Class 
 Book- 
 
 SMITHSONIAN. DEPOSIT 
 
A QUANTITATIVE STUDY OF 
 RHYTHM 
 
 THE EFFECT OF VARIATIONS IN INTENSITY, 
 RATE AND DURATION 
 
 HERBERT W^OODROTN^, A.M. 
 
 I<ectnrer in Barrnard College, Columbia University 
 
 SUBMITTED IN PARTIAL FULFILMENT OF THE RE- 
 QUIREMENTS FOR THE DEGREE OF DOCTOR OF 
 PHILOSOPHY IN THE FACULTY OF PHIL- 
 OSOPHY, COLUMBIA UNIVERSITY 
 
 REPRINTED FROM THE ARCHIVES OF PSTCHOLOGX, 
 
 NO. 14 
 
 NEW YORK 
 
 THE SCIENCE PRESS 
 
 1909 
 
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A QUANTITATIVE STUDY OF 
 RHYTHM 
 
 THE EFFECT OF VARIATIONS IN INTENSITY, 
 RATE AND DURATION 
 
 BT 
 
 HERBERT T^OODROA^. A.M. 
 
 Liectnrer in Bajrnard Colleg'e, Columbia University 
 
 SUBMITTED IN PARTIAL FULFILMENT OF THE RE- 
 QUIREMENTS FOR THE DEGREE OF DOCTOR OF 
 PHILOSOPHY IN THE FACULTY OF PHIL- 
 OSOPHY, COLUMBIA UNIVERSITY 
 
 REPRINTED FROM THE ARCHIVES OF PSXCHOLOGX, 
 
 NO. 14 
 
 NEW^ YORK 
 
 THE SCIENCE PRESS 
 
 1909 
 
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 CONTENTS 
 
 I. Historical 5 
 
 II. Apparatus and Procedure 12 
 
 III. Intensity 30 
 
 IV. Rate and Intensity 34 
 
 V. Duration 38 
 
 VI. The Meaning of Rhythmical Grouping 53 
 
 VII. Summary 63 
 
Digitized by the Internet Archive 
 in 2011 with funding from 
 The Library of Congress 
 
 http://www.archive.org/details/quantitativestudOOwood 
 
CHAPTER I 
 
 HiSTOEICAL 
 
 To produce an impression of rhythm, it is necessary to have a 
 series of stimuli. These stimuli may be sounds, as in the ease of 
 poetry and music, muscular contractions, as in dancing and beat- 
 ing time, or lights and electrical shocks, as in some laboratory 
 experiments. The stimuli which give the impression of rhythm, 
 whatever their nature, may vary in intensity, in duration, and 
 in quality, and may be separated by intervals of varying length. 
 A fundamental task of the experimental investigation of rhythm 
 is to investigate the part played by each of these factors. Only 
 after each of them has been studied separately, may we study the 
 effect when two or more of them are simultaneously involved, 
 and when more complicated factors are introduced, as in melody 
 and harmony. 
 
 The aim of the present study is to examine quantitatively the de- 
 pendence of the rhythmical impression on the intensity and dura- 
 tion of the stimuli. Such an investigation is evidently along the same 
 lines as much of the experimental work of Meumann, Bolton, R. 
 McDougall, and others, who have studied the objective conditions 
 of rhythm. It is necessary, therefore, to review the work that 
 has already been done on the perception of rhythm as influenced 
 by variations in the intensity and the duration of the stimuli. 
 
 Meumann^ found that in listening to a series of sounds, some 
 of which were louder than others, there was a strong tendency 
 towards the formation of rhythmical groups. He studied the effect 
 of accented sounds on the intervals preceding and following them. 
 The most general conclusion at which he arrived is that the effect 
 of the more intense sound may be very different according to its 
 position in the rhythmical group. ^ He found that sometimes the 
 interval following the accented sound is overestimated and some- 
 times underestimated, and also that sometimes the interval pre- 
 ceding the accented sound is overestimated and sometimes under- 
 estimated, and in the cases in which he used more than one sub- 
 ject, he gets quite different results under the same objective con- 
 ditions. Meumann states, also, that, with most subjects, the sudden 
 introduction of a loud sound into a series of weaker ones causes 
 an underestimation of the interval preceding, and an overestima- 
 tion of the interval following, the loud sound ;^ but he does not 
 
 ^ Philos. Stud., g, 264-306, 1894. 
 ' Ibid., 9, 303 and 10, 311, 1894. 
 ^ Ibid., g, 276, 1894. 
 
6 A QUANTITATIVE STUDY OF RHYTHM 
 
 say how many observers gave this introspection, what was the 
 introspection of those who did not give it, how many judgments 
 were made by each observer, or how they were instructed. Meu- 
 mann made no investigation of the effect of duration in rhythm. 
 
 Bolton^ presented sets of sounds of different intensities and 
 durations, which recurred always in the same order, and asked 
 the subject to point out where the series was grouped. In this 
 way, he sought to determine what was the most natural order in 
 which the different intensities and durations occurred in the group. 
 These experiments led him to state the following general principle: 
 "In a series of auditory impressions, any regularly recurrent im- 
 pression which is different from the rest, subordinates the other 
 impressions to it, in such a way that they fall together in groups. 
 If the recurrent difference is one of intensity, the strongest im- 
 pression comes first in the group and the weaker ones after. If 
 the recurrent difference is one of duration, the longest impression 
 comes last."^ 
 
 Bolton calls attention, further, to the long interval which ap- 
 peared between the groups, the intervals being objectively equal. 
 The pause seemed to be due to the fact that a long interval gener- 
 ally preceded the accented sound. At the same time some sub- 
 jects, especially 10 and 15, make a short interval after the strongest 
 sound. But in another place, Bolton writes: "The accented long 
 sound frequently appeared more prolonged than the unaccented 
 of the same length: the accent had the effect both to increase the 
 length of sound and of the interval which followed." And con- 
 sulting his table of results,^ we find that his subjects often found 
 the interval preceding the accented sound longer than the others, 
 but more often did not. As regards the effect of duration, most 
 of Bolton's subjects remarked upon the long interval or pause 
 which seemed to follow the long sound, and for this reason it was 
 found difficult to make the close of the group come at any other 
 place. 
 
 Ettlinger^ has criticized Bolton for his tendency to generalize 
 his results on duration, which, being limited to the single case 
 in which one sound is twice the duration of the other, do not permit 
 of much generalization. So far as they go, however, his results 
 indicate that the effect of increasing the length of any regularly 
 recurrent sound is to produce an overestimation of the followdng 
 
 ^ Amer. J. of Psychol., 6, 222, 1894. 
 
 ^ Ibid., p. 232. 
 
 ^ Ibid., p. 228. 
 
 * Ztschr. f. Psychol., 22, 132-133, 1900. 
 
HISTORICAL 7 
 
 interval, while the effect of increasing the intensity is uncertain. 
 To most of Bolton's subjects, the strongest sound seemed longer 
 than the rest, and the long sound frequently seemed accented. 
 
 Schumann^ asked his subjects to compare the second of two 
 intervals enclosed within a series of three sounds with the first 
 interval. He found, in the case of four subjects, that when the 
 third sound was louder than the preceding, the second interval 
 was underestimated as compared with the first. Three of those 
 same subjects were also tested with regard to the effect of a loud 
 sound which was unexpectedly introduced in a series of weaker 
 sounds. In the case of all three, the interval preceding the louder 
 sound was apparently shorter than the other intervals. Two 
 subjects, on the other hand, obtained the opposite result in both 
 experiments, that is, the interval preceding the accent was over- 
 estimated as compared with the other intervals. Schumann ex- 
 plains this apparent contradiction on the ground that the two last 
 mentioned subjects perceived the sounds rhythmically. He made 
 no investigation of the effect of the regular recurrence of a more 
 intense sound every second or every third time in a long series, 
 nor did he study the effect of variation in the durations of sounds 
 on rhythm or on the judgment of intervals. 
 
 McDougalP found that a loud sound introduced into a uniform 
 series of six beats causes a considerable underestimation of the 
 interval following the loud sound, while it less often and less con- 
 siderably lengthens the preceding interval. As regards the over- 
 estimation of the interval preceding the accent, of the four tables^ 
 of results which are presented to prove this, one^ shows an uDder- 
 estimation; another^ shows practically no constant error, but an 
 underestimation rather than an overestimation, while a third^ 
 does not show that accent has any effect on the interval im- 
 mediately preceding, but that a longer interval causes over- 
 estimation of the interval preceding that longer interval. As 
 regards underestimation of the interval following the ac- 
 cented sound, we find one table^ which shows that when the 
 interval following was 20 per cent, shorter than the interval pre- 
 ceding the accent, and 10 per cent, shorter than the remaining 
 intervals, it was judged less than the remaining intervals 26 times, 
 
 ^ Ztschr. /. Psychol., i8, 30-36, 1898. 
 
 ^ Harvard Psychol. Stud., i; Monog. Sup. Psychol. Rev., 4, 309-412, 1903. 
 
 3 Op. cit.. Tables XXVIII, XXIX, XXXI and XXXII. 
 
 4 Table XXIX. 
 s Table XXXI. 
 
 « Table XXXII. 
 7 Table XXIX. 
 
8 A QUANTITATIVE STUDY OF RHYTHM 
 
 but as equal to them 31 times, and greater six times. The table 
 shows, therefore, that the interval following the accented sound, 
 which was 10 per cent, less than any of the other intervals, was 
 judged either equal or greater in 59 per cent, of the cases. This 
 is about as strong evidence of overestimation as the other three 
 tables give of underestimation. Still more doubt is thrown upon 
 McDougall's conclusion in this connection from the fact that, though 
 five subjects are said to have participated in the experiments repre- 
 sented in these four tables, no separate record is kept of these 
 individuals, and even then the total number of judgments on any- 
 one set of intervals is very often not over six. These experiments 
 were made as already stated with series of six sounds, one of which 
 was louder than the others. Other experiments were made by 
 McDougall in which series which might be called rhythmical were 
 presented to the subject, that is, series in which the accented sound 
 recurred regularly every other time or every third time. In these, 
 it is quite evident that the interval preceding the accented sound 
 is overestimated as compared with the other intervals. McDougall 
 moreover determines the magnitude of this relative overestimation 
 of the interval preceding accent in rhythmical groups of both two 
 and three sounds. To accomplish this, he ascertained at what 
 temporal spacing the grouping disappeared, that is, what relative 
 length of the intervals before and after the accented sound was 
 necessary to produce the impression of temporal uniformity in the 
 series. He refers to the point at which temporal uniformity takes 
 the place of rhythmical grouping as the indifference point. Con- 
 cerning this point, he writes: "At a certain definite stage in the 
 process the tendencies toward the two forms of apprehension bal- 
 ance each other, so that with the slightest change in direction of 
 attention the rhythmical figure inverts and reverts to the original 
 form indifferently."^ As regards the effect of duration, McDougall 
 made no investigations corresponding to those on intensity. 
 
 Miner^ investigated the effect of intensity and of duration in 
 visual rhythms, using lights in place of sounds. His subjects mis- 
 took a difference in the duration of the lights for one of intensity. 
 The more intense light regularly recurred every second or every 
 third time. The intervals between the lights were always equal. 
 The subjects were asked to group the lights first, in groups in which 
 the brighter came first, and second, in groups in which the brighter 
 came second. While doing this, they had, further, to judge which 
 interval was the longer, and which the shorter. We have no guaran- 
 
 ^ Op. cit., p. 382. 
 
 * Monog. Sup. Psychol. Rev., 5, No. 4, 1903. 
 
HISTORICAL 9 
 
 tee that the subjects actually held to this forced rhythm while ob- 
 serving the relative length of the intervals. In fact, Miner, him- 
 self, distinguishes two attitudes on the part of these subjects, 
 though all were similarly instructed. These attitudes he calls the 
 rhythmical and non-rhythmical. He reports no introspections, 
 however, on the part of the subjects, on which to base this division 
 of them into rhythmical and non-rhythmical, and it is made 
 merely as an explanation of the disagreement between the results 
 of different individuals. His results do not admit of much gen- 
 eralization. The following statements, however, can be made. 
 The interval which was most often judged the longest, no matter 
 how the sounds were grouped, was that which separated the groups. 
 The interval next most frequently judged the longest was that 
 before the brighter light. Like Meumann, Schumann, and Mc- 
 Dougall, Miner also investigated the effect of only one intense 
 stimulus in a longer series of less intense. He found that twelve 
 out of seventeen subjects judged the interval after the bright flash 
 to be the longer. 
 
 Miner says that the discrepancy between his own results and 
 those of McDougall may be due to the difference in the quality 
 of the stimulus or the length of the intervals. But this would not 
 explain the discrepancy between Meumann's statements and those 
 of McDougall. Of more importance, probably, is the fact that, 
 in Miner's experiments, the subjects did not know at what point 
 in the series to expect the intense stimulus, whereas, in McDougall's, 
 they did. Miner says that the bright flash was brought in at an un- 
 expected time. Moreover, the subjects were told to notice any 
 difference in the appearance of the interval or the light that followed 
 the bright flash, that is, their attention was directed to the interval 
 and stimulus following the more intense stimulus. With such 
 instructions, we do not know whether the interval following the 
 brighter light appeared longer because of the brighter light or be- 
 cause the subject's attention had been specially directed to the 
 interval and light following the bright flash. McDougall's sub- 
 jects, on the other hand, knew when the loudest sound was to ap- 
 pear. He writes: "As a single hearing very commonly produced 
 but a confused impression due to what was reported as a condi- 
 tion of unpreparedness the method adopted was 
 
 to repeat each series before asking for a judgment 
 
 In order to define the direction of attention on the part of the ob- 
 :server, it was made known that the factors to be compared were 
 the durations of the intervals adjacent to the louder sound in re- 
 
10 A QUANTITATIVE STUDY OF RHYTHM 
 
 lation to the remaining intervals of the series."^ This difference 
 in the instructions given the subjects may account for the differ- 
 ence in the results. In other words, the disagreement between 
 the results of Miner and McDougall may be interpreted as merel}^ 
 a piece of evidence of the importance of the direction of attention 
 in the estimation of intervals. The results of both investigations 
 agree in that they find an overestimation of the interval preceding 
 that stimulus which, there is reason to suppose, is the object of 
 the greatest amount of expectant attention, namely, the more 
 intense, when the subject knows when to expect it,^ or the one after 
 the more intense, when they are not informed when to expect the 
 more intense, but have had their attention especially directed, 
 as a result of their instructions, to the stimulus following the more 
 intense. The apparent discrepancy between the results of Miner 
 and McDougall, therefore, goes to confirm Schumann's theory 
 concerning the effect of strain of attention in the estimation of 
 small intervals. 
 
 Whether Miner's results are to be considered as contradicting 
 those of Bolton on the effect of an increase in duration is not very 
 clear. Miner found that his subjects mistook an increase in the 
 duration of a light for an increase in intensity, and finds that this 
 apparent increase in intensity has about the same effect on the 
 rhythmical grouping as an actual objective increase in the intensity 
 of the stimulus. But as we have already pointed out. Miner's 
 experiments do not permit us to generalize concerning this effect. 
 And, in fact, we have no way of being sure which of the large num- 
 ber of varieties of effect he mentions were due to apparent intensity 
 and which to the attempt of the subjects to group the sounds in 
 the way that they were instructed to group them. 
 
 Notwithstanding many apparent contradictions among the in- 
 vestigations I have reviewed, it is yet possible to indicate the general 
 trend of results so far obtained concerning the effect of variations 
 in the duration and intensity of certain stimuli in a series. There 
 
 ^ Monog. Sup. Psychol. Rev., 4, 362-363, 1903. 
 
 ^ G. F.°Arps und O. Klemm, "Der Verlauf der Aufmerksamkeit bei ryth- 
 mischen Reizen." Psychol. Stud., 4, 518-528, 1909. These authors, using the 
 sensitivity to temporal displacements of each of the members of a dactylic 
 group (every 3rd sound of the objective series being accented and the intervals 
 all equal) as a measure of the degree of attention bestowed on each of the 
 members, conclude that the greatest degree of the attention occurs at the 
 accented sound and the least at the second unaccented sound. The greatest 
 change in the level of attention occurs therefore during the interval preceding 
 the accented sound. Apart from this work we have the generally recognized 
 fact that intensity is one of the "objective conditions" of attention. 
 
HISTORICAL 11 
 
 can be no question but that the effect of a more intense stimulus 
 in a series of less intense is different, according as the more intense 
 occurs unexpectedly, or at a time when it is expected, as when 
 the more intense stimulus regularly recurs. The general con- 
 clusion indicated, if I may neglect, for the sake of simplicity, the 
 special conditions of the above-mentioned researches, is that a more 
 intense stimulus, if unexpected, causes a relative underestima- 
 tion of the interval preceding it; if expected (or regularly recurrent), 
 a relative underestimation of the interval following it. 
 
 As regards duration, about all that can be said, neglecting special 
 conditions again, is that in the case of regularly recurrent differ- 
 ences, one investigator has found an overestimation of the interval 
 preceding the longer stimulus, another, indications of an overes- 
 timation of the interval following the longer stimulus. 
 
CHAPTER II 
 
 Apparatus and Procedure 
 
 The following experiments were performed in the psychological 
 laboratory of the University of Michigan, under the guidance of 
 Professor Pillsbury and Dr. Shepard, whom I am glad to thank 
 for their advice and aid. The work was done mostly during the 
 summer vacations of 1907 and 1908. The subjects used, thirteen 
 in all, were, with one exception, advanced students doing original 
 work in experimental psychology. I take this occasion to express 
 to all those who acted as subjects my appreciation of their patience 
 and conscientiousness. 
 
 The first question to be solved, in an investigation of this sort, 
 is how to produce a series of sounds in which it is possible to make 
 accurately measurable variations in the absolute and relative length 
 of the intervals between the sounds, their absolute and relative 
 intensity, their absolute and relative duration, and the relative 
 proportion of sound and silence. I will describe, first, the dispo- 
 sition of apparatus for work in which the effect of intensity was 
 studied and then point out the changes that were necessary for 
 the study of variations in the duration of the sounds and their rate 
 of succession. 
 
 Part of the apparatus used in the production of the series of 
 intensively changeable sounds is shown in the cut, opposite page 13, 
 which represents the operator's room. In addition, there was 
 in the subject's room a telephone, head-rest, and a microscope 
 for reading the amplitude of vibration of the telephone plate. This 
 reading was not used in the final estimate of the intensity of the 
 sound, but merely as a preliminary guide. 
 
 The sound used for producing the rhythm was that made by a 
 telephone receiver, through which passed an alternating current 
 of 60 alternations per second (a branch from the city lighting cir- 
 cuit), which current, in turn, was interrupted 250 times per second 
 by a tuning fork. The object in having the current interrupted 
 by a tuning fork was to get rid of a click which otherwise occurred 
 at the moment of breaking the current. It was found that, if a 
 current which produced an approximately pure tone was used, 
 there was a very slight click, both at the instant of making the 
 circuit and of breaking it, especially, however, at the break. One 
 could not be sure in judging by the ear alone that this click was 
 not an illusion due to contrast with the preceding or following 
 silence; but Professor Pillsbury succeeded in demonstrating its 
 
APPARATUS AND PROCEDURE 
 
 13 
 
 existence objectively through the aid of a string galvanometer 
 set up in circuit with a microphone placed in front of the telephone. 
 I tried many methods of eliminating this click, suggested by officers 
 of the departments of physics and of electrical engineering, but 
 they all failed. Finally, I noticed that in some experiments in 
 which I was using an induction coil in order to get a telephone 
 sound, there was no click at the break and make. As the ordinary 
 inductorium is not absolutely reliable as regards the regularity 
 
 T 
 
 Fig. I 
 
 M = Meumann's machine. 
 
 C1C2 = contacts. 
 
 S1S2 = adjustable shunt resistances. 
 
 RjRj = adjustable resistances. 
 
 WW = wall. 
 
 T = telephone. 
 
