V I Cornell University Library MT 50.S44 1880 The correct order of fundamental harmoni 3 1924 022 305 365 JJlhara, Nem $ork FROM THE BENNO LOEWY LIBRARY COLLECTED BY BENNO LOEWY 1854-1919 BEQUEATHED TO CORNELL UNIVERSITY c Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924022305365 e ^*%* THE CORRECT ORDER OF FUNDAMENTAL HARMONIES: A TREATISE Jfmrtramttttal passes, anfo %ir Jnfcndons aito JMstitotes. FROM THE GERMAN OF S. SECHTER, LATE IMrERlAL COURT-ORGANIST, AND PROFESSOR OF HARMONY IN THE CONSERVATORY OP MUSK AT VIENNA. THE WHOLE COMPILED AND ADAPTED FOR POPULAR USE By C. C. MULLER. FOURTH EDITION. NEW YORK: Published by WM. A. POND & CO., 25 Union Square. 1880. Entered according to Act of Congress, in the year 1871, by C. C. MULLER, 4n the Office of the Librarian of Congress, at Washington. PREFACE. The extensive patronage which is extended to the Musical Art by the American Public, and which has encouraged so many to apply themselves to the noble study of Musical Composition, may be pleaded as an apolocy for the appearance of a new treatise on Harmony. The present work, which has been compiled, and adapted for popular use, from the celebrated treatise by S. Sechter, entitled : Die richtige Folge der Grundharmonien, etc., aims solely to elucidate the formation of harmonic progressions, and to present the means of modu- lation ; leaving untouched, for the present, the subjects of Melody and Counterpoint. The treatise begins with the discussion of the harmonic progressions in the diatonic major scale. Next is shown the peculiar nature of the diatonic minor scale, and the means of diatonic modulation are pointed out. Hereupon follow, the elucidation of the chromatic progressions, both in the major and minor scales ; and lastly, that of the enharmonic changes. The circumstance that Sechter's work was intended chiefly for teachers and advanced students, necessitated, in this translation, certain deviations from the original, especially in Part I, such as, the insertion of the sections elucidating the elementary principles of harmony, involv- ing, again, a different disposition of the subject matter, with a view to 4 PREFACE. the most gradual development, and to the presentation in the clearest manner possible, of Sechter's system. Moreover, at the ends of Parts I and II, the student will find, among other directions, a table suggestive of exercises for practice, applicable also to Parts IV and V ; an addition which, it is hoped, will compensate for the omission of many repetitions, redundant examples, etc., which was deemed advisable in a work intended for popular use. The compiler would here acknowledge with the deepest gratitude the valuable assistance which Mr. J. H. Cornell has rendered him in the translation of Sechter's work. It will be in great part owing to his interest in the Treatise, his knowledge of the subject, and his indefatigable and conscientious pains-taking, if this work, the fruit of much love and many sacrifices, should meet with the success which the name of Sechter ought to assure to it The compiler's thanks are also due to Mr. J. W. Taylor, the stereotyper, for his promptness and many courtesies (his typography, by the bye, speaks for itself) — also to Mr. Wm. A, Pond, the publisher, for the liberality to which the public are indebted for the getting up of the work. C C. MULLEB. New York, January i, 1871 PREFACE TO FOURTH EDITION. In introducing the Fourth Edition of m^ English adaptation of S. Sechter's Fundamental Harmony, I wish to thank the musical public for the kind attention they have bestowed on this work. Several corrections and alterations have been made in the new edition, all of which will, I hope, meet the approval of the patrons and friends of the work, to whom I confidently entrust its future success. C. C. Muller. New York, May 15, 1880. CONTENTS. PART I. OENERAL REMARKS, AND PROGRESSIONS IN THE DIATONIC MAJOR SCALE. I.