages of such a picture ; or a number of tiny electric lamps standing at variable distances and capable of being brought to glow at any desired rate of succession in otherwise complete darkness. . Each Httle lamp is placed in a black box open towards the spectator, so that Ught carmot faU from one lamp upon the glass of another, and so by, reflection simulate the continuity of motion (cf. 62, 60; 139, 179). o o o o o o o o o o o o 1 2 3 4 5 6 7 8 9 10 11 12 Kg. 8. At a certain distance from one another and at a certain rate of succession, the illumination of the row of lamps gives the appearance of a single lamp appearing at point 1, and moving continuously onwards, as if a real lamp continuously glowing had been uncovered at point 1 and had actually been moved thence along the line 1, 2, 3, 4, 5, etc. When the rate of succession is lowered or the distance between the lamps increased, the motion first becomes jerky, as if one lamp moved quickly forward a space, then stopped an instant, moved forward again, and so on; with further decrease of rate Uttle gaps appear in this continuity, as if the one lamp had passed behind a series of opaque pencik, perpendicular to the Une of lamps, one between each two. As the distance increases or the rate of succession decreases, the breadth of these imaginary pencils seems to increase, until only a Kttle tremor is seen where each lamp stands, when it glows, as if it were jolted once from left to right. If the rate of succession is slow enough, each lamp glows up where it stands and goes out again, while there seem to be as many lamps as there are positions or lamps in reality. This stage is preceded by one in which each lamp appears quite stiU, but is connected with its neighbour by a most evanescent and unobtrusive experience of motion, a sort of mere 'going-over' or 'passing' from one position to another^. ^ For the special study of this last phenomenon v. 139, 922 ff. esp. 226. For the sake of simplicity of statement I have appended Wertheimer's observations to Marbe's, as if Wertheimer's special observations had been observed with Marbe's row of lamps. VI] MELODY 117 These phenomena aie all to be regarded as motion, the same motion as we see in vision when any object moves before our sight. The only difference in the various rates of succession and distance is the obtrusive- ness of the visual stuff or of the visual sensations which make up the motion. When there appears to be one moving lamp, the sensory stuff of the motion between the positions 1, 2, 3, etc. is as intense, fuU and intrusive as is that corresponding to each position, or that which is evoked directly from the retinal stimulation. As the rate or distance changes, this maximal intensity decreases, until it is so weak as to be no longer mistakable for the appearance of the lamp. The lamp then seems to move behind upright obstructions. These gaps increase till they correspond to the real distances between the positions, but even then there is still a motion to be observed between the positions, but it is borne on sensory stuff of minimal intensity or obtrusiveness and of very short duration (cf. 139, 247ff.). And motion is thus always continuous just as it is in our ordinary acquaintance with it. If a cause is to be sought for the appearance of motion under these very peculiar circumstances of stimulation, it is clear that it must be sought in the physiological connexions that arise between the stimula- tions given by the lamps^ at these rates and distances (cf . 139, 247ff.). A psychological cause seems excluded. We need not follow out the subject along this line of interest. Our problem is to see whether the motional phenomena described can be found in hearing. And in fact we find there what must be held to be a very close parallel as far as it goes. The phenomena of hearing call for special research, which is not nearly so easy to carry out in an experimental form, as in vision. We must be satisfied with establishing a probable case for motional phenomena in hearing, remembering what a weight is added to any mere probabiUty by the large extent of parallehsm between hearing and the other senses already established. The proper parallel to the series of lamps is a series of tones of different pitch, whose distance apart, or interval, and whose rate of succession can be controlled. The intervals we find in our musical scales are all too big to allow any tonal continuity to emerge. As far as I have been able to test the matter, continuity of rise of pitch is indistinguishable from a rise by discrete steps if these steps are very small and rapid enough. But the maximal step is probably considerably smaller than a semitone. When a chromatic passage is played rapidly, it gives a great impression of continuity, although it could never be mistaken for a gliding change of pitch. Its component points of pitch are easily 118 MELODY [OH. audible. The 'chromatic scale' may, of course, in spite of this discrete- ness, contain a less obtrusive gliding continuity that the ear cannot readily detach from the rest for adequate description. As the rate of succession of the steps becomes slower, the impression of continuity decreases while the pauses on each pitch become more prominent. At the rates at which tones follow one another in ordinary melodies, there is no ghde about the pitches at all, as far as they are concerned. Each one is obviously a steady pitch, at least on instruments with fixed tones; in singing, of course, there is probably often a considerable amount of gUding just before and after the intonation of any one tone and a fair amount of very brief and unobtrusive ghding between the tones, in so far as intonation is maintained between them. This gUding is reduced by training to the necessary minimvmi. Nevertheless a melody is still a continuity. It is not a mere series of tones of different pitch, but a series of connected tones. The ear hears at once the punctuation of a melody, as it were, the points where the connexion comes to an end momentarily, and where a new 'phrase' begins. In our national anthem, for example, the first break occurs after the first word 'Bang' is sung; the next tone begins afresh. It hardly forms an interval with the preceding tone and there is little continuity or passage between them. We can, if we will, hear some of this continuity, but any considerable amount of it distorts the melody. But it is probably not entirely absent in such a case. If it is to be suppressed entirely, a longer pause must be made between the two tones. This occurs in our national anthem at the end of the third line, where the fresh beginning is most noticeable. LI. This continuity of melody and the unity created by it are so obvious that melody was chosen as the best example by those who first drew attention to the additions to experience which accrue from