 F = 250 fork. 
 
 of vibration and resistance of its interrupter, at the suggestion of 
 Dr. Shepard, a tuning fork with a vibration frequency of 250 per 
 
14 A QUANTITATIVE STUDY OF RHYTHM 
 
 second was substituted. The platinum plate of the fork was pro- 
 vided with a specially devised micrometer screw arrangement, 
 which made it possible accurately to control the closeness of the 
 contact. Needless to say, this contact, as well as all others, were 
 kept bright as new throughout the investigation and frequently 
 renewed. The sound produced in this manner, according to all 
 the subjects who took part in the investigation, was perfectly even 
 and uniform, and was absolutely free from a click at either the 
 beginning or ending. 
 
 For the purpose of making and interrupting the sounds, a Meu- 
 mann's time-sense apparatus' was used, on which were arranged 
 ordinarily from 2 to 6 contacts. Each contact, at the time it was 
 closed, formed part of a separate circuit. By changing the amount 
 of resistance in these separate circuits, the intensity of any sound 
 could be varied independently of that of any other. This arrange- 
 ment is shown in the diagram on the preceding page. 
 
 Whenever any measurements of intensity were made, all the 
 contacts and all circuits were arranged as in actual use. This was 
 necessary inasmuch as part of each circuit constituted a shunt for 
 the other circuits. 
 
 As a measure of the intensity of the sounds, the distance to which 
 they were just audible was taken. It is not claimed that such a 
 method of measurement is very exact, but on the whole it was con- 
 sidered the most satisfactory. The measurements were all made 
 on the same day, Sunday, during the sum.mer vacation, about two 
 weeks before the opening of college, in Ann Arbor, in an open space 
 on the campus left by the removal of an old building. Only a few 
 series of measurements were taken for each sound, as Sunday was 
 the only day it was quiet enough to work, and it was found impossible 
 in the course of about two months to get more than one good Sun- 
 day, that is, one which was free from wind or rain, or the noise 
 of birds, crickets, etc. I obtained as many series of measurements 
 as was possible in one day, working from early in the morning until 
 after dark, with Mr. Dockeray, at the time assistant in psychology 
 in the University of Michigan laboratory. Environmental con- 
 ditions were constant throughout the day. The method of minimal 
 changes was used, and from 2 to 6 series were obtained with each 
 intensity. At each step, the sound was presented three times in 
 succession, each time for about 1 second. The operator gave 
 the subject a signal by waving a handkerchief a short but variable 
 time before the first sound, and the subject indicated his judgment 
 by raising a handkerchief whenever he thought he heard the sounds 
 
 ^ Phil. Stud., 12, 142-152, i8g6 
 
APPARATUS AND PROCEDURE 15 
 
 and keeping it lowered the rest of the time. The intervals between 
 the three successive presentations of the sounds were made quite 
 irregular. If the subject signaled at such times as to indicate that 
 he heard all three sounds or two out of three, he was considered 
 to have judged "sound audible;" if he signaled correctly for only- 
 one of the three sounds, his judgment was called "doubtful;" if 
 he got none right, he was marked as if he had judged "sound inaudi- 
 ble." The method used was therefore really a combination of the 
 method of minimal changes and the method of constant stimuli. 
 The following measurements were made of the sounds used in 
 this investigation, the distances being given in feet: 
 
 Distance audible 
 
 No. of series 
 
 M. v. (absolute 
 
 24 
 
 6 
 
 I .2 
 
 28 
 
 2 
 
 I.O 
 
 30 
 
 2 
 
 0.8 
 
 32 
 
 4 
 
 2.6 
 
 40 
 
 2 
 
 2.0 
 
 70 
 
 3 
 
 1.2 
 
 136 
 
 3 
 
 4-3 
 
 196 
 
 2 
 
 6.0 
 
 300 
 
 4 
 
 24.6 
 
 420 
 
 4 
 
 17-3 
 
 616 
 
 2 
 
 35-0 
 
 800 
 
 4 
 
 56.0 
 
 1 100 
 
 2 
 
 50.0 
 
 In the remainder of this work, whenever the intensity of a sound 
 is indicated, what is meant is the distance to which it was audible. 
 
 The rate of rotation of the Meumann's apparatus, and so the 
 rate of succession of the sounds, was controlled by a Helmholtz 
 motor and two ball-bearing speed-reducers. The rate of rotation 
 of the Helmholtz motor was kept as constant as possible through 
 the aid of a speed counter and stop watch, and an adjustable re- 
 sistance in the circuit passing through the motor. An accurate 
 record of the rate of rotation as well as the duration of each sound 
 and each interval could be obtained by placing in the same circuit 
 with the telephone a time marker writing on a drum alongside 
 another time marker of 100 single vibrations per second. It was, 
 of course, out of the question to measure individually every one 
 of the sounds and intervals used, as the total number was over 
 half a million. What was done was to take several drums of records 
 of each different rhythm used, both at the beginning and end of 
 the hour, and to keep testing the rate of revolution of the time- 
 sense apparatus by a stop-watch and speed counter during the 
 
16 A QUANTITATIVE STUDY OF RHYTHM 
 
 hour. How great a degree of regularity of speed was obtained 
 is indicated by the mean variation of the following measurements 
 of the times of successive revolutions: 
 
 Duration of i revolution N. M. V. per cent. 
 
 1.50 sees. 50 0.3 
 
 3.00 sees. 56 0.5 
 
 A part of this variation is, of course, due to inaccuracies in esti- 
 mating fractions of a vibration of the fork. I will show later (Chap. 
 IV) that my results would not be changed even by very consid- 
 erable variations in the rate of revolution. Such extremely small 
 mean variations in the rate of revolution as those just indicated 
 have, therefore, no significance for the results of this investigation, 
 and for that reason, in the tables to follow, no mention is made 
 of the mean variation in the rate of revolution of the time-sense 
 apparatus. Changes in the rate were made by the aid of the speed- 
 reducers. A range of rate was obtained from one revolution in 
 0.5 seconds to one revolution in 26.0 seconds. 
 
 When variations in the duration of the sounds were desired, 
 contacts of different length were used. The accurate measure- 
 ment of the duration of the sounds was accomplished in the manner- 
 described above. Finally, it was possible, within certain limits, 
 to arrange for any desired combination of intensity and duration 
 in the sounds composing the rhythmical series. 
 
 Given a series of sounds which may be varied in intensity and 
 duration, the question arises, how are we to investigate the effect 
 of these variations? The question to be solved is, what values, 
 have intensity and duration for the impression of rhythm which 
 is obtained in listening to the series of sounds? If the study is 
 to take on a quantitative aspect, there must be some measure of 
 the magnitude of the rhythmical attribute of the total impression;, 
 there must be some index to show us whether more or less rhythm 
 is felt, some way of telling which of two rhythms, both of which 
 may perhaps be qualitatively alike, in the sense that they are both 
 trochaic or both iambic, is the stronger or more emphatic. The 
 members of a group may be thought of as being held together more 
 or less securely, by stronger or weaker bonds; and what we desire 
 is a measure of the force with which these members of a group are 
 held together — a measure of what McDougall calls "the rhythmic 
 integration of the stimuli." 
 
 A very direct method would be to ask the subject to introspect 
 regarding the relative amount of rhythm produced by two different 
 series of sounds. This task is similar to that required of subjects 
 
APPARATUS AND PROCEDURE 17 
 
 in experiments on the so-called intensity of sensation. The intro- 
 spection in the case of rhythm, however, is more difficult, and I 
 doubt if reliable results can be obtained from an investigation car- 
 ried out in this way. For instance, suppose we compare two trochaic 
 rhythms, one produced by an alternation in intensity and one by 
 an alternation in duration. It may be very apparent that both 
 rhythms are what we call trochaic, but the total impression is very 
 different in the two cases, and it is very hard to isolate the intensive 
 aspect; and consequently it is difficult to say in which case the 
 rhythm is the stronger; and it would be still more difficult to say 
 which is the stronger if one were trochaic and the other iambic. 
 I found that such judgments were not impossible, but I have pre- 
 ferred a more indirect and more objective method, one that de- 
 mands something far less difficult on the part of the subject. 
 
 The following description of the method used in this research 
 may be found rather difficult to follow by those who are unac- 
 customed to the terminology of rhythm; but I believe that, if the 
 reader will take a pencil and paper and follow out by the aid of 
 symbols the procedure below outlined, he will find no ambiguities. 
 It should be remembered that in all the rhythms here dealt with 
 every second or every third sound is either louder or longer than 
 the others. Also that rhythm is characterized by an apprehension 
 of the sounds in groups and that when there is no grouping there 
 is no rhythm. The method consists in taking as the measure of 
 the amount of rhythm the amount by which, measuring from the 
 point at which all intervals are equal, the internal intervals, or 
 intervals within the group, must be increased or decreased with 
 respect to the external intervals, or the intervals between the groups, 
 in order to cause a disappearance of the rhythm, that is, a disap- 
 pearance of apparent grouping. Roughly speaking, I have used 
 as the measure of the amount of rhythm the amount of work that 
 had to be done on the intervals to destroy the rhythm. Suppose, 
 for instance, that a certain series of sounds in which every other 
 one is accented, produces an impression of trochaic rhythm, the 
 accented sound seeming to begin the group. Now, such a rhythm 
 can be changed to an iambic one by increasing the interval fol- 
 lowing the accented sound, with respect to the interval preceding 
 the accented sound. I have found no exception to this possibility. 
 Moreover, as the process of increasing the interval after the accented 
 sound is going on, just before the rhythm becomes iambic, there 
 will be a point reached at which the rhythm can hardly be said 
 to be more iambic than it is trochaic. This point may be called 
 the iambic-trochaic indifference point. It is the point at which 
 
18 A QUANTITATIVE STUDY OF RHYTHM 
 
 the rhythm is destroyed, or at least reduced to a minimum. If, 
 to arrive at this point, it was necessary to produce only a very 
 slight increase in the duration of the interval after the louder 
 sound with respect to the duration of the interval before it, the 
 rhythm may be said to have possessed only a slight degree of tem- 
 poral segregration to begin with; and that slight degree of temporal 
 segregation to have been in the trochaic direction. This means 
 that, to begin with, the temporal grouping of the sounds was such 
 that the sounds within the same group were only slightly 
 more grouped than two successive sounds belonging to different 
 groups. 
 
 Now, if it is desired to determine the effect of intensity or duration 
 on temporal segregation, we must be able to separate the influence 
 toward temporal segregation exerted by intensity and duration 
 from that exerted directly by unequal temporal spacing of the 
 sounds. The temporal segregation we wish to measure is, of course, 
 the apparent, or subjective, temporal segregation — the tem- 
 poral segregation presented by the rhythm consciousness and not 
 that presented by the objective series. If the indifference point 
 occurs where the intervals are objectively equal, then any differ- 
 ences in duration or intensity which may exist in the sound series 
 are evidently exerting no influence towards temporal segregation. 
 To measure the amount of temporal segregation produced by changes 
 in intensity or duration we must determine the amount of change 
 in the intervals, from objective equality, necessary to arrive at the 
 indifference point, or, what amounts to the same thing, take as 
 the measure of the temporal segregation produced by intensity 
 or duration the difference at the rhythm indifference point be- 
 tween the intervals before and after the accented or the longer sound. 
 If, when the intervals are equal and every other sound accented, 
 the rhythm is heard as trochaic, but if this trochaism^ can be de- 
 stroyed by increasing the interval after the accented sound, then 
 it is clear that the accent is exerting some influence toward tem- 
 poral segregation in the trochaic direction. Moreover, if in one 
 case the interval after the accented sound has to be increased 10 
 per cent., starting with intervals equal, and in a second only 1 per 
 cent., in order to arrive at the point of indifference, it is clear that 
 a greater degree of temporal segregation in the trochaic direction 
 exists in the first case than in the second. On the other hand, 
 if, with equal intervals, a series of sounds should produce the im- 
 pression of iambic rhythm, but this impression is changed to one 
 
 ^ This word was suggested to me by Professor Titchener as at least prefer- 
 able to "trochaicness." 
 
APPARATUS AND PROCEDURE 19 
 
 of indifference by increasing by 10 per cent, the interval before 
 the accented sound, then this rhythm could be said to present the 
 same amount of temporal segregation as the series which, with 
 equal intervals, produced the impression of trochaic rhythm, but 
 in which an increase of 10 per cent, in the interval after the accented 
 sound brought the listener to the indifference point. Both rhythms 
 could be said to present the same degree of temporal segregation; 
 but one would be in the iambic direction and the other in the trochaic. 
 The influence towards temporal segregation exerted by any such 
 factor as recurrent differences in accent, pitch, duration, etc., may, 
 therefore, be measured by the difference between the external 
 and internal intervals of the group at the indifference point. In 
 the tables of the following chapters this difference has been re- 
 corded in the columns headed A-B, which means the difference 
 in duration at the rhythm indifference point between the interval 
 after the accented or the longer second (A) and the interval before 
 the accented or longer sound (B). When A-B is positive, this 
 means that the series is heard as trochaic, when negative, as iambic, 
 providing the intervals are objectively equal. Further, the magni- 
 tude of A-B, when positive, is a measure of the degree of temporal 
 segregation in the trochaic sense; when negative, of the degree 
 of temporal segregation in the iambic sense — so far as this segrega- 
 tion or grouping is due to other factors than objective difference 
 of intervals. 
 
 But is measuring the amount of temporal segregation the equiv- 
 alent of measuring the amount of rhythm, the amount of trochaism 
 or iambism? I shall show, in Chapter VI, that rhythmical group- 
 ing is a temporal grouping. Rhythmical segregation implies (sub- 
 jective) temporal segregation. Then "more" or "less" applied 
 to rhythm means more or less temporal segregation; and, in meas- 
 uring the amount of temporal segregation, we obtain an index 
 of the quantity of rhythm. This same conclusion may be reached 
 by a different line of reasoning. The judgments, iambic, trochaic 
 and doubtful, have a certain range of distribution. If we admit 
 that a rhythm which is judged trochaic 95 per cent, of the judg- 
 ments is more trochaic than one which is judged trochaic in 60 per 
 cent, of the judgments, then we must also admit that our measure 
 of temporal segregation serves also as a measure of more or less 
 rhythm: because as the degree of temporal segregation in the trochaic 
 direction exerted by an alternation of more and less intense sounds 
 increases (because of increase in the ratio between the intensities) 
 the percentage of judgments "trochaic" also increases, the objective 
 intervals remaining equal. In other words, to destroy a trochaic 
 
20 A QUANTITATIVE STUDY OF RHYTHM 
 
 group which is strongly enough trochaic to be judged trochaic in 
 95 per cent, of the trials, a larger increase in the interval after the 
 emphasized sound is necessary than that required to destroy a 
 trochaic group which is only strong enough to be judged trochaic 
 in 60 per cent, of the trials. We may conclude, then, that the 
 magnitude of A-B, when positive, may be taken as a measure of 
 the degree of trochaism, when negative, of the degree of iambism, 
 in so far as this trochaism or iambism is due to other factors than 
 objective differences in the intervals between the stimuli. 
 
 No use has as yet ever been made of this rhythm indifference 
 point. McDougall, as I have already explained, made use of the 
 indifference point in the estimation of time intervals. The ques- 
 tion asked of his subjects "was invariably as to the apparent rela- 
 tive duration of the two intervals,"^ and the indifference point at 
 which he arrived represents "the quantitative proportion of the 
 two durations necessary to produce the impression of temporal 
 uniformity in the series."^ The subjects who took part in the 
 present investigation, however, were instructed, except in certain 
 cases mentioned later, first, to judge whether the sounds produced 
 an impression which they would speak of as rhythmical, and second, 
 in case there was rhythm, to indicate as best they could the nature 
 of the rhythm. In their judgment as to the quality of the rhythm, 
 they usually made use of the terms iambic, trochaic, etc.; but as 
 full an introspection as the subject was able to make was taken on 
 every rhythm to which he listened. Nothing was said to the subjects 
 about duration of intervals or temporal uniformity. The indiffer- 
 ence point here means a point at which the rhythm is no more one 
 type than another. The iambic-trochaic indifference point is the 
 point at which the impression is no more trochaic than iambic, but 
 at which a slight increase in the interval following the accented 
 sound causes the impression to become one of iambic rhythm, and a 
 slight decrease in the same interval changes the impression to that 
 of trochaic rhythm. That such an indifference point is also the 
 indifference point for the perception of the time intervals is by no 
 means self-evident, and the question can be settled only by experi- 
 ment. I shall show^ that the two indifference points correspond 
 very closely indeed, but are not quite identical. 
 
 I have defined the indifference point as regards rhythm as that 
 point at which the impression is no more that of one rhythm than of 
 another, ^. e., the point at which one rhythm is just as natural and 
 
 * Monog. Sup. Psychol. Rev., 4, 379, 1903. 
 ^ Ibid., p. 378. 
 3 Chap. VI. 
 
APPARATUS AND PROCEDURE 21 
 
 just as easy as another, instead of defining it as the point where no 
 rhythm at all exists, because rhythm may occur even at the indiffer- 
 ence point. As Professor Woodworth has pointed out, the same 
 series may be heard in different rhythms.^ The following intro- 
 spections from subjects who have had a great deal of experience 
 with rhythms near the indifference point are very definite. Subject 
 Ws. writes: "At the very point where iambic turns trochaic, there 
 seems to be no rhythm at all. The reason I put down 'doubtful' 
 is because I can determine no rhythm. If I could determine rhythm, 
 I could tell whether it was iambic or trochaic. At the turning point, 
 there is just a series of sounds, one louder than the other. It seems 
 to me that at one point there is formed a continuous, even, un- 
 divided series, and no rhythm exists. At this point, you can make 
 the rhythm either iambic or trochaic, in your mind." Subject Ww. 
 says: "My general conclusion concerning the indifference point is 
 that I can get any one of three things — no rhythm at all, iambic, 
 or trochaic — any of the three cases may occur depending upon the 
 way I attend to the sounds— the way I listen to them." Subject 
 Br, who always counted when he obtained the rhythmical effect, 
 except in cases where he purposely avoided it, gives the following 
 introspection: "At the time when the notes and intervals were all 
 equal, the idea of number, that is, the impulse to count one, two, 
 dropped out. An idea of mere succession remained." These 
 introspections seem sufficient to establish the fact, that, at the in- 
 difference point, rhythm may entirely disappear, but also that 
 either one of two rhythms may be obtained with about equal facility. 
 Notwithstanding the fact that the impression of rhythm produced 
 by a series of sounds depends to a large extent upon subjective 
 factors, it is none the less true that it depends largely on the nature 
 of the sound series, and it is by no means impossible to study the 
 relation between the series of stimuli and the ensuing impression. 
 When the series of sounds is not near the indifference point, the 
 rhythm perceived by any one subject as the result of any given 
 series of stimuli is practically always the same. And if the subjects 
 are instructed in all cases to indicate which rhythm is the most 
 natural or the easiest, it will be found that the indifference point is 
 really a quite narrow 7one, though of course a variable quantity. 
 I instructed the subjects in this manner; and throughout the follow- 
 ing work, whenever it is indicated that a certain series of sounds 
 produced a certain impression of rhythm, it is not meant that any 
 other rhythm was absolutely impossible, but that the subject found 
 the rhythm indicated to be the most natural. To indicate the 
 ' /. of Phil. Psych, and Sci. Meth., 4, 17, 1907. 
 
22 A QUANTITATIVE STUDY OF RHYTHM 
 
 manner in which the instructions were followed I cite the following 
 introspection from Ws: "Each time, before determining whether 
 the rhythm is iambic or trochaic, I try it each way. When the 
 loudest and longest note comes first, I put down trochaic; but when 
 it comes at the end of each rhythm, I put down iambic. When 
 I can not distinguish any definite rhythm or tell which comes first, 
 the loud or the soft tone, I put down uncertain. It is not like an 
 impression which comes immediately but which comes after listening 
 a while. I do not notice the length of intervals especially." Sub- 
 ject Wr writes that "Possibly the judgment iambic or trochaic does 
 not mean always that this is obtained the easiest, using easiest in 
 the sense of least effort, but perhaps it has occasionally meant 
 merely most satisfying." All subjects reported that there was no 
 difficulty in following the instructions. Whenever they were not 
 sure of the rhythm, they indicated the fact. 
 