— --Interval. . Half-step. Step, . . . II. — Scales. Major Scale, .... III. — Tones : Diatonic, Chromatic, Enharmonic, IV. — Intervals of the Major Scale, . V. — Figuring, VI. — Inversion of Intervals, .... VII. — Perfect System of Tuning in the Major Scale, VIII. — Consonances and Dissonances, IX. — -Melodic and Harmonic Passages, X. — Primary Chords, ..... XL— The Triad j Chord of the Sixch ; Chord of the Fourth and Sixth, XII.— The Chord of the Seventh; Chord of the Fifth" and Sixth; Chord of the Third and Fourth ; Chord of the Second, XIII.— The Chord of the Ninth, XIV. — Disposition and Names of the Voices, XV. — The different Positions of a Chord, .... XVI. — Different kinds of Motion, PACT »4 «$ 16 J8 »9 20 21 21 23 25 26 26 28 CONTENTS. SBCTION PACT XVII. — Exchanges, in the Triad, JO XVIII. — do. in the Chord of the Seventh, . . .31 XIX. — Passing Tones, . . ..... 32 XX. — Returning Motions, '34 XXI. — Passing Seventh, ....... 35 XXII. — Continuation of Section XXI, . . . . -35 XXIII. — Passing Ninth, 37 XXIV. — Regular Progressions of Component Tones of Chords, when the Fundamental descends by a Fifth, . . . -38 XXV. — Covered Fifths and Octaves, 39 XXVI. — Essential Qualities of a good Harmony, . . . .40 XXVII. — Disposition of the most important Triads, and Regular Progression of the Fundamentals, . . . . 41 XXVIII. — Closing Cadence (Dominant to Tonic), . . , .42 XXIX. — Ascent of the Fundamental by a Fifth or Third, . . 43 XXX. — Continuation of Section XXIX, 44 XXXI. — do. do. 44 XXXIL— do. do. 4S XXXIII. — Ascent of the Fundamentals by Degrees, ... 46 XXXIV. — Irregular Progressions of the Voices, . . -47 XXXV. — Descent of the Fundamentals by Degrees, ... 49 XXXVI. — Unprepared Entrances of the Dominant Chord of the Seventh, . . . . . . . . 51 XXXVII. — Unprepared Entrances of the Diminished Triad of the 7th Degree, . . . . . . . . . 51 XXXVIII. — Different Closing Cadences, 53 XXXIX.— Leading Tone, 53 XL. — Suspensions, ........ 53 XLI. — Delayed Resolution of the Ninth, . . . . . . 60 XLII. — Prolonged Delay of the Resolution of the Ninth. Chang- ing Tones, ........ 61 XLHI. — Progression of the Octave and Thirteenth to the Seventh and Fifth, 6t XLTV. — Irregular Passing Motions, ..... 6a CONTENTS. XLV. — Application of the Suspensions to the Requirements of the Measure, ......... 63 XLVI. — Anticipations, ........ 64 Supplementary Remarks, ....... 65 Table for Exercises, ....... 66 Harmonic Successions to the Triad and Chord of the Seventh of the 7th Degree, 66 PART II. HARMONIC PROGRESSIONS IN THE DIATONIC MINOR SCALE. Introduct9ry, ...... I. — Order of Successive Degrees of the Minor Scale, II. — New Intervals, ..... III. — Perfect System of Tuning for the Minor Scale, IV. — Triads of the Minor Scale, V. — Chords of the Seventh of the Minor Scale, VI. — Dominant Chord of the Seventh and Ninth, VII. — Chord of the Seventh of the Raised 7th Degree, VIII. — Exchanges, and Passing Tones, . IX. — Returning Motions, ..... X. — Various Passing Motions, . . . XI. — Fundamental Progressions, .... XII. — The Raised 6th and 7th Degrees as Fundamentals, ,\ill. — Closing Cadence, ..... XIV. — Natural Succession of Triads, .... XV. — Entrance of, and Succession to, Conditional Triads, XVI. — Preparation of the Chords of the Seventh, XVII. — Progressions arising from the Ascent of the Fundamentals Degrees, .- by 78 78 79 80 81 82 83 84 85 87 88 89 89 90 90 92 95 101 8 CONTENTS. section nam XVIII. — Continuation of Section XVII, 103 XIX. — Suspensions, ........ 106 Supplementary Remarks, .. . . . . .106 Table for Exercises, . . . . . . . 107 PART III. DIATONIC MODULATION. SECTION FAGS Introductory, . . . . . . . .108 I. — Recognition of a Triad as Tonic Triad, . . . 108 II. — Different Situations of the same Triad, . . . • . 109 III. — Modulations from C major into its Relative Scales, and back, no IV. — Modulation from A minor into its Relative Scales, and back, 112 V. — General Summary, . . . ... . .114 VI. — Modulations from C major into the other Major Scales, . 114 VII. — Modulation from A minor into the other Minor Scales, . 