the collocation of a number of experiences of the same group (v. 13). Melody was held to be a clear case of a figure-experience, a much more obAHOus case than are the figures of vision, squares, triangles, etc. A melody is recognisable even by those who can recognise neither pitches nor intervals. It can be given high or low in the musical range, and so is obviously something more than the tones that compose it, since these can all be different, while it remains the same. No doubt melody is a figure in this sense, just as a square is; and our ordinal classification of pitch makes the parallel much better than could have been expected by those who first saw the similarity between the two. VI} MELODY 119 But melody is not only a figure in the static sense suggested by squares and triangles; it is a motional figure as well; it is a imity because of the fossage that arises between the tones of each phrase or sentence. In fact it seems highly probable that this motional imity must have been the first melodic unity; the static unity can only have become promiuent when scales of definite intervals were formed by means of which the motions of melody were restrained to certain figures and so made more thoroughly subject to attention and expectation (29, 400; 30, 252). But it must not be supposed that the adoption of scales has suppressed the motional figure ; in fact this seems to be still the essential ingredient in what we collectively call melody. Melodic motion is thus easily distinguishable from the full ghding motion of tones. But, as in the case of vision, it seems necessary to suppose that the two are varieties of one and the same process, the difference between them being one of fullness and obtrusiveness of the sensory stuff which bears the motion in each case. The sensory stuff in melodic motion is so unobtrusive that it is greatly influenced by the direction of attention. One can often 'think' pause into and out of melodic sequences, as we have already seen. The punctuation of a melody which is produced by the conclusion of a phrase with tonic or dominant also rests upon this influence of attention or attitude upon the experience of 'passage.' This influence of the attention is found also in the unobtrusive motions of vision (139, 2i8f.). If two visual presentations are given successively, say an upright line and a horizontal one, which if projected simultaneously would form a figure like an inverted T, the motion restilting between the lines will take various directions according both to objective conditions and to the direction of attention. If the upright line slopes a little to the left, it will seem to fall into the horizontal line to the left ; if it slopes more to the right, it wiU move down to the position of the horizontal line to the right. If the slope of the upright line be made in successive trials more and more from one side to the other, it will depend upon the habit of atten- tion thus set up or upon voltmtary direction of the attention in which direction the upright line shall fall into the horizontal one. In hearing, the influence of attention upon melodic motion extends even to simultaneous tonal masses. If a chord is played, e.g. e, g, c', e', the attention can pick out each component pitch in turn, and so the melody can often be heard into the mass. Thus the above chord can be heard as the beginning of Brahms's song "Ihr wunderschonen Augenblicke." Stumpf calls this sort of analysis "singing by ear" 120 MELODY [ch. (112, 291). If it is practised and observed carefuUy, it may seem to be practically identical with the motion of really successive melodies. Such practice is of course easier with the fairly pure tones of tuning forks or bottles and with masses of only two pitches. The phenomenon then becomes very prominent, much more prominent than it is usually in ordinary melodies. One may surge down to the lower pitch, when the tone seems to swell out towards its pitch, as if it were actually growing louder and gliding at the same time. Accentuation of the upper tone in the movements of attention makes the upper tone seem to swell and glide into place. But one hardly gets the impression of a continuity between the pitches. The gliding is distinct only near the pitches, its extent on the side of the other tone being rather indefinite. A fast rate of oscillation of the attention seems more favourable for the development of the motion. In vision this oscillation of attention in a stable presentation does not produce any such motional phenomena. Several reasons may account for this. The chief of these is undoubtedly the fact of the very subordinate part played by variations of visual intensity. We have reason to beUeve that the extremes of white and black are more intense than the intermediate grays, for a mixture of each with a positive colour shows that the extremes swamp the colour more than do the grays (81, 33). But between the extremes and the mean there is very much less variation of intensity than one might expect, apart altogether from the fact that we ordinarily think of black, not of the medium grays, as the minimum intensity of colourless vision. In sound, on the other hand, volumic outline, over which the attention has to wander to pass from one pitch to another, is essentially a variation of the intensity of the elements of different order which compose it. This reason might lead us to suppose that any semblance of motion derived from movement of the attention in a two pitch mass is really an illusion, due to the successive clearness which the movement of attention gives to the various parts of the whole mass. This may well be so, but the similarity of the changes thus produced to those produced by a gliding tone is important. Another reason which might be advanced, asserts that the attention to any component of a tonal mass intensifies that component, so that a movement of the attention from one component to another would be the same as a shght and alternating increase of the intensity of the components. Thus oscillation of the attention would produce the same melodic phenomena as would be produced by the presentation of VI] MELODY 121 successive tones of the intensity of the intensive increment due to attention. But as already (above, p. 102) indicated, this intensive effect of attention is by no means so certain as is usually allowed. It is admitted to be a specially auditory phenomenon and I suggest that its admission in hearing is due largely, if not solely, to the effect of the qualitative classification of pitch, which excludes any other interpreta- tion of the prominence given by attention to one pitch in a mass over against any other pitch in the mass as a whole. If the volumic theory of tones is admitted, the assumption of greater intensity becomes unnecessary and therefore highly improbable. The whole effect of attention