 The method used to arrive at the indifference point was the method 
 of minimal changes. The subject was first presented a series of 
 sounds concerning the rhythmical nature of which there was no 
 doubt, a series concerning which he could judge "plainly iambic," 
 or, "plainly trochaic," etc. The time allowed for this judgment 
 in the early part of the investigation was 45 seconds, i. e., the rhythm 
 was allowed to run along unchanged for 45 seconds. In the greater 
 part of the work, however, the rhythm was continued until the sub- 
 ject made up his mind, v^^hich event was indicated to the operator 
 by an electric bell signal. In case the subject had not made up 
 his mind, however, at the end of one minute, the rhythm was stopped 
 anyway, and the subject, in these cases, wrote "doubtful." If the 
 judgment "doubtful" occurred on the first rhythm of the series, 
 this judgment was thrown out and the series commenced at some 
 other point where the judgment was that the rhythm was plainly 
 one thing or another. The theory of the method of minimal changes 
 seems to require that we take as our starting-point some point 
 where the judgment is not in doubt. The judgment "doubtful" 
 on the first member of the series occurred very seldom, not over 
 twenty times in the whole investigation; and most of these cases 
 occurred in the presentation of the first series or two of a new rhythm, 
 before the operator had any definite idea as to where the indifference 
 point was located, or, consequently, where best to start the series. 
 The very few remaining judgments of doubtful on the first member 
 of the series occurred in cases where the series was started nearer 
 than usual to the indifference point, either in the hope of hurrying 
 the progress of the investigation or of preventing fatigue on the part 
 of the subject. 
 
APPARATUS AND PROCEDURE 23 
 
 In case the first judgment was "trochaic," the interval following 
 the more intense or the longer sound was slightly lengthened, and 
 that preceding shortened by the same amount. This second rhythm 
 was then given for 45 seconds, or until the subject had made up his 
 mind. This procedure was continued until the subject judged 
 "plainly iambic." This whole series was then repeated many times 
 either in the same or reverse direction. In case the first judgment 
 was "iambic," then naturally the succeeding rhythms of that series 
 were produced by shortening the interval following the louder or 
 longer sound and lengthening the interval following the weaker 
 or shorter sound. Similarly, to pass from dactylic rhythm to ana- 
 paestic, the interval following the emphasized sound was increased 
 while the others were decreased. The number of rhythms, i. e., 
 the number of steps, in any one series, varied from four to fifteen, 
 but was usually between five and nine. The subjects knew that 
 the change from one rhythm to another was effected by changing 
 the intervals, and that the direction of this change was such that 
 the rhythm would finally go over to something else. They were 
 ignorant of the starting point and the size of the steps. They were 
 informed that the size of the steps might vary considerably or might 
 not vary at all. As a matter of fact, their size varied in different 
 series from .02 to .06 second, in the case of rhythmical measures 
 the total duration of which was 1.5 seconds, and correspondingly 
 for longer or shorter groups. This means that in rhythmical groups 
 the total duration of which was 1.5 seconds the same interval in 
 any two successively presented rhythms of the same series would 
 vary only by from .01 to .03 second, inasmuch as one interval was 
 lengthened as the other was shortened. The most usual variation 
 between any two consecutive steps was ,016 second. Very often 
 the subject could notice absolutely no difference in two successive 
 rhythms. I believe that, under the conditions of this investigation, 
 the size of the steps was usually about what would correspond to a 
 just noticeable difference. For any one series of minimal changes, 
 the steps were the same throughout. 
 
 It is quite conceivable that under some conditions the judgment 
 concerning rhythm might be strongly influenced by which sound 
 was heard first. There exists, e. g., a tendency, under certain 
 conditions, for persons to hear a series of sounds as trochaic in case 
 the louder note first reaches the ear, though the same series is judged 
 iambic in case the weaker note is the first to reach the ear, or, as. 
 I believe may be said with greater truth, the first to receive attention. 
 There can be no doubt of this tendency. I remember that, at the 
 beginning of this investigation, I was rather surprised to hear a. 
 
24 
 
 A QUANTITATIVE STUDY OF RHYTHM 
 
 subject judge a rhythm to be trochaic which I had judged to be 
 plainly iambic; but, on listening a second time to the same series, 
 I, too, got the impression of trochaic rhythm; and I soon found that 
 which I got depended on which note I heard first. I think it probable 
 that this effect is due to the fact that a certain direction of attention 
 is prescribed by the way the rhythm starts out, inasmuch as this 
 same shifting of the rhythm may occur within one and the same 
 series by shifting the direction of attention, at least according to 
 the introspection of all the subjects I have questioned on this point. 
 The instructions given to my subjects prevented this influence, of 
 the way the rhythm starts out, from affecting my results. Inasmuch 
 as the subjects were to judge which rhythm was the more natural, 
 in those cases in which there could possibly be any doubt concerning 
 the rhythm, and, with some subjects, in every case (as introspec- 
 tions quoted above show), the subjects tried both rhythms to see 
 which was the more natural. Hence it would not matter which 
 way the rhythm started. I have proved that this is the case by 
 keeping throughout quite a period a record of which sound began 
 the rhythmical series. The following table proves that it did not 
 matter for the purpose of this investigation which sound began 
 the series. The table represents the readings in degrees, on Meu- 
 mann's machine, of the movable contact, at the rhythm indifference 
 point, first, using series in which the longer sound was given first 
 in each case, and second, using series in which the rhythms were 
 always begun on the shorter sound. The temporal value of 1 degree 
 during this work was .00833 second. N refers to the number of 
 series of minimal changes. 
 
 Subject 
 
 I<onger sound 
 given 1st. 
 
 N. 
 
 Shorter sound 
 given I St. 
 
 N. 
 
 Ws 
 
 102.5 
 
 10 
 
 102.5 
 
 II 
 
 
 106.8 
 
 9 
 
 106.3 
 
 II 
 
 
 1 10. 8 
 
 9 
 
 112. I 
 
 16 
 
 
 122.9 
 
 15 
 
 121. 3 
 
 14 
 
 
 III. 9 
 
 4 
 
 III. 4 
 
 7 
 
 
 84.6 
 
 17 
 
 85.0 
 
 16 
 
 
 101.5 
 
 17 
 
 102.0 
 
 II 
 
 
 97.1 
 
 6 
 
 95-8 
 
 5 
 
 
 97.0 
 
 5 
 
 96.0 
 
 4 
 
 
 73-3 
 
 14 
 
 75-0 
 
 14 
 
 
 III. 3 
 
 14 
 
 III. 7 
 
 13 
 
 
 107 -5 
 
 14 
 
 107 -5 
 
 14 
 
 Py 
 
 105.0 
 
 II 
 
 1 14.0 
 
 16 
 
 
 113. 8 
 
 14 
 
 103.4 
 
 15 
 
 
 102.0 
 
 6 
 
 102.0 
 
 6 
 
 Wr 
 
 102.5 
 
 10 
 
 102.5 
 
 II 
 
 
 lOI.O 
 
 8 
 
 103.4 
 
 7 
 
 
 86.0 
 
 16 
 
 83.1 
 
 6 
 
APPARATUS AND PROCEDURE 25 
 
 Notwithstanding that the above results show that, under the con- 
 ditions of this research, the results were not affected in any very 
 appreciable degree by the question as to which sound came first, 
 nevertheless the precaution was taken throughout to begin about 
 equally often with each note, both in the case of two-membered and 
 three-membered rhythms. 
 
 While the method of minimal changes has been used almost 
 exclusively in this research, I have also occasionally used a method 
 of constant stimuli, and cite below results on the same point by the 
 two methods. Both methods, of course, have advantages and 
 disadvantages, but for the purpose of this research no other method 
 than that of minimal changes could even be seriously considered, 
 because of the greater time required, and it would have been ab- 
 solutely impossible to get any of the subjects to work another 
 hour longer than they did. As it was, three of the subjects, Ws, 
 Dy, and Ww, served for over 300 hours each. From the experi- 
 ments I performed with the method of constant stimuli, I estimate 
 that 50 per cent, more time would have been required to cover 
 the same ground; and, as far as I can see, the results, even then, 
 would have been less reliable. 
 
 It may be well to point out that, while I have spoken of the 
 method used as that of minimal changes, there was no standard 
 stimulus with which a changeable one was compared, and that 
 consequently the procedure is different from that of the method 
 of minimal changes as ordinarily described. The act required 
 of the subject was merely one of introspective description of a single 
 stimulus, the stimulus consisting of a series of sounds. To a certain 
 extent, the judgment was a matter of identification. Further, it 
 may be well to repeat in this connection that the same objective 
 rhythm continued for about 45 seconds, during which time, of 
 course, a large number, say about 30, separate rhythmical groups 
 were presented. The judgment of the subject occasionally under- 
 went several fluctuations before a final decision was reached. Sev- 
 eral different and more or less opposed judgments might be made 
 during the continuance of the same rhythmical stimulus; but, 
 unless one of these seemed much less forced than the others, such 
 rhythms were judged "doubtful." 
 
 The middle point of this doubtful zone is what is given as the in- 
 difference point. In most series, the doubtful zone was quite small. 
 In many series, it was less than the size of the steps used in changing 
 the rhythm. Thus the judgment "iambic" w'ould be given up to 
 a certain point and then a sudden change on the very next step to 
 "trochaic" would occur. In these cases, the indifference point 
 
26 A QUANTITATIVE STUDY OF RHYTHM 
 
 is considered as lying halfway between the last "iambic" and the 
 first "trochaic" judgment. The treatment of series in which the 
 judgments do not follow a regular course has always been a matter 
 for difference of opinion. Some authors advocate considering all 
 judgments as "doubtful" between those judgments which are 
 alike for three successive judgments, while others advocate neglect 
 of all return to the judgment "doubtful" or to the judgment which 
 began the series after one sure judgment has been made which is 
 different from the sure judgment which began the series. Both 
 procedures are plainly unjustifiable theoretically. The question 
 might possibly be settled on an empirical basis, but never has been. 
 The method I have used for treating such series, while one that has 
 never been rigidly advocated, I believe to be the only one that 
 does not lead to absurdities. The method used consists in shifting 
 the doubtful and reverse judgments all to the middle before taking 
 the average. In part, this procedure follows conventional usage. 
 Thus, the rhythm indifference point in the following series would, 
 by most persons, I presume, be considered as lying between numbers 
 5 and 6, i. e., half way between numbers 3 and 8. 
 
 Place in series 
 I 
 
 Judgment 
 iambic 
 
 2 
 
 iambic 
 
 3 
 
 iambic 
 
 4 
 
 trochaic 
 
 5 
 
 iambic 
 
 6 
 
 trochaic 
 
 7 
 
 iambic 
 
 8 
 
 trochaic 
 
 9 
 
 trochaic 
 
 lO 
 
 trochaic 
 
 We should get the same result if we shifted judgments 4, 5, 6, 
 and 7 in such a way that all the iambic lay together and all the 
 trochaic together, i. e., changed the places of the trochaic judg- 
 ments 4 and 6 with the iambic judgments 5 and 7. For the sake 
 of consistency, then, any series of judgments should be treated in 
 the same manner. The following two cases will serve as illustrations. 
 
 Place in series Judgment 
 
 1 iambic 
 
 2 iambic 
 
 3 iambic 
 
 4 doubtful 
 
 5 iambic 
 
 6 trochaic 
 
 7 trochaic 
 
 8 trochaic 
 
 Indifference point in above series is at 5. 
 
APPARATUS AND PROCEDURE 27 
 
 e in series 
 
 Judgment 
 
 1 
 
 iambic 
 
 2 
 
 iambic 
 
 3 
 
 iambic 
 
 4 
 
 trochaic 
 
 5 
 
 iambic 
 
 6 
 
 trochaic 
 
 7 
 
 trochaic 
 
 8 
 
 doubtful 
 
 9 
 
 trochaic 
 
 lO 
 
 trochaic 
 
 II 
 
 trochaic 
 
 Indifference point in above series is at 5. 
 
 In order to save time and prevent fatigue, I told all subjects 
 to consider the series ended when they had obtained two judgments in 
 succession different from the beginning judgment. As a matter of 
 fact, the results of this research would hardly be noticeably affected 
 by the method used in calculating the indifference point as long as 
 any recognized method was followed. But I consider it advisable 
 to have some one definite rule at the start which gives a reasonable 
 result in any conceivable case. 
 
 I have not considered it worth while to give in the tables which 
 follow the range of the zone of doubtful cases. Nor have I tried 
 to work out the correlation between the size of any hypothetically 
 defined doubtful zone and the mean variation of the indifference 
 point in successive series. Inasmuch, however, as neither the 
 method of minimal changes nor that of constant stimuli has ever 
 before been systematically applied to the direct investigation of 
 rhythm, that is, to investigations in which the judgment rendered 
 by the subject is a designation of the nature of the rhythm he gets, it 
 may be worth while, in order to give some general idea of the dis- 
 tribution of judgments, to cite the following records selected as 
 typical : 
 
 Method of minimal changes. Subject, Py. Rhythmical stimuli 
 consisting of an alternation of two sounds of unequal, but constant, 
 length, and of equal intensity. Longer sound = .09 second, 
 shorter = .06 second. In the column headed "after," is indicated 
 the duration of the interval after the longer sound; in that headed 
 "before," the duration of the interval before the longer sound. 
 
 The table below represents the results for only one series of mini- 
 mal changes. The total number of series of minimal changes used 
 in this investigation was about 24,000. The indifference point in 
 this series is taken as the point at which the interval after the 
 longer sound is .53 second. 
 
28 
 
 A QUANTITATIVE STUDY OF RHYTHM 
 
 
 Duration 
 
 of 
 
 inteiA'als 
 
 before and 
 
 after 
 
 Subject's 
 
 
 Ion 
 
 gcr sound, in 
 
 seconds 
 
 
 judgments 
 
 
 AiUr. 
 
 
 
 
 £jfore. 
 
 
 
 G/ 
 
 
 
 
 
 .3 9 
 
 iambic 
 
 
 .59 
 
 
 
 
 
 .4? 
 
 iambic 
 
 
 .5 7 
 .55 
 
 
 
 
 
 .43 
 .■45- 
 
 iambic 
 
 iambic 
 
 
 .53 
 
 
 
 
 
 .47 
 
 • staccato 
 
 
 .51 
 .49 
 
 
 
 
 
 .4-9 
 .SI 
 
 trochaic 
 
 ' trochaic 
 
 
 .47 
 
 
 
 — 
 
 .53 
 
 trochaic 
 
 The distribution of judgments, as well as the relation between 
 the methods of minimal changes and constant stimuli, is indicated 
 in the following results from subject Ws. The rhj-thmical stimulus 
 consisted of an alternation of a loud with a soft sound. Loud soimd 
 audible to a distance of 136 feet, soft sound audible to a distance of 
 24 feet. Both sounds 0.13 second in duration. Total duration 
 of one measm'e, that is, from the beginning of one loud sound to 
 the beginning of the follomng loud sound = 1.5 seconds. The 
 method of constant stimuli used consisted merely in giving the same 
 objective rhythms as in the method of minimal changes, but in 
 irregular order. 
 
 Method of constant stimuli. 
 
 Intervals before and after 
 louder sound, in seconds 
 
 Before 
 
 Aftei 
 
 
 66 
 
 62 
 
 •57 
 
 .62 
 
 
 59 
 57 
 55 
 
 .64 
 .66 
 .68 
 
 
 53 
 
 .70 
 
 
 49 
 
 •74 
 
 Subject's judgments (percent) 
 
 Trochaic 
 
 Doubtful 
 
 Iambic 
 
 N 
 
 100. 
 
 0.0 
 
 0.0 
 
 26 
 
 85.7 
 
 14-3 
 
 0.0 
 
 42 
 
 71-4 
 
 2S.6 
 
 0.0 
 
 56 
 
 31-3 
 
 59-4 
 
 9-3 
 
 64 
 
 0.0 
 
 16.0 
 
 S4.0 
 
 50 
 
 0.0 
 
 5-5 
 
 94-5 
 
 54 
 
 0.0 
 
 0.0 
 
 100. 
 
 36 
 
 Judging from the above table, the rhythm indifference point 
 is about at the point where the interval before the louder sound is 
 .57 second and that follo-^ing it .66 second. It ■v\-ill be observed, 
 also, that a change by one step from this point is sufficient to make 
 the judgment in over 70 per cent, of the cases either iambic or 
 trochaic depending upon the direction of the change. Moreover, 
 in this case, as in others with the same and other subjects, the results 
 obtained by this method agree very closeh' with those obtained 
 by the method of minimal changes. Thus, in the present instance, 
 
APPARATUS AND PROCEDURE 29 
 
 the indifference point for the same rhythmical series of sounds 
 on the same morning, by the same subject, was found, as the result 
 of ten series of minimal changes, to be at that setting which gives 
 as the interval before the louder sound .57 second (absolute M. V. = 
 .01 second) and that following it .66 second (absolute M. V. = .01 
 second) — the same result as with the method of constant stimuli. 
 
CHAPTER III 
 
 Intensity 
 
 The method used for the investigation of the effect of intensity 
 in rh3^thm consists in finding the indifference point for rhythm with 
 different ratios of intensity between the sounds. The method of 
 determining the indifference point and the method of measuring 
 the intensity of the sounds have been described in Chapter II. 
 Rhythms in which every alternate sound is objectively accented 
 were the ones given most attention. At the rate at which these 
 were usually given, one sound every .75 second, all the subjects 
 invariably got a two-group. Enough work was done on three- 
 membered groups, however, to show that the same general laws 
 hold in both cases. 
 
 In the experiments represented in table I., the duration of all 
 the sounds was kept constant and was .13 second, while the total 
 duration of one measure, that is, the time from the beginning of 
 one louder sound to the beginning of the following louder sound, 
 was also kept constant at 1.5 seconds. There were just two different 
 intensities of sounds used in any one rhythmical series, every second 
 sound being louder; and all the louder were of one intensity, and all 
 the weaker of the other intensity. The intensity of the sounds 
 is indicated by the distance, in feet, to which they were just audible. 
 For all the tables presented in this chapter, the weaker sound was 
 just audible at 24 feet. In the column headed A-B, is indicated 
 the duration, at the indifference point for rhythm, of the interval 
 following the louder sound minus the duration of the interval preced- 
 ing the louder sound. By "before" is meant the interval of silence 
 before the louder sound, i. e., the time from the end of the preceding 
 sound to the beginning of the louder sound; by " after" is meant the 
 interval of silence following the louder sound, ^. e., the time from 
 the end of the louder sound to the beginning of the following sound. 
 The quantity A-B, therefore, is obtained by subtracting the cor- 
 responding quantity in the column "before" from that in the column 
 "after." N refers to the number of series of minimal changes run 
 through in the determination of any one indifference point. The 
 mean variation (MV) is the mean variation in the duration of the 
 intervals at the indifference point. All the results obtained with 
 any one ratio of intensities are treated as one group in obtaining 
 the mean variation. Inasmuch as one interval was lengthened as 
 the other was shortened, and vice versa, the absolute M. V. of one in- 
 terval is also that of the other; and for that reason, I have given 
 
INTENSITY 
 
 31 
 
 the absolute M. V. rather than the relative. By "louder sound/' 
 is meant the distance to which the louder sound was just audible. 
 All durations are given in seconds. 
 
 Subject 
 Wr 
 
 Dy 
 
 Ww 
 
 Ws 
 
 TAB 
 
 tive Intensity on the 
 
 LE I 
 Iambic-trochaic 
 
 Indifference 
 
 Point. 
 
 ttd just audible at 24 feet. 
 