119 VIII. — Modulation connecting the extremes of the Table given in Section V, . . . . . . . . .124 IX. — Accelerated Resolutions of the Modulations, . . . 125 X. — Delayed Resolutions of the Modulations, . . . .126 XI. — Change of Scales required at the end and beginning of Periods, . . . . . . . . .127 XII. — Concluding Remarks, . . . . . . .127 PART IV. CHROMATIC PROGRESSIONS IN THE SCALE OF C MAJOR. »AG» I. — The Chromatic Scale of C major, 128 II. — Explication of the Chromatic Steps in the Scale of C major, 129 CONTENTS. ON III. — The Diatonic Tones of C major, as Fundamentals of Relative Scales, . . . . . . . .130 IV. — Chromatic Alterations of Diatonic Triads, Chords of the Seventh, and of the Seventh and Ninth, . . .130 V. — Different Triads of the same Degree . . . . 132 VI. — False Relation, . . • 133 VII. — Essential Condition for the formation of a Chromatic Har- mony, 134 VIII. — Returning Motions, 135 IX. — Chromatic Steps arising from the different Progressions of the Fundamental, . . . . . . .136 X. — Progressions arising from the Fundamental Descent by a Fifth, 139 XI. — Progressions arising from Two successive Fundamental De- scents by a Fifth, ....... 141 XII. — Progressions arising from Three successive Fundamental De- scents by a Fifth, ...... 143 XIII. — Successions of Dominant Harmonies, .... 145 XIV. — Fundamentals descending by a Minor Third, . . . 146 XV. — Continuation of Section XIV, 148 XVI. — Formation of Chromatic Chords of the Seventh and Ninth, XVII. — Resolution of Chromatic Chords of the Seventh and into a Triad Harmony, .... XVIII. — Further Chromatic Alterations, .... XIX. — Other Licenses in Chromatic Progression, . XX. — Complicated Chromatic Progressions XXI. — Diatonic Theme for Chromatic Variations, . XXII. — Organ-point, XXIII. — Concluding Remarks, Ninth 149 !5» »53 »54 156 l 57 1 6c 161 10 CONTENTS. PART V. CHROMATIC PROGRESSIONS IN THE SCALE OF A MINOR. SECTION PACB I. — Chromatic Scale of A minor, . . . . . .162 II. — Effects of the Variable Nature of the 6th and 7th Degrees, 162 III. — Chromatic Alteration of Diatonic Triads, Chords of the Seventh, and of the Seventh and Ninth, . . . 163 IV. — Diatonic Tones of A minor, as Fundamentals of Relative Scales, ......... 165 V. — Diatonic Progressions which admit no Chromatic Alter- ation, ......... 166 VI. — The most simple Chromatic Progressions in A minor, . 169 VII. — Progressions arising from Three successive Fundamental De- scents by a Fifth, ....... 170 VIII. — Exemplifications of Sections VI and VII, . . . 173 IX. — Use of the Natural, instead of the Raised 6th and 7th De- grees in the Fundamental, . . . . . .174 X. — Alteration of Triads, etc., into Dominant Triads, etc., . 177 XL — Formation of Chromatic Chords of the Seventh and Ninth, . 178 XII. — Direct and Delayed Resolutions of the above-mentioned Chords, . . . . . . . . .183 XIII. — First Theme. of Diatonic Triads for Chromatic Variations, . 184. XIV. — Second Theme, . . • . . . . . 187 XV. — Organ-point, .... . . . 189 PART VI. OF ENHARMONIC CHANGES. SECTION PACT I- — Nature of the Enharmonic Change. Enharmonic Circles, 190 II. — Shorter Enharmonic Circles, .... .104 CONTENTS. 11 SUCTION PAOB III. — Other Enharmonic Circles, ... .195 IV. — Enhaimonic Chords of the Seventh and Ninth, . . .197 V. — Modulations by means of Enharmonic Chords, . . 198 VI. — Delayed Resolution of the Ninth, applied to Enharmonic Modulation, ........ 199 VII. — Chromatic Chords of the Seventh and Ninth, applied to Enharmonic Modulation, . . " . . . . 200 VIII. — Alteration of Dominant into Chromatic Chords of the Seventh and Ninth, . ' . . . . . .201 IX. — Ascending Succession of Enharmonic Chords of the Seventh and Ninth, ........ 202 X. — The Suspension of the Thirteenth applied to Enharmonic Modulation, ..... 203 Concluding Remarks, . ..... 