can be got by a displacement of the attention from one part of the whole tonal mass to another or from the whole tonal mass to a part of it. In any case, even if attention in tonal masses does some- what intensify the component attended to, it seems clear that it cannot do so to anything Uke the extent we must suppose upon the qualitative classification. We should thus bring the effects of attention in hearing much more into Une with their effects in other senses. And similar behaviour in aU senses is surely the more probable a friori. In melodic sequences these two explanations would be of similar vahdity. For even if the attention did intensify tones, it would have to intensify successive tones equally, whereby their relative intensities would hardly be affected. The melodic passage between tones would therefore still have to be explained. On the other hand the succession of the tones would play much the same part in calling the attention away from the pitch just heard to the next, and so in making the attention move towards that second pitch over its volumic outline to its predominance. This would give the second tone that apparent extension which, when the attention moved over it in one direction, would appear as a gHding of the second pitch into its place. But melodic motion is found even when successive tones are separated by a sUght pause. This would not, of course, exclude the movement of attention over at least a part of the volume of the second tone to its predominance. But in either case, succession or simultaneity, the latter hypothesis would hardly sufl&ce to explain the apparent passage that is characteristic of melody from the pitch of one tone to the next. It must be admitted, in short, that the whole problem is one calling for subtle observation and experimentation before final judgment can be given. It is not so much a matter of whether hearing offers a parallel to the motion of the other senses or not. That parallel is undoubtedly present. It is only a question as to how far melody contains an unobtrusive passage 122 MELODY [co- phenomenon and how far this can be identified with the passage pheno- menon of vision^. LII. The identification favoured here is supported by certain forms of early music, v. Hombostel claims that the basis of melody has been much affected by the one-sided development of tonality and rhythm. The strong habits thus formed, he says, prevent us from apprehending melody in a really melodic way. We tend now always to think of it in terms of our own tonaUty and harmonic accompani- ments. The music of non-Exiropean people is with few exceptions purely melodic. The study of melody must therefore proceed from this purely melodic music, and not from our harmonic music, as Th. Lipps (57), Weinmann (138), and M. Meyer (73), attempted it. In a melody we have more than tones and intervals ; we hear ' motives' (32). V. Hornbostel points to certain interesting facts in support of his demand. In certain songs of unisonal music the intonation changes in the course of a melody, but in quite regular ways : the pitch rises continuously, the melody changes its niveau so to speak; or one tone remains fixed, but the melodic steps from this tone grow bigger or smaller at certain points of the melody; or changes are made that for us would spoil the melody altogether, e.g. gee for g^c. These melodic devices do seem from our point of view to show the pure motion of pitch, free of all the usual restraints of tonal systems. A delight ' Max Meyer has propounded a very strange and revolutionary theory of melody, which rests upon an alleged (melodic) relationship between tones, which is said to be present in tones of the ratio 2 : 3, but absent entirely in tones of the relation 7 : 11, or 11 : 10. I am quite unable to observe this difference, which Meyer merely asserts without any sort of record of observations from different persons and without adequate indication of what difference is meant. A construction on such a basis does not seem entitled to displace aU previous efforts at theory of melody (v. 73). For a criticism of Meyer's theory compare Th. Lipps (57). Lipps proceeds from a basis similar to that of Meyer, but more mystical, if anything. These theories derive any merit they may have solely from their attempt to explain either the dominance of the octave alone, or consonances in general. They have no factual or logical validity and would never be looked at alongside a really efficient theory of fusion and consonance. For an experimental test of Meyer's views of relationship compare W. D. Bingham, (147, 22) ; "the characteristic feeling of 'relation- ship' was nearly always still present when the interval had been increased or diminished 32 cents (a third of an equally tempered semitone)." 48 cents destroyed it in 74 per cent, of the judgments. Bingham thinks that the irregular results of his experiments upon the preference for the second of two pitches as a close or resting point are due to the emergence, now of one tonality, now of another, in the mind of the observers (p. 36). That seems most likely and confirms the view obvious from comparative considerations, that the study of melody should not be approached from a harmonic basis. Yi} MELODY 123 is taken in the mere -vsariations of movement over greater or smaller distances, by different points through the same distance, or from different niveaux. The identity in variation is maintained by the direction of movement, which as an element of form is capable of providing a certain amount of guidance for attention. No doubt our elaborations of form much exceed this in complexity, but it is a mistake to suppose, as v. Hornbostel perhaps suggests, that our music contains no merely motional melody. It contains much of that but it is always imder the restraint of our complexer forms of tonal systems — scales and tonahty. Changes of niveau, whose amounts are, of course, made definitely conformable to the steps of our scales, occur very frequently in melodies, as in the second and fifth lines of our national anthem, as compared with the first and fourth; intervals are extended in paany of these repetitions, and so on. But whatever the motional foundations of melody may be, they are always subordinated in our music to our static forms of tonality. CHAPTER VII THE FORMATION OF SCALES LIII. The study of scale formation follows properly upon the study of melody. For the early scales were melodic, not harmonic, constructions. That is to say, they were certainly not formed by the analysis of the fundamental harmonies and the spreading of their com- ponent pitches into a series. It is a well known fact that there is hardly a trace of harmony to be found in any music but our own modern European music; and that harmonisation is a notion almost quite foreign to the primitive mind, if not more or less