 
 
 
 der sound Before After 
 
 N 
 
 My 
 
 A — B 
 
 30 
 
 613 
 
 617 
 
 10 
 
 .008 
 
 .004 
 
 32 
 
 603 
 
 627 
 
 9 
 
 .005 
 
 .024 
 
 40 
 
 576 
 
 654 
 
 9 
 
 .008 
 
 .078 
 
 70 
 
 567 
 
 663 
 
 31 
 
 .025 
 
 .096 
 
 136 
 
 555 
 
 675 
 
 10 
 
 .010 
 
 .120 
 
 196 
 
 548 
 
 682 
 
 15 
 
 .016 
 
 •134 
 
 300 
 
 547 
 
 683 
 
 10 
 
 .012 
 
 .136 
 
 420 
 
 536 
 
 694 
 
 25 
 
 .024 
 
 .158 
 
 616 
 
 513 
 
 717 
 
 II 
 
 .013 
 
 .204 
 
 IIOO 
 
 509 
 
 721 
 
 17 
 
 .016 
 
 .212 
 
 28 
 
 592 
 
 638 
 
 6 
 
 .012 
 
 .046 
 
 30 
 
 573 
 
 657 
 
 II 
 
 .012 
 
 .084 
 
 32 
 
 564 
 
 666 
 
 II 
 
 .014 
 
 .102 
 
 40 
 
 532 
 
 698 
 
 18 
 
 .020 
 
 .166 
 
 70 
 
 532 
 
 698 
 
 33 
 
 .023 
 
 .166 
 
 136 
 
 506 
 
 724 
 
 15 
 
 .016 
 
 .218 
 
 196 
 
 503 
 
 727 
 
 23 
 
 .024 
 
 .224 
 
 300 
 
 499 
 
 731 
 
 28 
 
 .023 
 
 •232 
 
 420 
 
 503 
 
 727 
 
 10 
 
 .014 
 
 .224 
 
 616 
 
 484 
 
 746 
 
 22 
 
 .oig 
 
 .262 
 
 800 
 
 456 
 
 774 
 
 15 
 
 .022 
 
 .318 
 
 IIOO 
 
 457 
 
 773 
 
 34 
 
 .024 
 
 .316 
 
 28 
 
 594 
 
 636 
 
 22 
 
 .022 
 
 .042 
 
 30 
 
 581 
 
 648 
 
 28 
 
 .016 
 
 .068 
 
 32 
 
 573 
 
 657 
 
 20 
 
 .oig 
 
 .084 
 
 40 
 
 568 
 
 662 
 
 19 
 
 .026 
 
 .094 
 
 70 
 
 558 
 
 672 
 
 23 
 
 .024 
 
 .114 
 
 136 
 
 554 
 
 676 
 
 25 
 
 .015 
 
 .122 
 
 196 
 
 546 
 
 684 
 
 21 
 
 .022 
 
 .138 
 
 300 
 
 546 
 
 684 
 
 21 
 
 .020 
 
 .138 
 
 616 
 
 534 
 
 696 
 
 21 
 
 .028 
 
 .162 
 
 IIOO 
 
 534 
 
 696 
 
 20 
 
 .016 
 
 .162 
 
 28 ■ 
 
 614 
 
 616 
 
 20 
 
 .022 
 
 .002 
 
 30 
 
 600 
 
 630 
 
 20 
 
 .oog 
 
 .030 
 
 32 
 
 590 
 
 640 
 
 20 
 
 .012 
 
 .050 
 
 40 
 
 582 
 
 648 
 
 38 
 
 .019 
 
 .066 
 
 70 
 
 577 
 
 653 
 
 20 
 
 .021 
 
 .076 
 
 136 
 
 568 
 
 662 
 
 58 
 
 ■033 
 
 .094 
 
 196 
 
 568 
 
 662 
 
 56 
 
 .025 
 
 .094 
 
 300 
 
 566 
 
 664 
 
 41 
 
 .021 
 
 .098 
 
 420 
 
 559 
 
 671 
 
 44 
 
 .028 
 
 .112 
 
 616 
 
 558 
 
 672 
 
 46 
 
 .028 
 
 .114 
 
 800 
 
 563 
 
 667 
 
 35 
 
 .oig 
 
 .104 
 
 IIOO 
 
 550 
 
 680 
 
 38 
 
 .018 
 
 .130 
 
32 A QUANTITATIVE STUDY OF RHYTHM 
 
 From the preceding table, it is obvious that with equal intervals 
 and every second sound accented the rhythm is trochaic, that is, 
 the effect of accent is to cause the accented sound to appe.ar grouped 
 with the following weaker sound. In chapter II it was shown 
 that when the quantity A-B is positive, its magnitude measures 
 the trochaism of the sound series. It is obvious, therefore, from 
 Table I, that the rhythm becomes more and more trochaic as the 
 ratio between the intensities of the louder and weaker sounds in- 
 creases. It is further obvious that the trochaism of the series in- 
 creases at first very rapidly and later very slowly relative to the 
 increase in distance to which the louder sound is just audible, the 
 weaker remaining constant. The same statement can be made 
 concerning the increase in the intensity of a sensation as the energy 
 of the stimulus increases. It is therefore not impossible that the 
 increase in trochaism tends to be proportional to the increase in 
 the ratio of the sensation intensities of the sounds, but not enough 
 is known of the relation between the energy of sound stimuli and 
 the intensity of sensations to make speculation on this point profit- 
 able. 
 
 That the effect of an objective accent in three-membered groups 
 is similar to that in two-membered groups is shown by Table II, 
 which presents results obtained when every third sound was the 
 more intense. The indifference point for these tables is the dactylic- 
 anapaestic, and was obtained in the same way as in the case of 
 groups of two sounds each, namely, by shortening the interval 
 preceding the more intense sound and at the same time lengthening 
 the interval following it, or vice versa. The interval between the 
 two weaker sounds, which were both of the same intensity, (24 feet), 
 was kept constant throughout, and equal to .577 second. As 
 before, the quantities in the column headed A-B represent the 
 duration of the interval following the louder sound minus the dura- 
 tion of the interval preceding it. The duration of all the sounds 
 was .09 second, and the total duration of one measure, that is, 
 the time from the beginning of one louder sound to the beginning 
 of the succeeding louder sound, was 2.0 seconds. 
 
 TABLE II 
 Effect of Relative Intensity on the Anapaestic-dactylic Indifference Point. 
 Interval between the two weaker sounds (constant) = 0.577 seconds. 
 Weaker sound just audible at 24 feet. 
 
 ibject 
 
 Louder sound 
 
 Before 
 
 After 
 
 N 
 
 M V 
 
 A — B 
 
 Dy 
 
 32 
 
 •541 
 
 .613 
 
 ID 
 
 .028 
 
 .072 
 
 
 40 
 
 •525 
 
 .629 
 
 10 
 
 .022 
 
 .104 
 
 
 136 
 
 •517 
 
 •637 
 
 9 
 
 .013 
 
 .120 
 
 
 616 
 
 •479 
 
 .675 
 
 ID 
 
 .020 
 
 .196 
 

 
 INTENSITY 
 
 
 
 
 abject 
 
 I,ouder sound 
 
 Before 
 
 After 
 
 N 
 
 MV 
 
 A — B 
 
 Ww 
 
 32 
 
 •540 
 
 .614 
 
 10 
 
 .016 
 
 .074 
 
 
 40 
 
 ■523 
 
 .631 
 
 8 
 
 .013 
 
 .108 
 
 
 136 
 
 •499 
 
 .655 
 
 12 
 
 .016 
 
 .156 
 
 
 616 
 
 •475 
 
 .679 
 
 15 
 
 .017 
 
 -204 
 
 33 
 
 This same law, namely, that the accented stimulus tends more 
 and more strongly to begin the group with increase in the ratio of 
 intensities of the stimuli, was found to hold good of rhythms in which 
 the stimuli were electrical shocks instead of sounds. The same 
 apparatus was used as in the investigation of sound rhythms except 
 that in place of the telephone, which had produced the sounds, 
 was substituted a pair of sponge electrodes, which were tied to 
 motor points of the arms. Subject Dy gave the introspection that 
 the impression of rhythm was "fully as strong in the case of elec- 
 trical stimulation as in the case of sound." My own introspection 
 is that there is about as much rhythm in one case as the other. 
 Every third stimulus was the stronger and consequently each 
 group consisted of three movement percepts (each shock producing 
 a twitch of the muscles of the arm): and the indifference point 
 obtained was the dactylic-anapaestic. The results for electrical 
 stimulation are presented in Table III. The total duration of one 
 measure is 2.0 seconds, and the duration of each stimulus equals 
 .09 second. The interval between the two weaker shocks, which 
 were both equal in intensity, was kept constant throughout and was 
 equal to .577 second. 
 
 TABLE III 
 
 Effect of Relative Intensity on Anapaestic-dactylic Indifference Point when 
 
 Electric Shocks Are Used as Stimuli. 
 
 Interval between the two weakest shocks (constant) = 0.577 second. 
 Subject stronger stimulus Before After N MV A — B 
 
 Dy weak 
 
 medium 
 very great 
 
 Ww weak 
 
 medium 
 very great 
 
 566 
 553 
 513 
 
 567 
 561 
 533 
 
 588 6 .022 .022 
 
 601 8 .017 .048 
 
 641 8 .027 .128 
 
 587 6 .014 .020 
 
 593 4 -016 .032 
 
 621 6 .018 .088 
 
 In the work on sound, all subjects frequently remarked on the 
 greater apparent duration of the louder sound. The difference both 
 in loudness and in duration seemed greater when the rhythm was 
 pronounced than when near the indifference point. 
 
CHAPTER IV 
 
 Rate and Intensity 
 
 It was desired to study here the effect on the rhythmical impres- 
 sion of changes in the rapidity with which the series was run off. 
 These changes are those produced by varying the rate of rotation 
 of the Meumann's time-sense apparatus. The absolute duration of 
 the measure and of all its parts, intervals and sounds, was varied, 
 while all relative durations were kept constant. One method of 
 changing the rate of a series of sounds is to change only the intervals 
 of silence between the sounds. Such a procedure introduces changes 
 in a factor other than rate, namely, the proportion between the 
 duration of the sounds and the duration of the intervals. In the 
 present experiments, the absolute duration of the sounds was varied 
 in direct proportion to the absolute duration of the intervals between 
 the sounds, so that the proportion of sound to silence and the pro- 
 portion of either sound or silence to the whole measure was kept 
 constant. The intensities of the sounds were also kept constant. 
 
 Tables IV to VI show the effect of variation in the rate of rhythmi- 
 cal series in which every second sound is the louder. The subjects 
 perceived the sounds in groups of two except for the measures whose 
 total duration is indicated as .75 second or .5 second. At these 
 last mentioned rates, the sounds were grouped by four, so that as 
 a matter of fact at these two rates the total duration of the groups 
 was twice that indicated in the tables as the duration of the measure; 
 but the time from the beginning of one of the loud sounds to the 
 beginning of the following is indicated as the total duration of the 
 measure, at these rates as in the other cases, for the sake of readier 
 comparison. In the following tables are presented the duration 
 of the intervals at the indifference point. In the column headed 
 "measure" is indicated the duration of the cycle from the beginning, 
 say, of one loud sound to the beginning of the following loud sound. 
 The expressions, N, M V, A-B, before and after, have the same 
 meaning as in the previous chapter. The quantity A-B, however, 
 which represents the amount by which the interval after the accented 
 sound is longer than the interval before it, at the indifference point 
 for rhythm, is expressed in the following tables as a percentage 
 of the duration, at the indifference point, of the interval before 
 the accented sound. All durations are given in seconds. 
 
RATE AND INTENSITY 35 
 
 TABLE IV 
 
 The Effect of Rate on the Iambic-trochaic Indifference Point. 
 
 Louder sound just audible at 70 feet; weaker, at 24 feet. 
 Duration of every sound relative to the total duration of one measure = 
 13 to 150. 
 
 ubject 
 
 Measure 
 
 Before 
 
 After 
 
 N 
 
 M V 
 
 A — B 
 
 Wr 
 
 I.O 
 
 •38 
 
 •44 
 
 13 
 
 .008 
 
 15.8 
 
 
 1-5 
 
 •57 
 
 .66 
 
 14 
 
 .014 
 
 15-8 
 
 
 2.0 
 
 •77 
 
 .87 
 
 14 
 
 .007 
 
 13.0 
 
 
 30 
 
 1. 14 
 
 1.32 
 
 19 
 
 ■054 
 
 15.8 
 
 
 4.0 
 
 1-52 
 
 1.76 
 
 14 
 
 .027 
 
 15.8 
 
 
 5-5 
 
 2. 12 
 
 2-39 
 
 14 
 
 ■055 
 
 12.7 
 
 
 7-5 
 
 3-05 
 
 3-II 
 
 17 
 
 .100 
 
 2.0 
 
 Ws 
 
 1.0 
 
 •39 
 
 •44 
 
 II 
 
 .006 
 
 12.8 
 
 
 1-5 
 
 •58 
 
 • 65 
 
 II 
 
 .011 
 
 12. 1 
 
 
 2.0 
 
 ■78 
 
 .87 
 
 ID 
 
 .016 
 
 II-5 
 
 
 ■ 2.5 
 
 •97 
 
 1 .09 
 
 8 
 
 .026 
 
 12.4 
 
 Dy 
 
 0.5 
 
 .18 
 
 •23 
 
 18 
 
 .007 
 
 27.8 
 
 
 0.75 
 
 .29 
 
 •34 
 
 19 
 
 .008 
 
 17.2 
 
 
 1-5 
 
 •57 
 
 .66 
 
 II 
 
 .021 
 
 15.8 
 
 
 2.0 
 
 •77 
 
 •87 
 
 15 
 
 .015 
 
 13-0 
 
 
 3-0 
 
 1^15 
 
 131 
 
 10 
 
 .032 
 
 13.9 
 
 
 4.0 
 
 1^55 
 
 1^73 
 
 10 
 
 .040 
 
 11.6 
 
 
 5-5 
 
 2.24 
 
 2.27 
 
 9 
 
 .051 
 
 1.3 
 
 
 7.0 
 
 3.08 
 
 3.08 
 
 8 
 
 .og6 
 
 0.0 
 
 TABLE V 
 
 The Effect of Rate on the Iambic-trochaic Indifference Point. 
 
 Subject, Ww. 
 
 Louder sound just audible at 136 feet; weaker, at 24 feet. 
 Duration of every sound relative to the total duration of one measure 
 13 to 150. 
 
 Measure 
 
 Before 
 
 After 
 
 N 
 
 M V 
 
 A — B 
 
 0.5 
 
 
 16 
 
 •25 
 
 12 
 
 .005 
 
 56.3 
 
 0.75 
 
 
 26 
 
 •36 
 
 ID 
 
 .010 
 
 38.5 
 
 1 .0 
 
 
 37 
 
 .46 
 
 ID 
 
 .006 
 
 24-3 
 
 1^5 
 
 
 53 
 
 .70 
 
 8 
 
 .oop 
 
 32.1 
 
 2.0 
 
 
 •73 
 
 •91 
 
 II 
 
 .018 
 
 24.7 
 
 2.5 
 
 
 90 
 
 1. 16 
 
 ID 
 
 .022 
 
 28.9 
 
 3^o 
 
 I 
 
 06 
 
 1 .40 
 
 12 
 
 .oog 
 
 32.1 
 
 3-5 
 
 I 
 
 •29 
 
 1.58 
 
 10 
 
 .021 
 
 22.5 
 
 5-0 
 
 I 
 
 •91 
 
 2. 19 
 
 14 
 
 .041 
 
 14.7 
 
 7-5 
 
 2 
 
 •97 
 
 3^i8 
 
 6 
 
 .104 
 
 7-1 
 
 10. 
 
 4 
 
 03 
 
 4.17 
 
 7 
 
 .083 
 
 3.5 
 
36 A QUANTITATIVE STUDY OF RHYTHM 
 
 TABLE VI 
 
 The Effect of Rate on the Iambic-trochaic Indifference Point. 
 Subject, Dy. 
 
 Louder sound just audible at 136 feet; weaker, at 24 feet. 
 Duration of louder sound relative to the total duration of one measure 
 
 37 to 150; of weaker = 13 to 150. 
 
 Measure 
 
 Before 
 
 After 
 
 N 
 
 M V 
 
 A— B 
 
 1 .0 
 
 .29 
 
 •38 
 
 16 
 
 .016 
 
 31.0 
 
 1-5 
 
 
 44 
 
 •56 
 
 16 
 
 .013 
 
 27.3 
 
 2.0 
 
 
 59 
 
 •71 
 
 16 
 
 .017 
 
 20.3 
 
 2.5 
 
 
 75 , 
 
 .92 
 
 18 
 
 .040 
 
 22.7 
 
 3-5 
 
 I 
 
 05 
 
 1.28 
 
 16 
 
 .047 
 
 21.9 
 
 5-0 
 
 I 
 
 47 
 
 1.87 
 
 13 
 
 .072 
 
 27.2 
 
 7-5 
 
 2 
 
 30 
 
 2.70 
 
 16 
 
 ■054 
 
 17.4 
 
 10. 
 
 3 
 
 20 
 
 3-47 
 
 16 
 
 .106 
 
 8.4 
 
 In considering the above tables, it is necessary to bear in mind 
 the significance of the quantities in the columns headed A-B. As 
 I have previously stated, this quantity is a measure of the amount 
 of temporal segregation presented by the group, and accordingly an 
 index to the amount of rhythm got out of the sound series by the 
 subject. We may therefore describe the effect of variations in rate 
 as follows. Within certain limits the degree of temporal segregation 
 presented by the groups remains very nearly constant. With 
 slower rates, the rhythm entirely or almost entirely disappears. 
 The limits within which the degree of temporal segregation remains 
 relatively constant vary with individuals and with the nature of 
 the objective series. Thus, in the case of subject Dy, in the case 
 presented in Table IV, where there was merely a difference in in- 
 tensity between the sounds, the rhythm dropped off very suddenly 
 between a rate which gave as the total duration of one measure 
 4.0 seconds and one which gave as the total duration of one measure 
 5.5 seconds. With a different sound series, however, one in which 
 the louder sound was also the longer (Table VI), a certain amount 
 of rhythm persisted even at a rate which gave as the total duration 
 of one measure 10.0 seconds; though even in this case there was 
 already a very considerable decrease at the rate of 7.5 seconds for 
 one measure. In general, the tables indicate that, from a rate of 
 1.0 second for one measure up to a rate of between 4.0 and 7.0 
 seconds for one measure, the degree of rhythm remains about con- 
 stant. 
 
 Sufficient observations have not been made with very fast rates 
 to generalize concerning the effect of increasing the rate beyond 1.0 
 second for one measure. There is some indication, however, that 
 with rates faster than 1.0 second for one measure there is a marked 
 
RATE AND INTENSITY 37 
 
 increase in the amount of temporal segregation presented by the 
 rhythm. As already stated, grouping by four instead of by two 
 comes in at rates faster than 1.0 second for one measure. Inasmuch 
 as subjective rhythm comes in very markedly at rates of about 
 1 second for two beats or faster, the increase above noted in the 
 rhythmical effect for the faster rates can hardly be regarded as the 
 effect of an increase in rate on the rhythmical effect of intensity or 
 duration. It seems rather that we have in the case of these rates 
 a subjective factor which is niore or less independent of any differ- 
 ences which may prevail in the intensity or duration of the sounds 
 composing the series, since at these rates grouping occurs when all 
 the sounds are equal both in intensity and duration.^ 
 
 The rate at which intensity and duration completely fail to exert 
 any influence favoring rhythm is indicated as being somewhat slower 
 than that usually given as the limit of rhythm, being in some cases 
 beyond 10.0 seconds for one measure. While a marked decrease in 
 the amount of rhythm got from a series occurs in the neighborhood 
 of from 4.0 to 7.0 seconds for one measure, the limit at which varia- 
 tions in intensity and duration fail to exert any influence making 
 for temporal segregation is indicated as varying from 7.0 to over 
 10.0 seconds for one measure. Subject Wr, an experienced in- 
 trospectionist, told me he got rhythm (when the intervals were not 
 too nearly equal) from a rate which made the total duration of one 
 measure 26.0 seconds. What he meant was that when the intervals 
 were not too nearly equal the sounds seemed grouped by twos, and 
 one sound was louder than the other. Now, grouping is usually 
 held to be one of the chief characteristics of rhythm. Yet Table IV 
 shows for this same subject that, so far as the effect of intensity 
 goes, making one sound stimulus about eight times as energetic as 
 the other failed, practically, to exert any grouping effect when 
 the rate was slow enough to make the total duration of one measure 
 equal to 7.5 seconds. It is evidently impossible to fix the upper 
 (slow) limit of rhythm until we have a definition of rhythm, but 
 by the method I have used it is possible to fix the upper limit at 
 which intensity or duration exert any rhythmical effect in the sense 
 of promoting temporal segregation. 
 