204 PART I. GENERAL REMARKS, AND PROGRESSIONS IN THE DIATONIC MAJOR SCALE. SECTION I. A treatise on harmony properly commences with the definition of /Intervals. I The term interval expresses that relation in which one tone stands to another with regard to the difference of pitch. Note. — We use the word tone in its strict sense, as signifying a sound, not an interval. Instead of the word tone, as generally, but incorrectly used, we adopt the expression ivhole step, or simply step. The interval is directly determined according to the respective positions of the two tones upon the staff. Thus, we have two tones, each written upon a different degree of the staff, the lower of which is generally reckoned as standing, for the time being (i. e. in respect to the higher tone), upon the first degree. The relation of the higher to the lower tone — in other words, the interval- — is determined by the number of degrees involved in their dif- ference of pitch. We have, for instance, the two tones, D and the G above it. Counting D as one, we pass through the degrees E (two) and F (three), till we arrive at G, four. G is therefore a fourth in respect to D below it The smallest interval is that which forms a half-step. The half-step is either chromatic or diatonic. The chromatic half-step is a progression from one tone to another, both of which are written upon the same degree (a). The diatonic half-step is a 14 THE CORRECT ORDER OF progression from one tone to another, one of which is written on the degree immediately above the other (6), e. g. . (a) Chromatic Half-steps. (b) Diatonic Half-steps. fe|^^"$ "^2S7- p. iillli etc. A step is a progression from one tone to another, comprising a chromatic and a diatonic half-step, e. g. : ^ &+ e> *£> p & h w-fr^ --Ji-e-t-fyz- iiii etc SECTION II. An ascending or descending intermingled succession of five steps and two diatonic half-steps constitutes a scale, e. g. : * Step. Step. Step. Step. Step. Step. Step . Step . Step . Step . -ISE 3=E zp Half-step. Half-step. Half -SUp. Hdlf-step. The most simple scale, characterized by the major third and major sixth, is that which ascends from the ist to the 2d degree, one step ; from the 2d to the 3d degree, one step ; from the 3d to the 4th, one diatonic half-step ; from the 4th to the 5th, one step ; from the 5th to the 6th, one step ; from the 6th to the 7th, one step ; and from the 7th to the 8th, one diatonic half-step, and descends again in the same order, e. g. . % Half- Half- Step. Step. step. Step. Step. Step. step. Half- Half- step. Step. Step. Step. step. Step. Step. Degree: 1 2 -*—Jr This scale, in which the two diatonic half-steps occur between the 3d and 4th, and between the 7th and 8th degrees, respectively, is called the major scale, to the discussion of which this first part is exclusively devoted. That major scale which begins with C, as being the simplest, requiring no flats or sharps in its construction, will be taken as the model scale, in which all the examples in this work will be given. FUNDAMENTAL HARMONIC. 10 Each degree has its own name : the 1st is called Tonic ; the 2d, Second; the 3d, Mediant ; the 4th, Sub-dominant ; the 5th, Dominant ; the 6th, Sub- mediant ; and the 7th, Leading tone. The most important of these are : the tonic, the dominant, and the sub-dominant. Each degree may be changed into a tonic of a major scale, provided the order of successive degrees, as explained above, is adhered to. This change is made by means of the accidentals : sharps (Jf), and flats Q?), e. g. .- ^ip^nr; £ ^Ej^t^EfcE i E3e ?=M=E EE ^s -* — f- SI ii=p= EEEEE 1 etc. etc. SECTION III. Tones which form an essential part of the scale are called Diatonic tones, e. g. : >_— ?= siESfe^ai^— *=* : f=>= 1 Tones which are foreign to the Diatonic scale, are named Chromatic tones, e. g. : m ± g^N^=^P ig=^ * m |fc --3=z^=.*=*i ' m ^=± fr=$BJMi t=T- Tones standing on different degrees, yet producing the same sound, are known as Enharmonic tones, «. ^. .• i |^— i»— J -j|» — ^ — x ~¥ — ffl= Part I will treat of the Diatonic major scale only. - \f& etc. 16 THE CORRECT ORDER OF SECTION IV. The major scale contains the following intervals : ist. — The perfect prime {unison), the two tones of which must invariably stand on the same degree, e. g. : is etc. Note. — It is evident that in the case of the perfect prime there is, strictly speaking, no interval. 