impossible without a reconstruction of their scales. Even polyphony is unknown amongst many tribes and peoples. Besides our own scales have been considerably affected by the development of harmony. The intervals of primitive music are not usually very precise; and the precision that attaches to our scales is undoubtedly a reflex of the harmonic processes which evolved out of polyphony. At the same time we must not overlook the fact that relations similar to those of the harmony of simultaneity are established in 124 THE FOEMATION OF SCALES [oh. melodic sequences. We have already considered this matter and have recognised the existence of a natural basis for similarity of relations, namely the positions of the defining points of the related tones in the series of elemental orders of hearing. The basis of the relations in simultaneous and consecutive tones is identical, then, but we do not therefore expect the relations which emerge to be the same in all respects — even if only because in the one case they hold for simultaneity, in the other for succession. A scale is not a sequence of dissonances, as Krueger aptly remarked. Nevertheless certain, pairs of tones acquire a natural affinity in succession, just as they do in simultaneity. These pairs are the octave and the fifth or fourth. The octave is the basis of standardisation of all scales whatever. No scale is known in which the octave is not the first division, as it were, unless it be some degenerate remnant of an earher scale. A scale or a part of a scale may have passed from one place to another without the transference of any proper understanding of its nature and inven- tions^. Such a suggestion is, of course, no basis upon which to start the work of theoretical construction, but only a hypothesis to accommo- date sUght and improbable renmants. The octave is not the outcome of parallel polyphony alone, as has been often suggested for want of a better explanation. For parallels are possible at any interval: third, sixth, seventh, ninth, or any irra- tional interval. Nor is it likely that women and men should naturally tend to sing exactly at the interval of an octave, unless that interval somehow exerts a strong attraction upon voices that naturally fall in its neighbourhood. Helmholtz's explanation that in the octave we have again a part of what we hear in the fundamental, presupposes that the explanation given for the unity of partials in a pitch-blend is sufficient, which is not the case in Helmholtz's theory. His theory apparently succeeds so beautifully, because it pushes the explanation back into a region into which we can hardly follow it. Our full knowledge of the facts of partial tones was the gift of Helmholtz himself. So the primitive mind that knows nothing of them, could only be supposed to appreciate the greater or less coincidence of series of partials in some sort of 'sub- conscious' manner. And even if we concede the unity of pitch-blends, it is not clear that any of the same vmity should attach to the objectively similar case of the octave parallel, in which the presence of two tones ^ Of. 120, 35. Many apparently primitive instruments are d^enerate forms of earlier vn] THE FORMATION OF SCALES 125 in paiallel shovild ex hypothesi be perfectly obvious. For they are sung loudly and clearly, by different persons standing in different places in different 'timbres,' not always in exact coincidence, etc. Moreover Helmholtz's theory suffers from the very assumption it made, that the tones of all scales coincide with any or many of the members of the series of partials. Of course the imperfections of tuning must be conceded to a theory which rests upon similarities whose basis remains unconscious or subconscious. But if we find numerous scales which, apart from the octave, show to contact with the series of partials at all, it then becomes more than possible that those scales which do cling to the series of partials do so, not directly, but indirectly, and that the universal prevalence of the octave is not due to the coincidence of series of partials. And even of those scales which seem to show a considerable amount of connexion with the series of partials, a number of cases do not really bear this interpretation. They have been identified with our notes which are indirectly related to partials only because the minds of those who observed or studied them for us, could only hear them in the manner of these notes of ours. Thus exotic notes get assimilated to our notes with which they may have no real relationship by derivation or intention. LIV. Parallels, then, explain nothing, but presuppose an explana- tion upon which they rest. That explanation has been given above (pp. 63 ff., 90 ff.). And from it flows not only the nattiral predominance of the octave, but also that of the other almost universal interval — the fifth-fourth. The conjunction of these two seems required in fact, as well as being natural in itself. The fifth is a naturally predominant interval. And if it is given along with an octave, the fourth is thereby given as well. What we have to settle, is not that the fourth should be as prominent an interval as the fifth, but that it is preferred in practice and construction to the fifth, and why. The preference in practice and construction seems indisputable. EUis says : " all Greek music, and hence all modern European, as well as all old and medieval music, is founded on this interval. The Fifth seems to have been rather appreciated as the defect of the Fourth from the Octave, though modem tuners find the Fifth much easier to appreciate than the Fourth" (143, 525). In his translation of Helm- holtz's work EUis speaks of the "predominance of the Fourth, and mere evolution of the Fifth, in Greece, Arabia, India, and Japan" (143, 524). It is true that the Fourth seems to be used more in tuning. 