 All subjects found the work on the very slow rhythms extremely 
 fatiguing, and it was hardly possible at the slowest rates to run 
 through more than 3 or 4 series of minimal changes at one sitting. 
 Extremely close attention is required at these slow rates to get any 
 rhythm at all. 
 
 ^ Meumann, Phil. Stud., lo, 302, 1894. 
 
CHAPTER V 
 
 Duration 
 
 In the investigation of the effect on rhythm of variations in 
 the duration of the sounds, I did not find so much agreement between 
 different individuals as in the case of variations in intensity. In- 
 dividual differences, while quite marked in the case of duration, 
 are yet not so great as to prevent certain generalizations. 
 
 The investigation of the effect of duration in rhythm was carried 
 on in the same way as the investigation of the effect of intensity, 
 except that instead of using sounds of different intensity, the in- 
 tensity of all sounds was kept constant and their duration alone 
 varied. As has already been shown, in the case of two-membered 
 rhythms, if we start out with a trochaic rhythm, in which the louder 
 sound begins the measure, by gradually increasing the interval 
 after the louder sound, we arrive at an iambic rhythm, in which 
 the louder sound ends the measure. I found no exception to the 
 possibility of changing in this manner from trochaic to iambic. 
 Similarly, in the case of rhythm in which we have differences in dura- 
 tion of the sounds but not in intensity, if we start with a trochaic 
 rhythm with the longer sound beginning the measure, by increasing 
 the interval after the longer sound and decreasing the interval 
 before it, we will arrive at an iambic rhythm, with the longer sound 
 second. In this case, the longer sound ordinarily seems to the 
 subject to be accented. I found no exception to the possibility 
 of changing in this manner from trochaic to iambic in the case of 
 any of the subjects that I have worked on, and I have tested thirteen 
 subjects in this way. In the case of one subject however, subject 
 Sh, the 7nost usual result was not a change from trochaic to iambic. 
 With a sufficient increase in the interval following the longer sound, 
 the rhythm, instead of changing to iambic, remained trochaic, but 
 a trochaic in which the shorter sound was described as accented. 
 In other words, in the case of this subject the first sound of the group 
 was usually accented, irrespective of whether the first sound was 
 the longer or the shorter. While I did not get this result in any 
 other case, three other subjects occasionally stated that they had 
 a tendency to hear a trochaic with the short sound accented instead 
 of an iambic with the long sound accented, when the interval after 
 the longer sound appeared the longer. When the interval before 
 the longer sound appeared greater than that following it, the longer 
 sound always seemed accented, that is, in no case was any tendency 
 
DURATION 39 
 
 found to accent the second sound of a two-membered group, when 
 the second sound was the shorter and of equal intensity with the 
 first. 
 
 This result seems to indicate that there are two separable ob- 
 jective factors tending to produce subjective accent, at least in 
 some subjects. There is a tendency to accent the longer sound 
 and also a tendency to accent the sound which seems to begin the 
 group. In some subjects there is no evidence of the latter tendency, 
 and only in one subject out of thirteen, subject Sh, was it as strong 
 as the tendency subjectively to accent the longer sound. Only 
 on the supposition of these two separable tendencies does it seem 
 possible to explain the fact, that, when the longer sound appears 
 to begin the group, it invariably receives an accent, whereas when 
 the longer sound ends the group, it occasionally appears less ac- 
 cented than the shorter sound. In the one case the two factors 
 work together; in the other, in opposition. 
 
 The apparent intensity of stimuli which, objectively, differ only 
 in duration, is a point of considerable interest. Miner found in 
 his investigation of rhythm that an increase in the duration of a 
 light was mistaken for an increase in intensity. It is, further, 
 a well-known fact, that, in the case of sound, the apparent loudness 
 of a sound increases with the increase in duration, the objective 
 intensity remaining constant. This increase in apparent loudness 
 occurs at first very rapidly and then slower, as the duration is in- 
 creased; but the maximum of apparent loudness, for sounds of the 
 objective intensity used in this investigation of duration in rhythm, 
 would hardly be reached before the duration had reached at least 
 one second.^ 
 
 The significance of this fact of increase in apparent intensity 
 with increase in duration is very definitely indicated by the follow- 
 ing introspection from subject Wr, on a rhythm in which the longer 
 sound was .25 second and the shorter .13 second, and the rhythm 
 given first with both sounds rather loud and then with both sounds 
 rather weak. ''There is no doubt but emphasis goes with length: 
 tendency to accentuate short note practically disappeared with 
 weaker intensity." This tendency to accentuate the short note 
 disappears with weak intensities because, when the sounds are 
 weak, a slight difference in duration is of more significance for 
 their apparent relative loudness than when both are quite loud. 
 The results we have just spoken of above, however, show that the 
 greater apparent intensity of one or the other sound is not due 
 
 ' Kafka, "Uber das Ansteigen der Tonerregung," Psychol. Stud., 2, 256- 
 292, 1907. 
 
40 A QUANTITATIVE STUDY OF RHYTHM 
 
 merely to the fact that a sound increases in apparent intensity with 
 increase in duration. In the case of subject Sh, and others, in which 
 the first sound of the group was sometimes accented whether the 
 longer or not, we have an indication that some process in the central 
 nervous system is exerting an effect on the apparent loudness of 
 the sounds. I have also many series of introspections which show 
 that the difference in apparent loudness of the two sounds, when 
 the apparently louder is second, is greater as the interval before 
 the second sound is shortened and the interval before the first 
 or apparently weaker is lengthened. The reverse introspection 
 was never obtained. That is to say, in an iambic rhythm, with 
 the second sound apparently louder than the first, the apparent differ- 
 ence in loudness is often reported as decreasing as the intervals become 
 more and more equal. Similarly in a trochaic rhythm, the first 
 sound often seems to decrease in relative loudness as the intervals 
 are made more nearly apparently equal. These introspections were 
 given by subjects Br, Ww, Py, Dy and Ws. When the intervals were 
 nearly equal, such expressions as "very slight accent," ''emphasis 
 exceedingly light," "accent doubtful, depending on will" were 
 obtained, while when the intervals were unequal, such introspections 
 as "accent very pronounced," "2nd (or 1st) clearly louder" etc., 
 were given. Concerning a series in which the sounds were unchanged, 
 but the intervals varied in the usual way, subject Ws writes: "The 
 notes are more nearly of the same length as you approach the chang- 
 ing point ' (indifference point). Subject Br, concerning a rhythm in 
 which the longer sound was 0.07 second and the shorter 0.06 second: 
 ''The longer notes seem more intense and slightly higher than the 
 shorter. When the intervals are about equal, the length and in- 
 tensity of the notes also seem about equal. According as the at- 
 tention seizes on one note or the other, is the position of accent and 
 the grouping determined. At the time the intervals are equal, dif- 
 ference in length and intensity practically disappears. The difference 
 became more pronounced whenever the rhythm was more pro- 
 nounced." 
 
 These results show that, in some cases, at least, the greater ap- 
 parent intensity of one or other of the sounds is due to some central 
 process, a conclusion similar to what seems inevitable in regard to 
 the subjective accent in purely subjective rhythm. I think that 
 this process is probably that of attention, but I do not care to insist 
 on this at present, as it would require too much space to give all 
 the grounds for such a belief. It is some process, however, which 
 is quite separate from that involved in the increase in apparent 
 intensity resulting from an increase in duration. 
 
DURATION 41 
 
 The general conclusion, then, is that the longer sound should 
 naturally appear the louder, as we might say, for sensory reasons, 
 apart from more complicated or more central processes. At the 
 same time, there is some central process, possibly attention, 
 which has some slight effect on apparent intensity. When the longer 
 sound appears the louder there is no necessity for considering this 
 as due to greater attention directed to the longer. On the contrary, 
 the shorter might be receiving the greater amount of attention and 
 yet not appear as loud as the longer, the physiological or sensory 
 effect exerted by duration on intensity being able to swamp the slight 
 effect of the central process on intensity, in case the two processes 
 work in opposition. In those cases, however, where the shorter 
 sound seems the louder, the central process is evidently having 
 a stronger effect than the sensory. It is possible that all the subjects 
 have not used the words intensity, and accent, and emphasis in the 
 same sense, and that the effects referred to in some cases as differences 
 in intensity were differences in some other sort of emphasis; but 
 this makes it all the more desirable to distinguish between the un- 
 questionable effect on intensity of increase in duration and the 
 effect on something referred to as accent or as intensity, exerted 
 equally unquestionably by some other apparently more central 
 process. 
 
 The results obtained with regard to the indifference point in 
 two-membered rhythms produced by alternation of a longer and 
 shorter sound of equal objective intensity and of the same quality 
 are presented in Tables VII and VIII. By the total duration of a 
 measure, or in Table VIII, by "Measure," is meant the time, say, 
 from the beginning of one long sound to the beginning of the next 
 long sound. By N, is meant the number of series of minimal changes 
 run through in the determination of any one indifference point. 
 The tables show the length of the intervals at the indifference point 
 for rhythm. By the interval "before,'' is meant the duration of 
 the interval before the longer sound, i. e., the time elapsing from 
 the end of the short sound to the beginning of the long; and by the 
 interval 'Rafter," is meant the interval after the longer sound, i. e., 
 from the end of the long sound to the beginning of the short. The 
 mean variation referred to is the variation in the length of these 
 intervals. All the results obtained from any one ratio are treated 
 as one group in obtaining the mean variation. Inasmuch as one 
 interval was lengthened as the other was shortened, and vice versa, 
 the absolute M V of one interval is also that of the other; and for 
 that reason, I have given the absolute M V rather than the relative. 
 In the column A-B, I have indicated the difference in duration 
 
42 
 
 A QUANTITATIVE STUDY OF RHYTHM 
 
 between the interval following the longer sound and that preceding 
 it. In the column headed (A + L) — (B + S), is indicated the 
 difference between the duration of the iDterval of silence after the 
 longer sound + that of the longer sound and that of the interval 
 of silence before the longer sound + that of the shorter sound. 
 In other words, in this column the measure is considered as being 
 composed of two intervals each of which extends from the beginning 
 of one sound to the beginning of the following sound. The column 
 headed (A + L) — (B + S) presents the duration of the interval 
 extending from the beginning of the longer sound to the beginning 
 of the shorter minus the duration of the interval extending from the 
 beginning of the shorter to the beginning of the longer. The following- 
 scheme illustrates this method of treating the measure: 
 
 I measure 
 
 B 
 
 Iv 
 
 (A + L) 
 
 (B + S) 
 
 All durations are indicated in seconds. By "longer" or "shorter'" 
 is meant the duration of the longer or shorter sound. 
 
 TABLE VII 
 The Effect of Relative Duration on the Iambic-trochaic Indifference Points 
 Duration of shorter sound (constant) =0.13 second. 
 Total duration of measure (constant) = 1.5 seconds. 
 
 Ws 
 (July '08) 
 
 Ws 
 (July, '07) 
 
 3 
 
 5 
 
 V 
 
 16 
 
 58 
 
 ■ 19 
 
 58 
 
 . 22 
 
 58 
 
 •25 
 
 55 
 
 .28 
 
 55 
 
 •31 
 
 55 
 
 ■37 
 
 55 
 
 •43 
 
 55 
 
 .48 
 
 54 
 
 .60 
 
 54 
 
 .18 
 
 50 
 
 .28 , 
 
 55 
 
 •34 
 
 54 
 
 •54 
 
 51 
 
 < 
 
 IZ! 
 
 
 M 
 1 
 
 •4 
 < 
 
 63 
 
 31 
 
 .014 
 
 + 
 
 .05 
 
 + 
 
 08 
 
 60 
 
 58 
 
 .024 ■ 
 
 + 
 
 .02 
 
 + 
 
 0& 
 
 57 
 
 26 
 
 .017 
 
 — 
 
 .01 
 
 + 
 
 09 
 
 57 
 
 28 
 
 .022 
 
 + 
 
 .02 
 
 + 
 
 13 
 
 54 
 
 24 
 
 .020 
 
 
 
 .01 
 
 + 
 
 14 
 
 51 
 
 26 
 
 .010 
 
 
 
 .04 
 
 + 
 
 14 
 
 45 
 
 28 
 
 .014 
 
 
 
 .10 
 
 + 
 
 14 
 
 39 
 
 27 
 
 .020 
 
 — 
 
 .16 
 
 + 
 
 13 
 
 35 
 
 22 
 
 .010 
 
 
 
 .19 
 
 + 
 
 16 
 
 23 
 
 23 
 
 .018 
 
 
 
 •31 
 
 + 
 
 16 
 
 56 
 
 32 
 
 .020 
 
 + 
 
 .06 
 
 + 
 
 II 
 
 53 
 
 18 
 
 .016 
 
 — 
 
 .02 
 
 + 
 
 1 3. 
 
 48 
 
 41 
 
 .030 
 
 
 
 .06 
 
 + 
 
 13 
 
 •31 
 
 22 
 
 .025 
 
 — 
 
 .20 
 
 + 
 
 21 
 
DURATION 
 
 TABLE Yll— {Continued) 
 
 43 
 
 V 
 
 iff 
 
 n 
 
 03 
 
 u 
 
 il 
 
 M 
 C 
 
 ►4 
 
 
 
 pq 
 
 u 
 < 
 
 ^ 
 
 \ 
 
 ^ 
 
 fq 
 1 
 < 
 
 + 
 < 
 
 Vv^w 
 
 16 
 
 57 
 
 64 
 
 15 
 
 .016 
 
 + .07 
 
 4-. 10 
 
 (July, '08) 
 
 19 
 
 57 
 
 61 
 
 16 
 
 ■ 015 
 
 + .04 
 
 + .10 
 
 
 22 
 
 57 
 
 58 
 
 16 
 
 .023 
 
 + .01 
 
 + .10 
 
 
 25 
 
 54 
 
 58 
 
 16 
 
 ■ 015 
 
 + .04 
 
 + .16 
 
 
 28 
 
 54 
 
 55 
 
 10 
 
 .014 
 
 + .01 
 
 + .16 
 
 
 31 
 
 53 
 
 54 
 
 10 
 
 .010 
 
 + .01 
 
 + ■19 
 
 
 37 
 
 51 
 
 49 
 
 15 
 
 .016 
 
 — .02 
 
 4-. 22 
 
 
 43 
 
 47 
 
 47 
 
 10 
 
 ■ 015 
 
 .00 
 
 4-. 30 
 
 
 48 
 
 43 
 
 46 
 
 10 
 
 .024 
 
 + .03 
 
 + •38 
 
 
 60 
 
 40 
 
 37 
 
 16 
 
 .013 
 
 — .03 
 
 + •44 
 
 
 72 
 
 36 
 
 29 
 
 10 
 
 .011 
 
 — .07 
 
 + .52 
 
 
 84 
 
 33 
 
 20 
 
 10 
 
 .013 
 
 — .13 
 
 + .58 
 
 Ww 
 
 18 
 
 56 
 
 62 
 
 31 
 
 .025 
 
 + .06 
 
 + .11 
 
 (July, '07) 
 
 26 
 
 52 
 
 58 
 
 23 
 
 .036 
 
 + .06 
 
 + •19 
 
 
 34 
 
 50 
 
 52 
 
 28 
 
 .030 
 
 + .02 
 
 + ■23 
 
 
 42 
 
 48 
 
 46 
 
 21 
 
 ■034 
 
 — .02 
 
 4- .27 
 
 
 54 
 
 43 
 
 39 
 
 25 
 
 .020 
 
 — .04 
 
 + .36 
 
 Dy 
 
 16 
 
 53 ■ 
 
 68 
 
 42 
 
 .040 
 
 4- .15 
 
 4-. 18 
 
 
 19 
 
 51 
 
 67 
 
 35 
 
 ■037 
 
 + .16 
 
 + .22 
 
 
 22 
 
 53 
 
 62 
 
 33 
 
 .020 
 
 + .09 
 
 4-. 18 
 
 
 25 
 
 51 
 
 61 
 
 45 
 
 .o2y 
 
 4- .10 
 
 4-. 22 
 
 
 31 
 
 46 
 
 60 
 
 33 
 
 .026 
 
 + .14 
 
 + .32 
 
 
 37 
 
 43 
 
 57 
 
 37 
 
 .031 
 
 + .14 
 
 + .38 
 
 
 43 
 
 40 
 
 54 
 
 35 
 
 .028 
 
 + .14 
 
 4-. 44 
 
 
 48 
 
 39 
 
 50 
 
 28 
 
 .ojg 
 
 + .11 
 
 + .46 
 
 
 60 
 
 36 
 
 41 
 
 25 
 
 .038 
 
 + .05 
 
 + •52 
 
 
 72 
 
 28 
 
 37 
 
 18 
 
 .041 
 
 + .09 
 
 4-. 68 
 
 
 84 
 
 24 
 
 29 
 
 15 
 
 .020 
 
 + .05 
 
 + .76 
 
 Wr 
 
 16 
 
 61 
 
 60 
 
 15 
 
 .02^ 
 
 — .01 
 
 4- .02 
 
 
 19 
 
 58 
 
 60 
 
 II 
 
 .Oil 
 
 + .02 
 
 4- .08 
 
 
 22 
 
 58 
 
 57 
 
 25 
 
 .020 
 
 — .01 
 
 + .08 
 
 
 25 
 
 54 
 
 58 
 
 16 
 
 .018 
 
 + .04 
 
 + .16 
 
 
 28 
 
 55 
 
 54 
 
 16 
 
 .023 
 
 — .01 
 
 + .14 
 
 
 31 
 
 53 
 
 53 
 
 17 
 
 .Oil 
 
 .00 
 
 4-. 18 
 
 
 37 
 
 54 
 
 46 
 
 15 
 
 .014 
 
 — .08 
 
 + .18 
 
 
 43 
 
 51 
 
 43 
 
 17 
 
 .023 
 
 — .08 
 
 + •24 
 
 
 48 
 
 49 
 
 40 
 
 19 
 
 .014 
 
 — .09 
 
 4-. 26 
 
 
 60 
 
 44 
 
 33 
 
 1-9 
 
 .024 
 
 — .11 
 
 4-. 36 
 
44 
 
 A QUANTITATIVE STUDY OF RHYTHM 
 
 TABLE VIII 
 The Effect of Relative Duration on the Iambic-trochaic Indifference Point. 
 