2d. — Seconds are intervals from one degree to the next one above. They are (a) minor, if the distance between the two tones forming the second is that of a diatonic half-step, or {b) major, if it is that of a step, e. g. : (a) Minor Seconds. (b) Major Seconds. i =S«5C ^21 -^&- -e&- -&&. I=5|l=^ll 3d. — Thirds are intervals from one degree to the second above. They are : (a) minor, if the distance between the two tones forming a third is that of a step and a diatonic half-step, or ib) major, if it is that of two steps, e. g. : (a) Minor Thirds. (b) Major Thirds. E|^5S*r=li=& $§!S^ 4th. — Fourths are intervals from one degree to the third above. They are : (4-^ 7th. — Sevenths are intervals from one degree to the sixth above, and are {a) minor, if the distance between the two tones is that of four steps and. two diatonic half-steps, or (&) major, if it is that of five steps and one diaton- ic half-step, e.-g. : (a) Minor Sevenths. (b) Major Sevenths. fe-af I22Z 1 $=^1=! 8th. — Octaves are intervals from one degree to the seventh above. In the diatonic major scale they only occur perfect, the distance between the two- tones being that of five steps and two diatonic half-steps, e.g. : Perfect Octaves. -rP-f -,-f- 12= -0-4- ^4- ■qr=L etc.- 9th. — Ninths are intervals from one degree to the eighth above. They are : (a) minor, if the distance between the two tones is that of five steps- and three diatonic half-steps ; or (5) major, if it is that of. six steps and. two diatonic half-steps, e. g. : (a) Minor Ninths. (b) Major Ninths. i ■#+- ■>4- -t4- ■*?' +4- fT-r-t- ■f-f- &4- =^m 7T>+ &4- •4 18 THE CORRECT ORDER Of The extension of a major scale by an octave will produce every desirable proportion, e. g. : 1 234 96 78 9 10 11 12 13 14 15 i ^^^ 3E * ». -m- — *- F T 2 3 The tones from the 8th degree to the 15th, bear the same names as those from the 1st, to the 8th degree; a peculiar relationship, however, will sometimes arise between the tones of an upper and those of a lower octave. This will be discussed under the head of suspensions. SECTION V. The intervals are indicated by ciphers. E. g., if one voice gives the tone : ' F ^ 1 an d another is to sound its perfect prime, then the figuring will be : m when its effect is : TO 1 If one voice has to sound a second to another voice, the figuring is BLJ and the effect ured : pi Effect. -$ 32?= The interval of a third is fig- ^3L and so on to the fifteenth : i 15 Effect. which is the greatest interval that occurs. This way oi expressing notes by ciphers is known by the name of Figuring, or Thorough-Bass notation. SECTION VI. An inversion arises when of two tones the lower one is transposed above Interval. Inversion. the upper one, e. g. r "'^ 3 changing the upper tone to be bass, and the fundamental to be upper tone. All intervals, within the space of an -octave, may be inverted in this manner : FUNDAMENTAL HARMONIES, 19 Interval. Inversion, ion. f i T"f — » — tf> S3 46r- — s> — •iB- C2a 1»- ~-r3t7~ "B — & — l» * a Expressed in ciphers : The inversion of i, 2, 3, 4, 5, 6, 7, 8. gives 8, 7, 6, 5, 4, 3, 2, 1. Inverted perfect intervals remain perfect; major intervals become minor; minor intervals become major ; the diminished become augmented; and the augmented become diminished. SECTION VII. The character of the major scale depends upon the perfect tuning of the perfect upper fifth and major upper third of the tonic, dominant and sub- dominant. E. g., if C is the tonic of the major scale, the dominant (G), which occurs as its perfect upper fifth, has to be tuned perfect. The next important tone of the scale is the sub-dominant (F), which occurs as perfect under-fifthof C, and must also be tuned perfect, as must also the fifth (D) of the dominant (G). Then the major thirds, e, of the tonic, b, of the dominant, and a, of the sub-dominant, must be tuned perfect. But since a was tuned as major third of F, and D as perfect fifth of G, without regard to each other, it