126 THE FORMATION OF SCALES [oh. but it seems to be altered as much as is the Fifth. Thus in 26 cases of more or less 'observed' fourths and 28 cases of similar fifths in Ellis's list (30, 5i4fl.), I find for the average value (in cents, v. p. 131 f., below) and the mean variation, respectively, 499 and 31 as against 690 and 18; the 'just' values of the fourth and fifth are 498 and 702 cents. LV. While we admit the preference of the fourth in tuning, we may, therefore, well doubt the assertion of disparity of age of the two intervals. If the octave precedes and regulates all scale con- struction and the fifth is the only other naturally obvious interval, then fifth and fourth are necessarily given together, although the latter does not originate of its own impulse, so to speak, but only as a consequence of the fifth. We are not called upon to find a special source for the fourth or to evolve the fifth out of the octave and the fourth, but only to show some reasonable ground for the preference of the fourth in practice. That the fourth could be retained as an interval with ease in spite of its lack of an impulse towards self-realisation, so to speak, is quite clear from our previous study of interval. Any interval can be retained with precision, provided it is somehow given often enough. And the fourth is given repeatedly, as we have just seen. Octave and fifth can be found with precision quite apart from any power to remember them as intervals with precision. The latter power, of course, does exist, and will naturally join itself to the spontaneous impulse of these intervals towards self-revelation. The use of the fourth on the basis of the memory of its volumic proportions only presupposes sufficient repetition of the fifth and attention to its consequence the fourth. Helmholtz says: "the relationship of the fifth, and its inversion the fourth, to the fundamental tone, is so close that it has been acknowledged in aU known systems of music. On the other hand, many variations occur in the choice of the intermediate tones which have to be inserted between the terminal tones of the tetrachord" (29, 405; 30, 255). If we abstract from this statement its connexion with Helmholtz's theo- retical basis in the series of partials, we can take it over as correct. LVI. Its preferabiUty in practice may have had several groimds. ElHs says: "the quartering of the string may have had much to do with its adoption." If a string is touched at half its length, it yields the octave ; if at a quarter of its length, it yields the fourth of the tone vii] THE FORMATION OF SCALES 127 ^v«ii by the open string. Quartering is a much easier division than is tripartition which would yield the fifth; and in the descent from ootave to fourth of the open tone, the successive fifth is given, as well as in the ascent from the open tone to the fifth above. In playing, it is more natural to proceed from the open tone to higher tones by shortening the string from the open tone than to proceed to lower tones by lengthening the string from the haH-division of the octave. Moreover, if the octave has been divided from one end into a fifth and a fourth, it is a fairly obvious idea to divide it so that the intervals are reversed, when a third interval — the whole tone (9 : 8) is obtained : o—f—g — c'. That is after all but a natural development of the first inevitable triad of intervals — ^the octave, fifth, and fourth. The intervals thus obtained, in their serial order from either end, would then be fourth, tone, fourth, whereby the fourth would acquire a greater prominence. The actual course of development in any particular case would depend very much upon the instrument in use. A stringed instrument might well lead to the construction of another fourth upon the first (from /) and so to the appearance of the whole tone as a defect from the octave. These are problems, however, for ethnological research. It is our interest here only to establish the natural basis upon which scales must develop. Much has surely been due to the caprice of tonal architecture, as Helmholtz said (29, 369 ; 30, 234H.). But not all of the scale can have had that origin. Caprice is only possible within the Umits left over by the natural procUvities that are grounded in the psychological nature of hearing itself^. LVII. These three intervals — octave, fifth, and fourth — then are the foundations of scale formation. Upon them scales are built up in detail by the action of a few principles of development. (a) One of them is very familiar. It is the mere intercomplication of the foundations themselves. A series of alternate fifths up or fourths down from a given tone with the necessary displacement by an octave required to bring them all within one octave wiU give a series of tones which, allowing for the inevitable deficiencies of tuning and intonation, is a very close approach to our own diatonic scale. The many later forms of theory which grew upon this basis of reduplicated fifths or fourths created problems which could not readily occur to those who * If the pure tonal distance between pitches or 'tones' as pitches has been of any effect at all, which is doubtful, the fourth may have derived some advantage from the fact that its pitch divides the octave distance exactly into two equal distances. 128 THE FOEMATION OF SCALES [ch. had not devised a mathematical basis for intervals. The primitive tuner might find considerable difficulty in carrying out his method of tuning to his complete satisfaction, but he would be urged thereby only to devise subsidiary methods of satisfying a desire for system or proportion in his scale construction. Even when theory has won free scope for itself, as in our own case, we find ourselves compelled in many circumstances to adopt a method of attaining system or propor- tion, viz. equal temperament. LVIII. For it is to be noted that the method by which a scale is obtained need not have the slightest relevance to the musical utility of that scale. There can be no question that any derivation of our diatonic scale by a series of fifths or fourths is entirely extraneous to that scale itself as apprehended by anyone in sequence or as the frame- work of a musical composition. In considering the relation of keys to one another the idea of a series of fifths may be a helpful guide. And the key of the fifth is an easy sequence to a given key, because the dominant is symmetrical to the tonic and comes to be closely associated with it, so that the change of key is favoured. But all that is of no significance for the scale actually adopted. Nor is it of any significance for the scale of jtist temperament that it is actually derivable from the series of partials. For the series of partials can act as its source during musical activity only under excep- tional circumstances; as e.g. the 'natural' tnimpet (80) and the Swiss Alpine Horn (120, 38). It is not evident whether this origin would directly favour the construction of melodies in our scale. I should doubt it. The eleventh and thirteenth harmonics do not fit in well (80, 134). The series of partials could be for our music only the de facto source from which the exact relations of our scales have been derived. Once obtained, these relations are learnt and are then on an equal footing with any other scale of any other source. Nor does it necessarily follow that the intervals of just temperament will fuse better than do those of other scales, apart, of course, from the natural intervals of the octave and fifth, about which there can be no doubt, as they precede and determine all scales. Of course in har- monic music the intervals of just temperament in combination will produce fewer beats amongst the upper partials than will the intervals of equal or various other temperaments. But harmony is not an essential ingredient of music, as all primitive