 Ws 
 
 Ws 
 
 .64 
 
 Py 1.49 
 
 Vt I. II 
 
 Br I . 50 
 
 • 05 
 
 13 
 
 .08 
 
 h-r 
 
 w 
 
 .06 
 
 24 
 
 .07 
 
 22 
 
 .08 
 
 23 
 
 . 10 
 
 21 
 
 • 15 
 
 22 
 
 . 20 
 
 23 
 
 • 25 
 
 22 
 
 •13 
 
 39 
 
 •15 
 
 39 
 
 .16 
 
 38 
 
 .20 
 
 38 
 
 •23 
 
 38 
 
 • 25 
 
 37 
 
 •29 
 
 38 
 
 ■34 
 
 37 
 
 .18 
 
 55 
 
 ■34 
 
 48 
 
 •54 
 
 40 
 
 •13 
 
 39 
 
 . 20 
 
 37 
 
 •25 
 
 37 
 
 . 10 
 
 60 
 
 .18 
 
 60 
 
 27 
 
 59 
 
 ■35 
 
 57 
 
 •43 
 
 58 
 
 ■52 
 
 55 
 
 60 
 
 55 
 
 
 
 
 
 
 + 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 1 
 
 1 
 
 + 
 
 
 < 
 
 Z 
 
 ^ 
 
 1 
 
 < 
 
 < 
 
 
 30 
 
 ID 
 
 .017 
 
 + 
 
 .06 
 
 + 
 
 07 
 
 30 
 
 12 
 
 .018 
 
 + 
 
 .08 
 
 + 
 
 10 
 
 29 
 
 10 
 
 .012 
 
 + 
 
 .06 
 
 + 
 
 09 
 
 27 
 
 10 
 
 .020 
 
 + 
 
 .06 
 
 + 
 
 II 
 
 22 
 
 II 
 
 .019 
 
 
 .00 
 
 + 
 
 ID 
 
 17 
 
 II 
 
 .011 
 
 — 
 
 .06 
 
 + 
 
 09 
 
 12 
 
 II 
 
 .018 
 
 — 
 
 .10 
 
 4- 
 
 II 
 
 49 
 
 10 
 
 .017 
 
 + 
 
 .10 
 
 + 
 
 13 
 
 47 
 
 10 
 
 .016 
 
 + 
 
 .08 
 
 + 
 
 13 
 
 46 
 
 10 
 
 .oig 
 
 + 
 
 .08 
 
 + 
 
 14 
 
 44 
 
 II 
 
 .016 
 
 + 
 
 .06 
 
 + 
 
 16 
 
 40 
 
 ID 
 
 .015 
 
 + 
 
 .02 
 
 + 
 
 15 
 
 39 
 
 10 
 
 ■ 015 
 
 + 
 
 .02 
 
 + 
 
 17 
 
 34 
 
 12 
 
 .017 
 
 — 
 
 .04 
 
 + 
 
 15 
 
 31 
 
 ID 
 
 .012 
 
 — 
 
 .06 
 
 + 
 
 18 
 
 63 
 
 32 
 
 .032 
 
 + 
 
 .08 
 
 + 
 
 II 
 
 54 
 
 26 
 
 ■039 
 
 + 
 
 .06 
 
 + 
 
 29 
 
 43 
 
 32 
 
 .031 
 
 + 
 
 •03 
 
 + 
 
 45 
 
 49 
 
 10 
 
 .016 
 
 + 
 
 .10 
 
 + 
 
 13 
 
 45 
 
 II 
 
 .010 
 
 4- 
 
 .08 
 
 + 
 
 18 
 
 39 
 
 ID 
 
 .017 
 
 + 
 
 .02 
 
 4- 
 
 17 
 
 72 
 
 II 
 
 .030 
 
 + 
 
 .12 
 
 + 
 
 14 
 
 64 
 
 10 
 
 .027 
 
 + 
 
 .04 
 
 4- 
 
 14 
 
 57 
 
 10 
 
 .014 
 
 • — 
 
 .02 
 
 + 
 
 17 
 
 49 
 
 ID 
 
 .017 
 
 — 
 
 .08 
 
 4- 
 
 19 
 
 40 
 
 7 
 
 .017 
 
 — 
 
 .18 
 
 + 
 
 17 
 
 35 
 
 8 
 
 .017 
 
 — 
 
 .20 
 
 4- 
 
 24 
 
 27 
 
 8 
 
 .015 
 
 — 
 
 .28 
 
 + 
 
 24 
 
 As I have already indicated in Chapter II (pages 17-20), the mag- 
 nitude of A-B, when positive, may be taken as a measure of the 
 degree of trochaism, when negative of the degree of iambism presented 
 by the group when the intervals between the sounds are objectively 
 equal. A study of the columns headed A-B in the above tables 
 shows that without exception the rhythm becomes less trochaic 
 or more iambic as the ratio between the longer and shorter sounds 
 increases. In some cases, as with subjects Ww, Ws, Wr and Br, 
 the change is one from a trochaic rhythm through an indifference 
 
DURATION 45 
 
 point to a more and more pronounced iambic. In other cases, as 
 with subjects Py, Dy and Vt, the change is one from a very decided 
 trochaic to a less stable trochaic. In the case of Vt, however, 
 there is little doubt but that the rhythm would have become strongly 
 iambic with larger ratios between the durations of the sounds. 
 In the case of Dy, the change is very slight; but it is still evident 
 enough to render it unnecessary to regard this subject as an exception 
 to the general rule. The general law indicated, then, regarding the 
 effect of the relative duration of the sounds in auditory rhythms, 
 is that the rhythm becomes less trochaic or more iambic as the 
 ratio between the longer and shorter sounds increases. As I have 
 found this law to hold with several different absolute durations of 
 the shorter sound, it seems more or less general, though I am unable 
 to present data concerning its limits. 
 
 Subject Sh might prove an exception to this generalization. 
 I was unable to study the subject thoroughly enough to be sure 
 whether he would or not. In his case, I found that, with a ratio 
 of two to one and equal intervals, the rhythm was trochaic with 
 the longer sound accented. When the interval before the longer 
 sound was made considerably shorter than that after it, the rhythm 
 usually changed to a trochaic with the shorter sound accented. 
 I believe, consequently, that in generalizing the effects of duration 
 on rhythm, it is safer to omit the words iambic and trochaic. I 
 would therefore state the law of the effect on auditory rhythms, 
 composed of groups of two sounds each, exerted by changes in the 
 ratio of the durations of the sounds, as follows: With an increase 
 in the ratio between the duration of the longer and shorter sounds, 
 if the intervals between the sounds are objectively equal, there is 
 an increase in the tendency to obtain a rhythm with the longer 
 sound at the end of the group or a decrease in the tendency to obtain 
 a rhythm with the longer sound at the beginning of the group. 
 
 It is interesting to note that when the intervals are objectively 
 equal, the series is heard as trochaic rhythm when the difference in 
 the duration of the sounds is sufficiently small. Out of a total 
 of thirteen subjects, only two, Ws and Wr, ever showed any ex- 
 ception to this statement, and even in these cases the prevailing ten- 
 dency with the smaller ratios is towards trochaic. In every case, 
 with a ratio of durations of approximately 2 to 1 (25 to 13), with 
 equal objective intervals, trochaic rhythm was the result. 
 
 The most strongly trochaic rhythms produced by duration changes 
 are those produced by the smallest ratio of durations. It looks 
 as though, were the series continued up to a ratio of one to one, 
 the series would be more trochaic than ever. This is, however, 
 
46 A QUANTITATIVE STUDY OF RHYTHM 
 
 not the case. When there is no difference at all in duration (nor 
 in accent) the indifference point was found to be almost an3^where, 
 that is, all judgments become doubtful. The strongest trochaic 
 which it is possible to produce by duration changes alone, therefore, 
 is produced when the longer sound is longer, but only very slightly 
 so, than the shorter sound. It may be that the trochaic rhythm 
 of the smaller ratios was due chiefly to the difference in apparent 
 loudness between the longer and shorter sounds. A comparison 
 of the effect of slight differences in duration with slight differences 
 in intensity lends support to this idea. If we rank the four sub- 
 jects used in the investigation of the effect of intensity with respect 
 to the magnitude of the quantity A-B for the two small intensity 
 ratios of 30 to 24 and 32 to 24 (feet at which audible), and then 
 with respect to the same quantity for small ratios of durations, 
 we find that the correlation is perfect. In other words, if a slight 
 difference in intensity produces a strong trochaic rhythm for any 
 given subject, a slight difference in the duration of the sounds does 
 also; whereas if a slight difference in intensity produces only a very 
 weak trochaic, the same will be true of a slight difference in duration. 
 I believe, therefore, that the trochaic rhythm produced by small 
 ratios in the durations is due to the different effect as regards in- 
 tensity of a long and short sound. Now this difference in apparent 
 intensity should increase as the difference in duration increases. 
 But after a certain comparatively short duration is reached, the 
 increase in apparent intensity is very slow. It seems probable, 
 therefore, that, in rhythms produced by short sounds, we have 
 to deal with two separate factors, intensity and duration, even though 
 the sounds differ physically only in duration. The effect of intensity 
 is to make the longer sound begin the measure, an effect which may 
 be spoken of as trochaic, whereas the effect of duration, as such, 
 is to make the longer sound end the measure, which usually means 
 an iambic effect. With a small difference in the absolute durations 
 of the longer and shorter sounds, the effect of intensity is compara- 
 tively great; but as the longer sound is increased more and more, the 
 intensity difference between the two sounds becomes less and less 
 relative to the difference in the duration between the sounds. When 
 the difference in duration is sufficiently increased, therefore, we 
 find that the trochaic effect of intensity tends to disappear because 
 of the iambic effect of duration, that is, we get trochaic rhythm 
 with small ratios but tend more in the iambic direction with large 
 ratios (the absolute duration of the short sound remaining constant). 
 This conclusion is further confirmed by the results on absolute 
 duration. The difference in intensity between the longer and shorter 
 
DURATION 47 
 
 sounds for any given ratio of their durations will be smaller, the 
 longer the short sound. And my results on the effect of absolute 
 duration, presented in Table X,show that, if the ratio is kept constant, 
 the rhythm becomes less trochaic or more iambic, as the absolute 
 duration of the sounds increases. 
 
 The effect of the difference in intensity produced by a difference 
 in duration may also explain in part at least the case of subject 
 Dy, in which there was only a slight decrease in the trochaicness 
 of the rhythm with an increase in the ratio between the durations. 
 For, with Dy, differences in intensity had much more effect than 
 with any of the other subjects investigated for the effect of varia- 
 tions in intensity. The greatest magnitude obtained for the quantity 
 A-B, in the investigation of intensive differences with Dy, was 
 50% greater than the greatest magnitude of the same quantity 
 for any of the other three subjects. Inasmuch as the effect of 
 intensity is trochaic, we should expect to find the tendency towards 
 iambic resulting from large ratios between the durations to be less 
 in the case of Dy than in the case of the other subjects. 
 
 I have confined myself chiefly to the investigation of two-mem- 
 bered groups; but that the above mentioned general law concerning 
 the effect of variations in the relative durations of the sounds holds 
 for three-membered groups, as well as two-membered groups, is 
 indicated by Table IX, where the quantity A-B changes from 
 + .02 to — .06 with increase in the ratio of duration between 
 the longer and the two shorter sounds. The method which was 
 used here to determine the indifference point was to vary only the 
 intervals immediately preceding and following the long sound. 
 The two shorter sounds, both of equal duration, were kept the same 
 distance apart throughout, being separated by an interval such that 
 this interval plus the duration of one of the shorter sounds equals 
 one-third the total duration of the measure. By the interval "be- 
 fore," is meant the interval before the longer sound; by the interval 
 "after," the interval after the longer sound, as in the tables on 
 rhythms made up of groups of two sounds. 
 
 TABLE IX 
 'The Effect of Relative Duration on the Anapaestic-dactyHc Indifference Point. 
 
 Subject, Ww. 
 Duration of the two shorter sounds (constant) = o.o8 second. Total duration 
 of measure = 2.0 seconds. Interval between the two shorter sounds 
 (constant) = 0.58 second. 
 l,onger sound Before After N MV A — B (A + 1,)— (B + S) 
 
 . l5 .54 .56 ID .01 J +.02 +.10 
 
 .30 .50 .44 II .024 — .06 +.16 
 
48 A QUANTITATIVE STUDY OF RHYTHM 
 
 Before proceeding to the discussion of the effect of variations 
 in the absolute duration of the sounds, it is desirable to bring up 
 the question of the effect of change in the duration of the intervals 
 while the absolute and relative duration of the sounds remains 
 unchanged. This is necessary inasmuch as, in the above tables, 
 which show that there is a change in the iambic direction as the ratio 
 of the durations increases, there occurs simultaneously with the 
 increase in the ratio between the sounds a decrease in the total 
 amount of silence within the group. The reason for this is that the 
 total duration of the group was kept the same, and evidently, then, 
 as one of the sounds was increased in duration in order to get bigger 
 ratios, the total amount of silence remaining had to be decreased. 
 The question might therefore arise as to whether the change noted 
 above in the iambic direction was due to the increase in the ratio 
 between the durations of the long and short sounds, or whether 
 it was due to the decrease in the proportion of the total duration 
 of the group occupied by silent intervals. I have not sufficient 
 data to state in a definite way what is the effect on rhythm of varia- 
 tion in the duration of the intervals, the absolute and relative 
 duration of the sounds remaining constant; but I have data to show 
 that it is utterly impossible that the changes noted above, from 
 trochaic to iambic rhythm with increase in the ratio of the durations 
 of the sounds, could be due to the coincident decrease in the pro- 
 portion of the total duration of the measure constituted by the 
 intervals of silence. In the case of subject Wr, with a ratio of 
 durations of 19 to 13, and equal intervals of .59 second, the quantity 
 A-B== + .02 second; with the ratio 43 to 13, and equal intervals 
 of .47 second, the quantity A-B = — .08 second, and with a 
 ratio of 60 to 13, and equal intervals of .39 second, A-B = — .11 
 second. That this change of A-B from + to — (i. e., change 
 from trochaic to iambic rhythm) is not due to the shortening in the 
 intervals is shown conclusively by the following table, where it is 
 
 TABLE X 
 
 Effect of Proportion of Sound to Silence on Iambic-trochaic Indifference Point 
 
 Subject, Wr. 
 
 Duration of shorter sound = .13 second. 
 
 I^onger sound 
 
 Before 
 
 After 
 
 N 
 
 M V 
 
 A— B 
 
 •19 
 
 •58 
 
 .60 
 
 II 
 
 .Oil 
 
 + .02 
 
 ■43 
 
 •51 
 
 ■43 
 
 17 
 
 ■023 
 
 — .08 
 
 .60 
 
 •44 
 
 •33 
 
 19 
 
 .024 
 
 .11 
 
 ■43 
 
 1. 19 
 
 1.25 
 
 14 
 
 ■047 
 
 — .06 
 
 .60 
 
 1.07 
 
 1.20 
 
 18 
 
 .025 
 
 — •13 
 
 .72 
 
 .98 
 
 1. 17 
 
 16 
 
 .025 
 
 — .19 
 
DURATION 49 
 
 seen that the intervals are in every case about twice as great as in 
 the case of the ratio 19 to 13, where the intervals were .59 second 
 and the rhythm trochaic, and yet the rhythm in these cases, too, 
 becomes strongly iambic as the ratio between the sounds increases. 
 Evidently the change in A-B from + .02 second to — .19 second, 
 as the ratio between the durations of the sounds increases, can not 
 be due to a decrease in the intervals of silence, for these have not 
 decreased, but nearly doubled. 
 
 As regards the result of variation in the absolute duration of the 
 sounds, the following table seems sufficient to establish beyond a 
 doubt that in rhythms in which the sounds of a two-membered 
 group are of different durations, with a given ratio of these dura- 
 tions the rhythm becomes more trochaic or less iambic as the ab- 
 solute duration of the sounds decreases, and less trochaic or more 
 iambic as their absolute duration increases. The effect of change 
 in the absolute duration of the sounds is exceedingly great, so that 
 it is impossible to say of any given ratio of durations that it will pro- 
 duce a certain rhythm, even in the same subject: the absolute, as 
 well as the relative, duration of the sounds must be specified. 
 
 TABLE XI 
 
 Effect of Absolute Duration on the Iambic-trochaic Indifference Point 
 
 Subject, Ww. 
 
 Ratio between durations of long and short sound = 2 to i. 
 
 I^onger sound Before After N MV A — B 
 
 .09 .63 .81 20 .oop + .18 
 
 .25 .54 .58 16 .on + .04 
 
 .45 .41 .40 20 .01^ —.01 
 
 .90 .87 .77 20 .016 .10 
 
 In the above table, it is true that the intervals have not been kept 
 constant, but enough has been stated already concerning the effect 
 of intervals to show that the changes here shown from a marked 
 trochaic to a marked iambic are due to changes in the absolute dura- 
 tion of the sounds. 
 
 A question of fundamental importance in the investigation of 
 the effect of duration in rhythm concerns the proper method of 
 presenting the results. Should the intervals between the sounds 
 be treated separately, or should only the temporal distances from 
 the beginning of one sound to the beginning of the next be con- 
 sidered? At first consideration, this question may seem to have 
 little significance, at least in the present investigation, inasmuch 
 as the duration of the sounds and intervals have been measured 
 separately, and consequently it is possible to present the resiilts 
 
50 A QUANTITATIVE STUDY OF RHYTHM 
 
 in both ways, and I have so presented them. And it is undoubtedly- 
 true that, if all we want is a description of what sort of rhythm is 
 heard from any series of sounds, both the length of the sounds and 
 the intervals between them, in case intervals exist, should be stated. 
 But if we wish to understand or to explain our data, it is of great 
 importance to know from what points to measure in presenting the 
 results. For instance, should a rhythmical group be considered as 
 being made up of "members" which extend, say, from the beginning 
 of one sound to the beginning of the following? Dr. Brown, in his 
 investigation of spoken verses writes: "On the whole I have decided 
 to consider only the beginning of the syllables."^ In the case of 
 spoken rhythms or of any other rhythms in which the sounds are 
 not of uniform intensity throughout, the question of measurement 
 is perhaps more difficult than in the case of sound rhythms such 
 as those used in this investigation. While in the present investiga- 
 tion the sounds were of uniform intensity throughout their length, 
 it is yet possible that what really counted to the listener was neither 
 the actual interval between the sounds nor the time elapsing from 
 the beginning of one sound to the beginning of the next, but the 
 interval between certain points of greatest subjective stress, the 
 subjective stress being due, perhaps, to the occurrence, at certain 
 points, of greater attention, of motor performances, or to unknown 
 factors; and these theoretically possible points of greatest subjective 
 stress might conceivably occur almost anywhere. The significance 
 which is to be attached to any data conceriiing the objective con- 
 ditions of rhythms in which variations in duration occur depends 
 upon the mode of tabulating the results. 
 
 I have obtained some data which seem to me to solve this ques- 
 tion concerning the proper treatment of results. As I shall show 
 in Chapter VI, the temporal spacing of the sounds at the indifference 
 point for rhythm very nearly coincides with that spacing at which 
 all the intervals appear equal, a fact which justifies the conclusion 
 that perception of rhythm and perception of time are closely allied 
 processes. This being the case, it is possible to tell whether in rhythm 
 the interval between the sounds or the interval from the beginning 
 of one to the beginning of the following is the more significant by 
 asking the subject, after having determined his indifference point 
 for rhythm, to compare intervals, first asking him to compare the 
 intervals between the beginnings of the sounds and then asking 
 him to compare the silent intervals between the sounds. Of the 
 two kinds of interval comparison, that one which gives results cor- 
 responding to the judgments on rhythm will be indicated as the 
 ^ Archiv. of Psychol., No. lo, 28, 1908. 
 