follows that the fifth, D-a, possesses not that perfect character of the other perfect fifths, the D having become too high by the ninth part of a step ; thus bringing the two tones that much nearer to each other, which fifth may then be designated as the Dubious fifth. Each of the fifths of the tonic, dominant, and sub-dominant, forms, in relation to the perfect major thirds of their fundamentals, a perfect minor third, but F having been tuned as perfect under-fifth of C, and D having been tuned as perfect upper-fifth of G, without regard to each other, they have as minor-third not that per- fect proportion, because their distance is short by one ninth part of a step, e.g.: % Perfect Fifth. Perfect Fifth. Minor Third. m ^9-Major Third. Minor Third. Major Third. Minor Third. Perfect Octave. Dubious Fifth. Major Third. Perfect Fifth. 20 THE CORRECT ORDER OF Nevertheless, it would be a great error to suppose for an instant, that the character of the fifth of the 2d degree is dubious, exclusively in the scale of C major, since every other major scale (speaking theoretically), would require to be tuned in the same manner, and since we should arrive at the same results. Although, according to the above explanation, the Piano-forte does not exactly correspond with the perfect system of the major scale, (the tones being tuned in general with regard to each other, and in such a way as to produce all the different scales in as nearly perfect a proportion as possible, which in fact is the sole characteristic of the tempered system, ) yet, the theory of the perfect system must be strictly adhered to. SECTION VIII. The intervals in Sec. IV are divided into, two classes, viz : Consonances and Dissonances. Consonances are intervals whose tones bear a satisfactory relation to each other. They are either complete, or incomplete. Complete consonances are the perfect prime, the perfect fifth (that of the 2d degree excepted), the perfect octave, and the perfect fourth (that of the 6th degree excepted, because the inversion of the dubious fifth of the 2d degree results in the dubious fourth of the 6th degree). Incomplete consonances, are the minor and major third, .and the minor and major sixth. The Consonances may enter free and unprepared ;* they require no resolution, and may be doubled, except the dubious fifth of the 2d degree, which must be prepared, and resolved one degree downward, and cannot be doubled. Dissonances are intervals whose tones do not bear a satisfactory relation to each other. To this class belong the minor and major second, the dimin- ished fifth, the augmented fourth, the minor and major seventh, and the minor and major ninth. The dissonances must be prepared and resolved. The preparation takes place when a tone occurs as consonance in the first, and remains as dis- sonance in the second chord. The resolution of a dissonance is effected by the progression of the dissonant tone to the next degree below or above. Dissonances admit of no doubling. Dissonances become the more dis- sonant the greater the distance between them and the fundamental. Hence, a seventh is more dissonant than a diminished fifth ; and a ninth is more dissonant than a seventh, etc. • With regard to the /mrih, «e Sect. XI. ai FUNDAMENTAL HARMONIES. 21 By maintaining its position, whilst other tones progress, a dissonance may become more dissonant (a), but it can never become a conso- nance (b). (a) (b) Faulty. SECTION IX. Two tones, progressing by degrees, form a melodic step* Several suc- cessive melodic steps form a melodic passage. Simultaneous sounds, of two or more tones, are termed chords. A succession of chords, formed accord- ing to theoretical rules, we call harmony I ox a harmonic passage. SECTION X. A primary chord is one in which the tones are arranged in thirds, one above the other. A simultaneous sound of three tones, arranged as above, forms a triad (a). A simultaneous sound of four tones, similarly arranged, forms a chord of the seventh (6). (a) . (J) i -* — j — &. — j The above two chords are the only primary chords : from these all others are derived. SECTION XI. The 1st', 5th, and 4th degrees of the major scale are fundamental to major triads, each having a major third and perfect fifth •: i =3^ They are the most important, and may enter free and unprepared. The 2d, 3d and 6th degrees are fundamental to minor triads, each having a minor third and perfect fifth : \% -JB1 • The word "step," in this sense, is used in contradistinction to the sk&. 