music shows, and consonance is not primarily based upon coincidence of upper partials. Only in VII] THE FOEMATION OF SCALES 129 the octave can ttere be complete coincidence of partials; and in all other intervals the innocence of the non-coilicident partials must rest upon their mutual fusional relations, so that here again any positive merit of just temperament reduces itself to the primary merit of the consonance of pure tones. In short; in the consideration of scales we must rid ourselves entirely of all older 'regulating' notions regarding the theoretical origin or method of attainment of scales, and pin our attention to the scale itself and to its inherent merits. If we do not do so, we are certain to fall into the grossest misconceptions regarding the nature of all scales other than that which seems to us most 'natural,' and to try to explain all these from that favourite one. LIX. A true notion of the inherent nature of a scale can be got from the theory of interval developed above. That theory so fits the facts as to persuade us to admit the facts straight as they are; and in the welter of conflicting notions about scales now prevalent, that alone is a considerable service. We saw above that an interval is a volumic outline, obtainable at any absolute pitch; it is a volumic figure whose essence is its proportions. Consequently a scale is a series of volumes whose proportions are constant, and from it can be built up a large number of more or less complex volumic figures of constant proportions. In the scale of just temperament, for example, the propor- tion between the volumes of c and d can be symbolised by x (cf. the ratio 8:9); the others in succession by y, z, x, y, x, z (cf . the ratios, 9 : 10, 15 : 16, 8 : 9, 9 : 10, 8:9, 15 : 16). This symhohsation, I may remind the reader, has been justified above on purely psychical grounds ; it is not based upon the physical ratios given. Now in actual practice we bear that series of proportions in mind as the series x, y, z, x, y,x,z; and we can sing or play the scale on any absolute tone, because the volume of that tone and the series of proportions are enough to deter- mine the whole series. Similarly for any other interval, thirds, fourths, etc. LX. In the Greek and the ecclesiastical scales, on the other hand, the series of proportions was made mobile, as it were; it could be begun at any point and continued to an end in a cychcal order (if we admit for the moment that the proportions involved were primarily the same as those of just temperament, which is more than doubtful. We must make allowance for the imperfections of tuning and the w. p. s. 9 130 THE FORMATION OF SCALES [ch. perfecting and remodelling influences of theory. But the present argument holds for any series of intervals). Thus the starting point of a series may be any member of the symboUc series given above, followed by all the others in a cyclical sequence, the several scales differing, not in the cycle of proportions, but in that member of the cycle with which the cycle starts. There is no more inherent psychical difficulty about remembering this psychical series and about beginning it at any point, than there is about remembering the series of digits and using them in addition as we do. For many people the process of rapid adding is very similar to moving along a series of lengths. Thus 7 and 4 immediately call up the idea of 11, because 4 'carries the mind forward' a definite length to 11. The addition of 8 carries it on another familiar length and there 19 is found. I do not, however, mean to imply that the Greeks or any others actually thought of the series of intervals of their scales cycUcaUy. They may well have done so, but it is not necessary that they should, any more than it is necessary for every adder to think of numbers as lengths. He may add by mere mechanical association, so to speak. So the Greek may have had his scales in mind as a number of series of proportions, of which he knew otherwise that they could all be got from one instrument. Even that knowledge may not have been present to many, any more than those who can sing a number of modem melodies need be aware of the connexions and differences of their underlying scales. We cannot but go wrong if we try to force our recognised scales upon any or aU of these, or try to see those that are strange through the more familiar. It may even be somewhat wrong to read exactly our scale feelings into objectively identical scales of other communities. For in spite of their objective identity, these scales may have had for their possessors a psychical atmosphere which was largely determined by interaction with their other scales. Or there may have been little or no interaction. It is dangerous to make any assimiptions about the psychical aura of the musical con- structions of any people until at least we know definitely the origin of what we ourselves experience, and can say whether a given aura wotild necessarily follow from a certain tonal construction or not. (b) A second principle of development is familiar m our own practice, though not in general. That is the principle of equal tempera- ment. It is obviously a secondary principle, applicable to proportions after they have already been supplied in forms so approximate to equality vti] THE FOEMATION OF SCALES 131 as to be able to suggest equality as desirable. Our own equal teitipera- ment is a device for tte reduction of a large systfem of keys involving many different absolute pitches to one which involves only a small number of notes, and so makes performances on a single instrument possible in any key. We have so modified the series of proportions which form our scales and the absolute pitches upon which we produce them, that we can obtain these series of proportions no matter upon which one of the absolute pitches or notes we begin. That is to say: a series of proportions, that can be symbolised as x, y, z, x, y, x, z, or (as X and y become identical in equal temperament) as x, x, z, x, x, x, z, can be begvm on any of the black or white digitals of the piano and correspondingly on other instruments. We have here a system of interaction between the scale as it is first produced by repetition of fifths or fourths and the notes given by applying that scale to each of its own notes and to the extra notes obtained by so doing. The effort at systematic completeness induces a modification of the basis of the system in order that the final system may be fully rounded off and exhausted. Complete transposibiUty is thus attained. The chromatic tempered scale represents a s.eries of volumes, each one of which differs from the preceding by one and the same proportion. Thus on the whole of the musical range of tones a definite series of points are determined. Any melody, whether harmonised or not, that rests upon a definite sequence