DURATION 51 
 
 process more nearly allied to the process involved in judging of 
 the rhythm; and consequently the more significant mode of tabu- 
 lating the results will also be indicated. In other words, if the re- 
 sults concerning the indifference point for rhythm correspond with 
 the results in determining the point where the intervals between the 
 the sounds appear equal, and do not correspond with those where the 
 intervals between the beginnings of the sounds appear equal, then 
 the data concerning rhythm should be presented in a way which 
 will make it clear what were the actual silent intervals between 
 the sounds. Four subjects were tested in the above manner. It 
 is unnecessary to insert at this point the detailed results of these 
 tests, as the data are given in Chapter VI. In the case of three 
 subjects, Dy, Ww, and Wr, it was found that the indifference point 
 for rhythm corresponded with that for the comparison of the in- 
 tervals from the end of one sound to the beginning of the following; 
 while in the case of the fourth subject, Ws, the results on rhythm 
 corresponded with the results on the comparison of intervals between 
 the beginnings of the sounds. There is some ground, then, for the 
 conclusion that usually the intervals which are most important 
 in deciding what will be the nature of the rhythmical impression 
 are the intervals between the end of one sound and the beginning 
 of the following. But, in the case of Ws, there is no doubt that 
 the rhythm judgment was more closely allied to the process of 
 comparing the intervals between the beginnings of the sounds than 
 to that of the comparison of the actual silent intervals. This 
 conclusion is borne out by the following introspection: "I do not 
 mean by intervals the time between the notes. The interval is 
 the difference between the times when the two notes strike." Again, 
 "Tones are more nearly of the same length as you approach the chang- 
 ing point, the point where iambic changes to trochaic. The tones 
 differ but the beats don't." Again, after an hour spent in attempting 
 to compare the actual silent intervals between the sounds, "It 
 is funny, because it is hard to catch the end of a sound and measure 
 intervals in such a manner." The subject plays the piano and 
 organ, and is the only piano player, in fact, the only practical mu- 
 sician of any sort who took part in this investigation. In piano 
 playing, of course, the beat producing any sound comes at the be- 
 ginning of the sound, at least the greatest amount of tactual and 
 kinaesthetic sensation would be received at the moment of pressing 
 down the keys; moreover, in music, especially organ music, there 
 are practically no silent intervals between the sounds. These 
 considerations suggest that possibly the reason why the judgments on 
 rhythm corresponded in this case with judgments on the intervals 
 
52 A QUANTITATIVE STUDY OF RHYTHM 
 
 between the beginnings of the sounds was that there was in this 
 case a motor performance coinciding with the beginning of the 
 sounds, a performance which may have been absent, insignificant, 
 or of a different character in the case of the other subjects. The 
 importance of motor phenomena in the origin of the impression of 
 rhythm has been sufficiently emphasized by previous authors. 
 
 The results of this investigation show very plainly the significance 
 of this question concerning the treatment of results. If we con- 
 sider the two-membered group made up of two periods, each ex- 
 tending from the beginning of one sound to the beginning of the 
 next, it is true without exception that, when these periods are equal, 
 the rhythm is heard as trochaic, and, providing these periods are 
 kept equal, the rhythm becomes more and more trochaic as the 
 ratio between the durations of the sounds increases. Thus, while 
 in the case of the ratio of durations of 2 to 1, with a rather long 
 absolute duration of the sounds, say a duration of the shorter sound 
 equal to .40 second, we might expect to find the rhythm strongly 
 iambic if the intervals between the sounds were equal, it would 
 without question be heard as trochaic if the intervals between the 
 beginnings of the sounds were equal. No subject was found, in this 
 investigation of rhythms produced by variations in duration only, 
 who in any case obtained iambic rhythm from any series of sounds, 
 the intervals between the beginnings of which were equal. As long 
 as the intervals between the beginnings of the sounds are equal, the 
 rhythm becomes more and more trochaic with an increase in the 
 ratio of the duration of the longer sound to that of the shorter. 
 In some subjects, this increase in trochaism is very nearly 
 proportional to the increase in the ratio of the durations, in 
 others, it is less rapid. It seems, therefore, impossible to state 
 the exact relation between the increase in the ratio of dura- 
 tions and the increase in the temporal segregation of the group 
 (in this case, the increase in trochaism) until the causes of individual 
 differences can be quantitatively estimated. 
 
CHAPTER VI 
 The Meaning of Rhythmical Grouping 
 
 Every one is agreed today that the essential thing in the percep- 
 tion of rhythm is the experiencing of groups. It is this experience 
 of groups which distinguishes rhythm in the psychological sense of the 
 word from rhythm in the sense of a regularly recurring event, such 
 as the revolution of the earth about its axis. Thus, Meumann 
 writes that the large number of well-trained observers, which he 
 used in his investigations of rhythm, gave without exception the 
 introspection that a subjective binding together of the impressions 
 into a whole is inseparable from the simplest case of the perception 
 of rhythm.^ Bolton found in the case of every one of thirty subjects 
 that grouping was the irremissible sign of rhythm. And Miner 
 states at the outset of his treatise on rhythm that he uses the word 
 rhythm only in the sense of rhythmic grouping. 
 
 But, while all recent writers on rhythm recognize that the ex- 
 perience of groups is an essential factor in the experience known 
 as the perception, or the feeling, of rhythm, there is wide diversity 
 of opinion as to what we mean when we speak of the members of 
 a series of impressions as being experienced in groups. I am un- 
 able to judge just to what extent this diversity of opinion goes, 
 because of the impossibility of ascertaining, in the case of many 
 writers, whether they are referring to the experience of groups, 
 or to the causes of this experience — whether they are describing 
 a state of consciousness, or presenting a theory as to the origin 
 of this state. To give an example, Bolton writes as follows: "The 
 conscious state accompanying each wave of attention grasps to- 
 gether or unifies all the impressions that fall within the temporal 
 period of the wave."^ If he had written merely that the attention 
 wave grasps the impressions together, one would believe that he 
 was talking of the origin of the group experience; but when he says 
 that it is the conscious state accompamjing the attention wave, it 
 seems as though he was giving an introspective description of the 
 group experience. Some of the interesting introspections reported 
 by Bolton show the same ambiguity. He says of himself, "The 
 puffs of a locomotive may now be grouped by two or three, but the 
 association of the drive-wheel making one revolution to four sounds 
 renders any other form of grouping than by four difficult."^ In 
 
 ^ Phil. Stud., 10, 271, 1894. 
 
 ^ Amer. Journ. of Psychol., 6, 220, 1894. 
 
 ^ Ibid., p. 205. 
 
54 A QUANTITATIVE STUDY OF RHYTHM 
 
 this case, did the idea of the drive-wheel, Hke the conscious state 
 accompanying an attention wave, grasp or tend to group four 
 impressions together, and so produce a grouping or did the group 
 experience consist merely in the simultaneous experience of the 
 four impressions and the idea of the drive-wheel making one revo- 
 lution? The introspections by his subjects are also often ambig- 
 uous, for instance, the introspection of subject No. 12, of whom 
 Bolton writes: "As to the nature of the group, the subject de- 
 scribed his feelings as a tendency to go back when he had heard 
 three or four clicks, as the case might be. He says he has a ' mouth- 
 ful'— a unity — and when he has one, he seeks to get another."^ 
 Another subject "noticed rhythms in the sound of mill wheels. 
 When he gave his attention to these sounds he visualized a series 
 of points on a line which he counted by four or two. When he 
 was asked to count a series of dots he said they were divided off 
 into twos by a bracket above them."^ In this case, the meaning 
 of rhythmical grouping seems to be a grouping by brackets, the 
 idea of bracketed points existing, apparently, more or less simul- 
 taneously with the percepts of the mill sounds. Subject No. 7 
 described some of his groupings as though grouping merely meant 
 counting to eight. At times, Bolton seems to consider that the 
 grouping is a temporal grouping, as when he writes as follows: 
 "The weaker or less accented sounds seem to run together with the 
 stronger, and to form organic groups which are separated from 
 one another by intervals which are apparently longer than the 
 interval which separates the individual clicks."^ But Bolton 
 nowhere tries to show how "the conscious state accompanying 
 each wave of attention" causes the impressions which it unifies 
 to appear separated by shorter intervals than the intervals separa- 
 ting two successive impressions which fall in different waves, i. e., 
 in different groups. 
 
 Meumann writes that for many observers the grouping was 
 always temporal, a temporal holding together, in which the mem- 
 bers of the groups appear to follow quicker upon each other, while 
 between every two groups lies a pause.* He regards grouping as 
 an intellectual act, which shows itself in the subordination of cer- 
 tain impressions to others, for instance, the subordination of the 
 more intense to the less intense. In verse, however, Meumann 
 
 * Amer. Journ. of Psychol., 6, 196, 1894. 
 ^ Ibid., p. 198. 
 
 ^ Ibid., p. 204. 
 
 * Phil. Stud., 10, 283 and 304, 1894. 
 
THE MEANING OF RHYTHMICAL GROUPING 55 
 
 says that grouping may be conditioned entirely by meaning.^ In 
 this case, then, rhythmical grouping means logical unity. 
 
 The position of McDougall seems to be that a rhythmical group 
 is an awareness of a temporal segregation of the impressions. "The 
 whole group of elements constituting the rhythmic unit is present 
 to consciousness as a single experience; the first of its elements 
 has never fallen out of consciousness before the final member ap- 
 pears, and the awareness of intensive differences and temporal 
 segregation is as immediate a fact of sensory apprehension as is 
 the perception of the musical qualities of the sounds themselves. "^ 
 At other times, he seems to regard the grouping as the experiencing 
 of impressions along with the experience of an ideal form, or 
 ''Gestaltsqualitat," as when he speaks of the experience of rhythm 
 as being supported by the conception of an ideal form which the 
 series of stimuli fulfils.^ He says further that the synthesis of 
 elements may be mediated by changes in the ideal significance 
 and relation of the various members; also by movements of the 
 head, jaw, throat, eyes, or by muscular strain;^ but he does not 
 show how these changes in ideal significance or these movements 
 of various parts of the body produce temporal segregation. 
 
 The view has been put forth by several recent authors that the 
 rhythmical group consists in the experience of muscular strains 
 along with the experience of the other impressions, such as 
 a series of sounds. According to this view, a rhythmical group 
 may be thought of as a series of impressions strung together by 
 one longer state of strain, much like fish strung together on a string; 
 as when Miner speaks of the separate sensations from the external 
 world as being strung in groups. Stetson represents this view 
 when he says that "the continuity of the rhythmic series, whereby 
 all the beats of a period seem to belong to a single whole, is due to 
 the continuity of the muscle sensations involved and the continuous 
 feeling of slight tension between the positive and negative muscle 
 sets;^ nowhere within the period does the feeling of strain die 
 out." Similarly, Miner says that "feeling the groups to be units 
 is an illusion due to the presence of movement or strain sensations 
 along with the sensations that are grouped." "These kinaes- 
 thetic sensations provide the factor by which unit sensations ap- 
 
 ^ Phil. Stud., 10, 396, 1894. 
 
 ^ Harvard Psychol. Stud., i, 1903 and Monog. Sup. Psychol. Rev., 4, 322, 
 1903. 
 
 3 Ibid., p. 468. 
 
 * Ibid., p. 343. 
 
 ^ Harvard Psychol. Stud., i, 1903 and Monog. Sup. Psychol. Rev., 4, 455, 
 1903. 
 
56 A QUANTITATIVE STUDY OF RHYTHM 
 
 pear bound into groups."* In speaking of visual rhythms, how- 
 ever, he says: "The units in the group seem crowded closer to- 
 gether and a longer interval appears before the next group starts."^ 
 This indicates that the grouping is temporal. We might expect, 
 therefore, some attempt on the part of the author to show that 
 strain sensations, which are said to hold the impressions together, 
 cause an underestimation in time; he makes no such attempt. And 
 while it might possibly be held that strain sensations and the per- 
 ception of time are identical, such a theory would rather lead us to 
 expect the strain sensations to force the separate impressions apart 
 from each other rather than hold them together, because accord- 
 ing to any theory which makes strain sensations the basis of the 
 perception of time, the greater the strain sensations during any 
 interval, the greater the apparent duration of that interval. 
 
 As I have already stated, in the present research in all the work 
 done on the effect of duration and intensity on rhythm, the sub- 
 jects were instructed to judge concerning the rhythm, and noth- 
 ing was said to them concerning intervals. In order to obtain 
 some idea of the nature of the rhythmical group, however, after 
 the work on rhythm was all over, I carried on some investigations 
 on the effect of the same variations in intensity and duration that 
 I had used on rhythm on the apparent length of the intervals. 
 The indifference points of which I have spoken in the chapters 
 on intensity and duration were indifference points for rhythm. 
 This rhythm indifference point should not be confused with what 
 is spoken of as the indifference point in the comparison of inter- 
 vals. The instructions given the subject and the experimental 
 procedure used in obtaining the various rhythm indifference points 
 have already been described. The procedure for obtaining the 
 point at which all the intervals appear equal, that is, for obtaining 
 the indifference points in the comparison of intervals, was in every 
 respect the same as that used in the study of the indifference point 
 for rhythm; but the instructions given the subject were different. 
 In obtaining the indifference point for intervals, the subject was 
 asked to ignore the rhytiim, and (every second sound being more 
 intense or longer) to judge merely whether the interval preceding 
 the louder or the longer sound was greater, equal, or less than that 
 following it. 
 
 When there is much of a difference in the duration of the sounds, 
 the subject's idea of what is meant by intervals will evidently have 
 a great influence on the results obtained. If the word interval 
 
 ^ Monog. Sup. Psychol. Rev., 5, No. 4, pp. 2, 20, 1903. 
 2 Ibid., p. 5.5. 
 
THE MEANING OF RHYTHMICAL GROUPING 57 
 
 means to the subject the time between the beginnings of the sounds, 
 the indifference point for the comparison of intervals will be quite 
 different from what is obtained when the subject is asked to con- 
 sider the intervals which exist between the sounds. No instruc- 
 tions were at first given to any of the subjects concerning what 
 they should consider to be the interval, as it was desired to see 
 what they did naturally. Rather curiously none of the subjects 
 seemed in any doubt as to what intervals to compare. The possi- 
 bility that interval might mean anything but one thing never oc- 
 curred to them, unless it was pointed out to them; and yet one of 
 the subjects, Ws, started in to compare the intervals between the 
 beginnings of the sounds, while three others compared the inter- 
 vals of silence between the sounds. In all four subjects the results 
 obtained by the comparison of intervals, leaving the subject to 
 follow his own ideas concerning the meaning of intervals, gave as 
 the indifference point for the comparison of intervals in any given 
 series of sounds practically the same point which had been pre- 
 viously obtained as the rhythm indifference point for the same 
 series. As long as the subjects compared intervals in the way 
 which seemed most natural to them, their results for the interval 
 indifference point and the rhythm indifference point corresponded 
 very closely; but when subject Ws was asked to compare actual 
 intervals, results were obtained very different from the results ob- 
 tained on the rhythm indifference point; and when subjects Dy, 
 Ww, and Wr compared the intervals between the beginnings of 
 the sounds, the interval indifference points thus obtained diverged 
 widely from the corresponding rhythm indifference points. By 
 "before" and "after" in the following table is meant, as usual, 
 the actual intervals of silence before and after the longer sound. 
 
 
 
 TABLE XII 
 
 
 
 
 Duration of shorter sound (constant) = o 
 
 13 second. 
 
 Total duration of 1 
 
 = 1.5 seconds. 
 
 
 
 
 
 
 
 
 r sounc 
 
 Intervals at indifference p 
 
 oint 
 
 N'&turc of iii~ 
 
 Subject I,onge 
 
 I Before 
 
 After 
 
 difference point 
 
 Wr 
 
 60 
 
 •44 
 
 
 33 
 
 
 rhythmic 
 
 Wr 
 
 60 
 
 
 46 
 
 
 31 
 
 
 silence 
 
 Wr 
 
 60 
 
 
 55 
 
 
 22 
 
 
 beginnings 
 
 Ws 
 
 60 
 
 
 54 
 
 
 23 
 
 
 rhythmic 
 
 Ws 
 
 28 
 
 
 55 
 
 
 54 
 
 
 rhythmic 
 
 Ws 
 
 28 
 
 
 54 
 
 
 55 
 
 
 beginnings 
 
 Ws 
 
 28 
 
 
 45 
 
 
 64 
 
 
 silence 
 
 Ww 
 
 37 
 
 
 51 
 
 
 49 
 
 
 rhythmic 
 
 Ww 
 
 37 
 
 
 51 
 
 
 49 
 
 
 silence 
 
 Ww 
 
 37 
 
 
 5§ 
 
 
 .42 
 
 
 beginnings 
 
58 
 
 A QUANTITATIVE STUDY OF RHYTHM 
 
 Results are presented for three different sorts of indifference point, 
 the iambic-trochaic indifference point, the indifference point for 
 the comparison of the actual intervals of silence between the end 
 of one sound and the beginning of the following sound, and the 
 indifference point for the comparison of the intervals between the 
 beginnings of the sounds. These three sorts of indifference point 
 are referred to in the column headed ''nature of indifference point," 
 as "rhythmic," "silence," and "beginnings." 
 
 The comparison of intervals thus affords a method of determin- 
 ing what intervals are the most significant in rhythm, for, evidently, 
 if comparing intervals in one way gives results the same as obtained 
 in judging rhythm, while comparison in a second way gives widely 
 different results, the process of judging of the rhythm must be more 
 closely allied to the first way of comparing intervals than to the 
 second. 
 
 TABLE XIII 
 
 
 ] 
 
 ntensity of 
 
 weaker sc 
 
 und (cc 
 
 )nstant) = 
 
 = 24 feet. 
 
 
 
 
 
 o-a 
 
 
 v> 
 
 3 V 
 
 
 
 
 
 
 
 
 
 15' 
 
 «J3 
 M 
 
 
 Indifference point 
 for interval 
 comparison 
 
 
 Indifference point 
 
 for rhythm 
 (iambic-trochaic) 
 
 
 3 
 
 CO 
 
 N 
 
 M V Before After Before After 
 
 N 
 
 MV 
 
 Ww 
 
 136 
 
 ■13 
 
 19 
 
 ■037 
 
 •57 
 
 .66 
 
 .55 .68 
 
 21 
 
 .015 
 
 Ww 
 
 136 
 
 .22 
 
 15 
 
 .026 
 
 •94 I 
 
 . II 
 
 90 I 
 
 16 
 
 10 
 
 .022 
 
 Ww 
 
 136 
 
 ■31 
 
 10 
 
 ■033 I 
 
 31 I 
 
 .56 I 
 
 29 I 
 
 58 
 
 10 
 
 .021 
 
 Ww 
 
 136 
 
 •45 
 
 16 
 
 .028 I 
 
 92 2 
 
 17 I 
 
 91 2 
 
 19 
 
 14 
 
 .041 
 
 Ww 
 
 136 
 
 68 
 
 16 
 
 ■055 2 
 
 93 3 
 
 22 2 
 
 97 3 
 
 18 
 
 6 
 
 .104 
 
 Ww 
 
 136 
 
 07 
 
 14 
 
 .008 
 
 30 
 
 32 
 
 26 
 
 36 
 
 10 
 
 .010 
 
 Ww 
 
 136 
 
 09 
 
 H 
 
 .Oil 
 
 36 
 
 46 
 
 37 
 
 46 
 
 12 
 
 .006 
 
 Ws 
 
 28 
 
 13 
 
 II 
 
 .018 
 
 62 
 
 62 
 
 61 
 
 62 
 
 20 
 
 .022 
 
 Ws 
 
 32 
 
 13 
 
 10 
 
 .018 
 
 60 
 
 63 
 
 59 
 
 64 
 
 20 
 
 .012 
 
 Ws 
 
 70 
 
 13 
 
 10 
 
 .018 
 
 60 
 
 63 
 
 58 
 
 65 
 
 20 
 
 .021 
 
 Ws 
 
 196 
 
 13 
 
 15 
 
 .010 
 
 60 
 
 63 
 
 57 
 
 66 
 
 56 
 
 ■02 5 
 
 Ws 
 
 300 
 
 13 
 
 10 
 
 .012 
 
 58 
 
 65 
 
 57 
 
 66 
 
 41 
 
 .021 
 
 Ws 
 
 1 100 
 
 13 
 
 10 
 
 .015 
 
 56 
 
 67 
 
 55 
 
 68 
 
 38 
 
 .018 
 
 Wr 
 
 136 
 
 13 
 
 18 
 
 .010 
 
 58 
 
 65 
 
 56 
 
 68 
 
 10 
 
 .010 
 
 Wr 
 
 1 100 
 
 13 
 
 II 
 
 .016 
 
 54 
 
 69 
 
 51 
 
 72 
 
 17 
 
 .016 
 
 Dy 
 
 136 
 
 13 
 
 19 
 
 .019 
 
 58 
 
 65 
 
 51 
 
 72 
 
 15 
 
 .016 
 
 Dy 
 
 420 
 
 13 
 
 14 
 
 .024. 
 