and grouping of these points wiU also rest upon such points and only such points, even when its propor- tions are in each case modified by the proportion between the volume of the tone it was first built upon and the volume of the tone it is now to be built upon. As Stumpf says: every melody can be transferred to any pitch "Hke a figure on a surface of constant curvature" (116, 89). That figure of speech is more applicable than Stumpf could have supposed. We are aU thoroughly practised in such transposition or modification of volumes. This principle of equal proportions is also found in exotic music. The highest attainment in a systematic sense would, of course, be when the intervals of all scales required no other notes than are required by the intervals of any scale. That case is realised in the pentatonic scale of Java and the heptatonic scale of Siam, first established and described for us by Alexander Ellis. The proportions of the pentatonic scale are defined by saying that the ratio of each successive interval is as the fifth root of two to one; the heptatonic requires the seventh root of two. Or in cents (of which there are 100 to each equal semitone, 9—2 132 THE FOEMATION OF SCALES [oh. the whole octave being divided up into 1200 equal— imaginary- intervals) 240 for the pentatonic and 171 for the heptatonic. LXI. The progress of Ellis's discovery of these scales is of con- siderable interest. In his first paper "On the musical scales of various nations" (143, 5io) only the Javese scale was given as one of equal temperament. The average values of the successive intervals observed were, in cents, 228, 256, 244, 232, 240, or in our scale c, d+, /-, g+, 5l7_^ c'. This scale is so unlike any scale similar to our own and the average values of the cents are so nearly equal, that the notion of an equally tempered scale once conceived was not to be rejected. More- over, EUis pointed out that his interpretation was more conformable to the habits of tuning this scale. But in his study of the heptatonic scale EUis failed to seize the true relation in spite of the fact that he actually measured a number of fairly well-timed instruments that might have suggested the idea of equal intervals, had that idea not been otherwise inhibited. Some of the instruments observed were indeed very much out of tune, owing to the loss of the lumps of wax by which they are tuned. In Table IX I have made a list of the measurements in cents of aU the instruments, recorded by ElUs, whose scale was known to be heptatonic and equal or may now be supposed by us to have been so. In the first column is given the name of the instrument and the page in vol. 33 of the Journal of the Society of Arts where the records appear. In Table X EUis's own interpretations of these observations are given. In Table XI will be found the average deviation (with mean variation) of the measured amounts (1) from the heptatonic interval of 171 cents and (2) from Elhs's suggestions. I do not mean to claim that all these instruments were really meant to give the heptatonic scale. But as there can be no doubt that nos. 9 and 10 were meant to give that scale, we can see at once that we might well assume from the measurements that all the others except perhaps nos. 5, 6, and 8 were meant to give it. Soon after the publication of EUis's paper giving aU these figures, except those of instruments 9 and 10, a band of Siamese musicians appeared in London and EUis proceeded to hear them play and to measure their instruments. While doing so, he was informed by the Siamese prince Prisdang that "the intention was to make all the intervals from note to note identicaUy the same" (143, iios). This information gave the key to the problem. In order to test its accuracy, Ellis made such a scale of seven equal intervals by calculation. This VII] THE FORMATION OF SCALES 133 the native musicians pronounced to be good. BUis then played them the scale of the Ranat Ek, no. 6 in Table IX, which was the specifically Siamese instrument measured for his leading paper. The musicians declared it to be out of tune. Thus the existence, both really and intentionally, of a heptatonic scale of equal intervals was assured. It has since been confirmed by Stumpf (116). Table IX. 's Measurements. (143) Cents oaloidated 171 171 171 171 171 171 171 1. Balafong (p. 505) ... 187 169 170 147 183 129 237 „ 2ud octave 180 167 181 189 160 159 — 2. Tar [ibid.) 175 179 158 208 176 166 175 „ 2nd octave 205 173 150 195 221 160 — 3. Balafong {ibid.) 169 181 193 166 185 146 165 4. Patala (506) 176 174 183 174 192 154 193 5. Balafong {ibid.) 114 236 200 137 151 194 164 „ 2nd octave — — — 147 154 208 181 6. Ranat Ek (ibid.) ... 2nd octave 129 201 148 103 231 218 45 258 225 7. Balafong (507) 152 135 245 191 166 149 161 8. (ibid.) ... 148 141 178 — — — — „ 2nd octave 195 94 224 173 110 212 201 9. Eanat Ek (1105) ... 177 219 127 150 149 148 167 10. {ibid.) ... 185 165 160 200 159 178 174 Now, instrument no. 6 is the worst tuned of all, its average error being 59 cents ± 22 or a quarter-tone. We might, then, well assume that all the instruments of the list are really heptatonic. Probably they were. But to make any claim merely from these figures would be rash. We should have to be able to identify the actual instrument with a well-tuned one giving the heptatonic scale, before we could be sure. The identification, that is to say, wotild have to be by outward form and material substance, not by the musical functions of the instrument. 134 THE FORMATION OF SCALES [CB. the Two other scales given by Ellis suggest the heptatonic scale scales set by Rajah R. P. Singh, 1st and 4th (143, 504). Cents observed: 183, 342, 533, 685, 871, 1074, 1230, 174, 350, 477, 697, 908, 1070, 1181. Calculated (assuming 171+ as the equal interval): 171, 343, 514, 686, 857, 1028, 1200. But it would be futile to pursue this similarity as it stands. Con- siderable uncertainty must attach to any results apart from the con- firmation of the most efl&cient native musicians ; and these latter should Table X. Ellis's Interpretations. Table XI. No. 2 200 150 150 200 200 150 150 3 150 200 200 160 200 150 150 4 150 200 200 150 200 150 150 6 100 250 200 150 150 200 150 6 150 100 250 200 100 200 200 7 130 150 250 150 200 150 150 8 200 100 200 200 100 2p0 200 (1) (2) 1 17 ±12 — 2 17±12 19± 7 3 12± 7 15± 5 4 12db 7 21 ±10 5 29 ±13 9± 6 6 59 ±22 35 ±16 7 26 ±16 16±16 8 34±19 12± 7 9 24±12 — 10 12± 6 Average 21 ±14 — be free of all the prejudice created so easily by 'improving' theories, especially those of a mathematical kind. The most we may take out of these comparisons is that probably the heptatonic scale is more widely distributed than EUis supposed. The course of Ellis's discovery is most instructive. It shows on the one hand how far out an interpretation may be that is made by a person who is accustomed to such a definite and special scale as ours is. On the other hand it emphasises the necessity, not only of actual observation of native music as played by native musicians, but also of the comparison of these records with the ideas of the best native musicians about their own music and instruments. These musicians must be such as can tune their own instruments in the very best way viil THE FORMATION OF SCALES 135 known in their community, not merely such as can play on them when tuned. Of course all understanding of the genesis of their own scale may be lost in a community. That is possible. But, so long as the scale is maintained in a fairly distinct form in the music of the com- munity, it is not probable. Theorists who neglect to build solely upon the psychical apprehension of a scale by the native mind, may give us a very fine scheme, capable of great development, but they can give us no guarantee that their theory and interpretation applies to its alleged object. It may have substituted its own theoretical object. Treatises on scales show how their authors have wrestled to force various scales into the scheme of their theory. Remarks from listeners to exotic music show how that music inevitably tends to be interpreted in terms of the famiUar scales and intervals. For a proper apprehension and understanding of exotic music we must put all our own habits out of mind, as far as we can. In theory we can do this perfectly, if we wilU- LXII. How were these scales obtained? All that we need to do in answering this question is to show by what means a set of intervals could be got of so nearly equal proportions as to suggest the construction of a series of thoroughly equal proportions. We start out as before from the octave, fourth, and fifth: c—f—g — c'. Bach of the extreme intervals is much larger than the middle one, so that there is in them room for subdivision. When that is done, and supposing it done perfectly, the larger intervals each being divided into two equal parts, we get a series of intervals in cents : 249 : 249 : 204 : 249 : 249. Four of these are the same, and that sameness could be appreciated. Of course, we must not suppose that a native tuner would get these intervals in such perfect division, but only approximately. But in so doing he would be liable to get an appreciation of their approximate equality. The middle interval being often a Uttle out of tune and sometimes approximating more than usual to the size of the others, would only favour the idea of equahty throughout. It remains then only for the tuner to conceive this idea of equaUty and to carry it out. And that can be done, without the use of logarithms, or abstruse physical knowr ledge, by ear alone, just as easily as we can sing up our diatonic scale from any pitch. Once the idea of equal intervals is established, the scale can be got at any time by tuning with fourths and by taking 1 For a statement of the demands put upon the ethnological study of music with special emphasis upon the use of the phonograph for recording observations, see 120, «2 if. 136 THE FORMATION OF SCALES [ch. the fourth from either end^ of the octave smaller than it should be (by 18 cents, less than a fifth of one of our semitones). That is, the fourth up from c is flattened a good deal. Then a fourth may be raised on this first one, and so on, by one or other method. Or the tuner might tune the whole scale by 'ear^,' each interval between two notes being repeated over and over again, and rectified, if need be, by the revision of the whole scale. It would be foolish to attempt to decide by theory on the actual method. That is a question for ethno- logical discovery. As for the heptatonic scale, the primary basis of its origin must be the same — the octave, fifth, and fourth. The interval between / and g is 204 cents. The other intervals are 498 cents. In the sub- diAHsion of the latter we need only suppose for this case that the interval between / and g had been sought in repetition between c and /. This would give three intervals : 204, 204, and 90 ; and a repetition of these between g and c'. Allowing, now, for the imperfections of tuning we can see good groimd upon which the idea of equality of intervals might appear. That equahty would presuppose the reduction of the f—g interval by a smaller amount than its deviation in the pentatonic scale (viz. 33 cents, or a third of our semitone as against 36 cents). Otherwise than by the idea of dividing the big intervals by taking the small one out of it repeatedly, the seven step scale may possibly have been suggested by the notion of perfection associated with that number (116, 90). Possibly it is merely to be laid to the account of a habit of tum'ng the fourth rather too large (by some 16 cents). In any case there is no improbability in its appearance amongst primitive people. For its psychological status as a case of equal temperament is part of the very essence of interval in any form — constancy of pro- portion. The occurrence of equal temperament is not confined to our intellectual race, that attained it only as the result of a long sustained and abeady highly elaborated effort to systematise thoroughly our musical constructions. We had begun by adopting a scale of imequal intervals, which has actually shown itself to be capable of higher evolution than the scales based primarily on equality of interval. ^ Of the method of tuning Sttimpf says: "First the octave is tooed and then intervals are taken from the two octave tones a fourth inwards and so on probably by fourths" (116, 96). ' As Hr Isawa said {v. 143, 5S2): a certain Japanese interval was got "not by con- sonance but by a certain melodic intuition." vii] THE FORMATION OF SCALES 137 There is in all this no contradiction with our own doings in the matter of scales. We need postidate no logarithmic inspirations or root extractions. Nor do we need any mysterious 'feelings' for equal intervals. That -feeUng' is given along with the consonance of fusion in any interval. The mistuning of the fourth and fifth required for the pentatonic scale is only 18 cents and for the heptatonic 16 cents ; certainly not a large amount. It is only five vibrations for a tone of 500 vibrations, or about the high contralto tenor c, i.e. (?. English tuners, as EUis found (143, 489), make errors even of eleven cents. LXIII. (c) The third principle of development is equal division of the playing string. "The principle is," says Ellis, "that equal divisions of the difference of two lengths of a string will give nearly equal intervals extending from one to the other" (143, 500). As examples of this Ellis cites "the mode in which the Persian and afterwards Zalzal's 'middle finger' was obtained, by halving the distance between the frets." On the Persian lute the middle finger had nothing to do. So a note was introduced for it between