 48 
 
 75 
 
 50 
 
 73 
 
 10 
 
 .014 
 
 The results obtained on the indifference point for the comparison 
 of intervals, are shown in Tables XIII and XIV, in which are also 
 given the results on the rhythm indifference point for the same 
 ratios of intensities and durations of the sounds. In Table XIII, 
 results are presented for sound series in which all the sounds of a 
 series were of the same length but every other sound the louder. 
 
THE MEANING OF RHYTHMICAL GROUPING 
 
 59 
 
 In Table XIV results are presented for series in which the sounds 
 were of equal loudness but every other sound the longer. In the 
 columns headed "before" and "after" are given the duration, 
 at the indifference point, of the intervals before and after the longer 
 or the more intense sound. 
 
 TABLE XIV 
 Intensity of all sounds = 24 feet. 
 
 
 
 15' 
 
 <n 
 
 
 
 01 
 
 u 
 
 Indifference point 
 for interval 
 comparison 
 
 
 
 
 [ndifference point 
 
 for rhythm 
 iambic-trochaic) 
 
 
 
 N MV Before After 
 
 Before 
 
 After 
 
 N 
 
 m'v 
 
 Ws 
 
 28 
 
 13 
 
 10 .oop 
 
 53 
 
 56 
 
 -55 
 
 ■54 
 
 24 
 
 .020 
 
 Ww 
 
 37 
 
 13 
 
 15 -015 
 
 51 
 
 49 
 
 
 51 
 
 
 49 
 
 15 
 
 .016 
 
 Ww 
 
 61 
 
 22 
 
 12 .022 
 
 76 
 
 90 
 
 
 75 
 
 
 91 
 
 12 
 
 .018 
 
 Ww 
 
 86 
 
 30 
 
 8 .028 
 
 98 I 
 
 34 
 
 I 
 
 GO 
 
 I 
 
 32 
 
 14 
 
 .028 
 
 Ww I 
 
 23 
 
 43 
 
 6 .033 I 
 
 53 I 
 
 80 
 
 I 
 
 47 
 
 I 
 
 86 
 
 6 
 
 .028 
 
 Ww I 
 
 65 
 
 65 
 
 6 . 030 2 
 
 35 2 
 
 65 
 
 2 
 
 21 
 
 2 
 
 79 
 
 6 
 
 ■037 
 
 *Dy 
 
 37 
 
 13 
 
 12 .019 
 
 43 
 
 57 
 
 
 40 
 
 
 60 
 
 10 
 
 ■045 
 
 Wr 
 
 19 
 
 13 
 
 10 .010 
 
 58 
 
 60 
 
 
 58 
 
 
 60 
 
 II 
 
 .Oil 
 
 Wr 
 
 25 
 
 13 
 
 9 .010 
 
 54 
 
 58 
 
 
 54 
 
 
 58 
 
 16 
 
 .018 
 
 Wr 
 
 31 
 
 13 
 
 15 .077 
 
 52 
 
 54 
 
 
 53 
 
 
 53 
 
 17 
 
 .oil 
 
 Wr 
 
 43 
 
 13 
 
 15 .018 
 
 47 
 
 45 
 
 
 51 
 
 
 43 
 
 17 
 
 .023 
 
 Wr 
 
 60 
 
 13 
 
 13 .027 
 
 46 
 
 31 
 
 
 44 
 
 
 33 
 
 19 
 
 .024 
 
 *In 
 
 this ca 
 
 36 the 
 
 longer sound ha 
 
 d an int 
 
 ensity 
 
 of 
 
 70 
 
 feet. 
 
 
 
 
 The results show that the absolute and relative durations of the 
 intervals at the iambic-trochaic indifference point are almost the 
 same as when the two intervals (the intervals before and after the 
 louder or the longer sound) appear to be equal in length. The 
 rhythm indifference points and the interval indifference points are 
 almost, but not entirely, identical. If we term the differences 
 in the durations of the intervals before and after the louder or the 
 more intense sound constant errors, then it may be said that con- 
 sidering both Tables XIII and XIV together, most frequently the 
 constant errors are less when the subject judges intervals than 
 when he judges rhythm. In the case of series in which the sounds 
 present only duration differences, however, there seems to be no 
 reliable difference, as in Table XIV the constant errors for rhythm 
 are in five cases greater, in three cases the same, and in four cases 
 less than the constant errors for comparison of intervals. In Table 
 XIII, however, it is fairly evident, perhaps, that the constant er- 
 rors due to differences in the intensity of the sounds are usually 
 slightly greater for the rhythm indifference points than for the 
 interval indifference points, as the constant errors for the rhythm 
 
60 A QUANTITATIVE STUDY OF RHYTHM 
 
 judgments are greater than those for the interval judgments in 
 fourteen cases and less in three cases. The two sets of results 
 are not exactly identical, but they are so close to it that the con- 
 clusion seems fairly safe that they are the result of mental opera- 
 tions M^hich have about the same basis. In other words, the state- 
 ment by the subject, that certain sounds form an iambic group, 
 is equivalent to the statement that he has perceived a shorter in- 
 terval before the louder or the longer sound than after it, and the 
 statement that the sounds form a trochaic group means that the 
 subject has perceived a shorter interval after the louder or longer 
 sound than before it. This is the equivalent of saying that the 
 experiencing of a rhythmical group is an experiencing of temporal 
 relations, that the meaning of rhythmical grouping is temporal 
 segregation. This conclusion in no way implies that the subject 
 makes judgments concerning intervals whenever he hears rhythm, 
 but merely that he perceives these intervals. Nor does it imply 
 that in comparing intervals one necessarily judges of rhythm; 
 nor possibly, even, that he has experienced rhythm, since it has 
 never been proved that the immediate experience of a temporal 
 grouping is of itself all that is necessary for the experience of rhythm. 
 In the hope of making clearer the relation between the percep- 
 tion of a temporal grouping and the perception of rhythm, I cite 
 a few introspections. They are quoted literally, and are to be 
 treated as data requiring interpretation, and not accepted at their 
 face value. Subject Br: "The rhythmical grouping seems a tem- 
 poral grouping without attention being paid to the temporal rela- 
 tions at the time the rhythm is heard." Subject Wr was asked, 
 after his first hour of work in comparing intervals, which inter- 
 vals he had been comparing, the intervals between the beginnings 
 of the sounds or the intervals of silence between the sounds. He 
 replied that he did not know which he had been doing. Subject 
 Dy: "Rhythm is not always present in comparing intervals, but 
 it sometimes comes. At such times it seems difficult to judge in- 
 tervals without the rhythm being the predominating factor. The 
 rhythm I notice in comparing intervals is not always the same as 
 when I put my attention on the rhythm itself, e. g., ii I get a tro- 
 chaic rh5^thm while judging intervals, and then turn my attention 
 to the rhythm I sometimes get iambic instead. An iambic never 
 changed to a trochaic in this way." Ws: "Rhythm becomes ap- 
 parent to the ear by unequal lengths of sound. The short inter- 
 vals come between the sounds which make the rhythm and the 
 long intervals between the groups of sounds." At another time, 
 "When the loudest and longest note comes first, I put down tro- 
 
THE MEANING OF RHYTHMICAL GROUPING 61 
 
 chaic; but when it comes at the end of each rhythm, I put down 
 iambic. I do not notice the length of intervals especially." Same 
 subject, summer, '07: "When there is absolutely no difference in 
 the length of intervals, the rhythm is trochaic." Summer, '08: "In 
 comparing the intervals, the intervals are equal in iambic rhythm." 
 After an experiment in which this same subject was asked to judge 
 in the case of each sound series, first, whether the rhythm was 
 iambic, doubtful, or trochaic, and second, whether the intervals 
 were equal or unequal, and if unequal whether the interval before 
 or after the louder sound was the longer. "In keeping a record of 
 both rhythms and intervals I went through two different processes 
 of judgment; the two judgments did not seem alike. I had never 
 before realized that the doubtful rhythms were those where the 
 intervals are equal." 
 
 The very slight difference already noted as existing between the 
 interval indifference point and the indifference point for rhythm 
 does not seem large enough nor reliable enough to invalidate the 
 conclusion that the grouping referred to when we speak of rhythmical 
 grouping is a temporal grouping. What slight tendency there is 
 for constant errors (in the sense used above) to be smaller when the 
 subject judges of intervals than when he judges of rhythm seems 
 to me to indicate merely the effect of the different attitude on the 
 part of the subject in the two cases, the difference being, to judge 
 from my own introspection, that, in comparing intervals, both 
 sounds are about evenly attended to, there is a comparatively 
 continuous strain, while in judging rhythm, some of the sounds, 
 e. g., the more intense, are attended to more than others. Such a 
 view enables us to understand the statement of Meumann that his 
 subjects only judged the interval between the groups as longer 
 than the intervals within the group, when they did not set out espe- 
 cially to judge intervals ("Wenn sie nicht darauf ausgehen, die 
 Intervalle bewusst zu vergleichen."^) 
 
 We are now in a position to understand in what consists the 
 grouping effect of duration and intensity, and consequently their 
 rhythmical effect in so far as rhythm consists in grouping. A regu- 
 larly recurring more intense sound has the effect of increasing the 
 temporal value of the interval preceding it as compared with that 
 following it. This means that if the intervals are equal the sounds 
 appear temporally grouped in such a way that the interval pre- 
 ceding the more intense sound separates the group, that is, that 
 the sounds, if heard as rhythm, produce the impression of trochaic 
 rhythm. This effect of the more intense sound may therefore be 
 
 ^ Phil. Stud., 10, 304, 1894. 
 
62 A QUANTITATIVE STUDY OF RHYTHM 
 
 spoken of as trochaic. The regularly recurring more intense sound 
 continues to exert this effect even though, because of an objective 
 shortening of the interval preceding the more intense sound, the 
 rhythm is heard as iambic. The more intense sound is exerting 
 an influence towards trochaism even when it occurs in an iambic 
 rhythm. Similarly, if the distinction between the duration effect 
 per se and the intensity effect of an increase in duration be admitted, 
 the results obtained on duration show that a regularly recurring 
 shorter sound also exerts a trochaic influence, and this, too, whether 
 the rhythm is heard as iambic or trochaic. In referring to the 
 trochaic effects of the shorter and the londer sounds, I am speak- 
 ing relatively. I mean that the effect of the regularly recurring 
 shorter or louder sound is trochaic compared to the effect of the 
 weaker or the longer sound. It is equally true, of course, if we are 
 speaking relatively, to say that an increase in duration or a decrease 
 in intensity exerts an iambic effect, or causes an underestimation, 
 of the interval preceding the longer or the weaker sound. 
 
SUMMARY 
 
 It is possible to pass from one rhythmical grouping to another 
 by changing the relative duration of the intervals between the 
 sounds. Thus, a trochaic rhythm, that is, one that is composed 
 of groups of two sounds each, the louder sound beginning the group, 
 may be changed to an iambic rhythm, one in which the louder 
 sound ends the group, by increasing the interval immediately 
 following the louder sound or by decreasing the interval immediately 
 preceding it. Similarly, a rhythmical group which begins with a 
 sound longer than the other sounds of the group may be changed 
 to one in which the longer sound ends the group by increasing the 
 interval immediately following the longer sound, or by decreasing 
 all the other intervals. As we pass in this way from rhythmical 
 groups beginning with the louder or longer sound to rhythmical 
 groups ending with that sound, we pass through a zone where the 
 tendency towards the two forms of grouping is equally strong. 
 The middle point of this zone may be termed the rhythm indifference 
 point. If the rhythm indifference point occurs where all the inter- 
 vals are objectively equal, then any differences which may exist 
 in the objective duration or loudness of the sounds are obviously 
 not exerting any effect towards grouping. But if, when the intervals 
 between all the sounds are objectively equal, grouping is still per- 
 ceived, the grouping must be regarded as brought about by other 
 factors than objective differences in the intervals. The amount 
 of this grouping effect can be determined by finding out what change 
 from objective equality of intervals is necessary in order to cause 
 the grouping to disappear, that is, by ascertaining the amount by 
 which one of the intervals has to be lengthened or shortened with 
 respect to the others in order to arrive at the rhythm indifference 
 point. We may say, therefore, that the influence towards rhyth- 
 mical grouping exerted by factors other than objective differences 
 in the intervals, that is, by such factors as recurrent differences 
 in accent, duration, pitch, etc., is measured by the difference, at the 
 rhythm indifference point, between the external and internal in- 
 tervals of the group. 
 
 When the intervals are equal, and every second stimulus the 
 stronger, the rhythm is trochaic, and when every third is the stronger, 
 dactylic. That is, a regularly recurring difference in intensity 
 exerts a tendency towards rhythmical groups with the more intense 
 sound at the beginning. In other words, accenting certain sounds 
 of a series has the same effect on the position of those sounds within 
 
64 A QUANTITATIVE STUDY OF RHYTHM 
 
 the rhythmical group as objectively increasing the interval preceding 
 them. For instance, if the sounds corresponding to the odd numbers 
 of an equally spaced series be accented, those sounds appear to 
 begin the groups; and similarly, without accenting them, but by 
 sufficiently increasing the interval immediately preceding them, 
 they may be made to begin the groups. 
 
 This trochaic or dactylic effect caused by an increase in the rela- 
 tive intensity of every second or every third sound, that is, the 
 tendency of the accented sound to begin the rhythmical group, 
 may be measured by the amount by which the interval immediately 
 following the stronger stimulus has to be increased in order to arrive 
 at the rhythm indifference point: and in this way the effect on 
 rhythm of variations in the relative intensity of the sounds may be 
 studied. With an increase in the ratio of the intensity of the louder 
 sound to that of the weaker, there is an increase, first rapid and then 
 slow, in the tendency of the more intense sound to begin the group. In 
 other words, with equal intervals, as the difference in intensity 
 between the louder and weaker sounds increases, the intensity of 
 the weaker sound remaining constant, the rhythm becomes more 
 and more trochaic, if composed of two-membered groups, or more 
 and more dactylic, if composed of three-membered groups. 
 
 I have stated that the rhythmical effect exerted by regularly 
 recurrent accents is measured by the difference between the in- 
 ternal and external intervals of the group at the indifference point. 
 This is an absolute measure. By dividing this by the total duration 
 of one measure we get a relative measure, that is, a measure of the 
 rhythmical effect relative to the total duration of one measure. 
 In comparing the rhythmical effect of intensity, duration, etc., in 
 sound series which are run off at different rates, the relative measure 
 is what should be taken into consideration. The rhythmical effect 
 of any given ratio of intensities between the louder and weaker 
 sounds, relative to the rate at which the series is run off, remains 
 constant, in two-group rhythms, for rates varying approximately 
 from one to four seconds for one measure, usually shows a marked 
 decrease by the time a rate of seven seconds for one measure is 
 reached, but in some cases does not entirely disappear at a rate of 
 ten seconds for one measure. 
 
 Measurements of the rhythmical effect of changes in the relative 
 and absolute duration of sounds, made by the same method as that 
 used in the case of intensive differences, lead to the following gen- 
 eralizations. With an increase in the ratio of the duration of the 
 longer sound to that of the shorter, there is an increase in the tendency 
 of the longer sound to end the group or a decrease in its tendency to 
 
SUMMARY 65 
 
 hegin the group. When the ratio of the duration of the longer sound 
 to the duration of the shorter is small, that is, when there is not 
 much difference in the duration of the sounds, and when, further, 
 the absolute duration of the sounds is also small, the longer sound 
 tends to begin the group. A small regularly recurrent increase in 
 duration, then, may have the same effect, providing the absolute 
 duration of the sounds is small, as a regularly recurrent difference 
 of accent. But whereas an increase in relative intensity has the 
 effect of increasing the tendency of the accented sound to begin 
 the group, an increase in relative duration has the effect of decreasing 
 the tendency of the longer sound to begin the group, and often 
 results in an exceedingly strong tendency on the part of the longer 
 sound to end the group. 
 
 If all intervals are kept equal, and every second sound is some- 
 what longer, we may have a trochaic rhythm. This seems at first 
 sight to indicate that the effect of a slight increase in duration is 
 a trochaic tendency whereas the effect of a considerable increase 
 is an iambic tendency. It is necessary, however, to make a dis- 
 tinction between duration per se and the increase in apparent in- 
 tensity of a stimulus due to an increase in duration. The apparent 
 intensity increases as the duration, at first fast but later very slowly. 
 The difference in the apparent intensity of a sound one second in 
 duration and of a sound two seconds in duration is small, very 
 small compared to the difference in duration. The difference in 
 apparent intensity of a sound one-fiftieth of a second in duration 
 and a sound two-fiftieths of a second in duration is very great, 
 even in comparison with the increase in apparent duration. As 
 we increase the duration of every second sound, therefore, we have 
 two separate and antagonistic effects to keep in mind. First, 
 there is an increase in the tendency of the longer sound to begin 
 the group, due to the effect of duration on apparent intensity: 
 second, there is a much more rapid increase in the tendency of the 
 longer sound to end the group, due to the effect of an increase in 
 duration per se. The increase in the second tendency is so much 
 faster than that in the first, that the second may overcome the first, 
 when the increase in duration is great, even though the first ten- 
 dency may have been the stronger when the increase in duration 
 was small. 
 
 The difference in the apparent intensity between two sounds 
 due to a difference in their duration decreases, compared to the 
 difference in apparent duration, as the absolute duration of the 
 sounds increases. As the absolute duration of the sounds is in- 
 creased, then, the trochaic effQct of duration due to the effect of 
 
66 A QUANTITATIVE STUDY OF RHYTHM 
 
 duration on intensity should decrease relatively to the iambic effect 
 of duration as such. And measurements show that, in fact, with a 
 constant ratio between the durations of the sounds, as their absolute 
 duration increases, there is a decrease in the tendency of the longer 
 sound to begin the group or an increase in its tendency to end the group. 
 The effect of both intensity and duration in rhythm may be 
 generalized as follows. If every second or third sound is made 
 more intense or is made shorter, the effect on grouping is the same 
 as if the interval immediately preceding that sound were increased 
 relative to the other intervals. The effect of the more intense 
 sound, when all the sounds are of equal duration, or of the shorter 
 sound, when all the sounds are of equal intensity, is a relative over- 
 estimation of the interval preceding the more intense or the shorter 
 sound. There is an objection to speaking of an overestimation 
 of any interval, however, in that the subjects in the experiments 
 so far considered were not estimating intervals, but were judging 
 rhythm. But when the subjects were instructed to estimate in- 
 tervals, it was found that, in fact, the interval preceding the regularly 
 recurrent more intense sound or the regularly recurrent shorter sound 
 is relatively overestimated. The rhythm indifference point and the 
 indifference point for the estimation of intervals are almost, though 
 not exactly, identical. This close correspondence between the 
 rhythmical grouping and the temporal grouping, or rather this 
 correspondence in the points where both disappear, indicates that 
 rhythmical grouping is a temporal grouping; that is, that rhythmical 
 grouping is determined by the duration of the subjective intervals, 
 not by the objectively measurable intervals, but by the subject's 
 consciousness of these intervals, that is, by the intervals considered 
 as mental magnitudes. 
 
VITA 
 
 The author was born in ChilHcothe, Ohio, February 25, 1883. 
 Matriculated in the University of Michigan 1900; A. B., 1904. 
 Matriculated in the University of Paris, Faculte des Lettres, 1904. 
 Demonstrator in Experimental Psychology, Princeton University, 
 1907. Lecturer in Psychology in Columbia University, 1907-1909; 
 appointed Tutor, 1909. For inspiration and encouragement the 
 author is indebted to Professors Pillsbury and liombard and Dr. 
 Shepard of the University of Michigan, to Professor G. Dumas and 
 ■the late Professor Egger of the Sorbonne, Paris, and to Professors 
 Cattell, Lee, Woodworth, and Thorndike of Columbia University. 
 
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