22 THE CORRECT ORDER OP The triads of the $d and 6th degrees may enter fret and unprepared ; but the dubious fifth of the zd degree must be prepared, and resolved one degree down- ward. It is prepared, if the triad of the 2d degree is preceded by that of the 6th or of the 4th degree, and resolved, if it is followed by that of the 5th degree, e. g. : 8 e£ The 7th degree is fundamental to a diminished triad, containing a minor third and diminished fifth : - fin a > — 1 The diminished fiflh,being a dissonance, must be prepared, and resolved one degree downward. It is pre- pared, if the triad of the 7th is preceded by that of the 4th degree, and resolved, if it is followed by that of the 3d degree, e.g. : rS — f^ 1 1 — If — ' <=> ■k^=- =H«= In forming a four-voiced chord from a triad, one of its three component tones must be doubled, or, most usually, added in the octave. The tone of the triad best adapted for duplication is the fundamental ; the doubling of the major third is best avoided. The third in the dominant harmony should not be doubled at all. The fifth can be doubled only when it is perfect. The most essential tones of the triad are- the fundamental and the third ; the fifth may be omitted. The triad needs no figuring. From the triad we derive the following inversions : 1st — If the octave of the fundamental is transposed above the third, leaving the third as bass, we have the chord of the sixth. It consists of a third and sixth of the bass, being marked 6, thereby including the third. % -»- Note. — In all cases of inversions, the fundamental will be indicated by capita) letters under the staff. It may enter free and unprepared. The interval best suited for duplica- tion is : The sixth (original fundamental). FUNDAMENTAL HARMONIES. 23 zd. — If the octave and the third are transposed above the fifth, leaving the fifth as bass, we have the chord of the fourth and sixth. It consists of the fourth and sixth of the bass, and is marked % . This chord can occur only when either the bass {original fifth) or its fourth is prepared, in which case the bass must resolve either one degree upward, or one degree down- ward, or it may remain. It is well to resolve the bass upward, if it is reached from below* or downward, if from above. This chord requires the doubling of the fourth (original fundamental). =g:i§b^cJ=: e§: as> &&-•• zzn zzzz These inversions do not alter the character of the dubious fifth of the 2d, or that of the diminished fifth of the 7th degree, which must be pre- pared and resolved, whether they occur in the primary chord or in an inversion. SECTION XII. The chord of the seventh is formed by the simultaneous sound of three thirds, placed one above the other. Hence it consists of the third, fifth and seventh of the fundamental, and is marked 7, which includes the third and fifth. Its component tones may be doubled, as follows : at first the funda- mental ; then the third, and lastly the fifth, provided the latter is not dubious or diminished. The seventh, being a dissonance, and requiring to be re- solved downwards, cannot be doubled . The fifth may be omitted. The chord of the seventh of the 5th degree is called the dominant chord of the seventh, and consists of a major third, perfect fifth, and minor seventh. It is the most harmonious of all chords of the seventh, and occurs oftenest, and therefore is called the primary chord of the seventh. It is the only one that can enter free, while all other sevenths must be prepared ; but all sevenths must be resolved one degree downward. 1 The major scale contains the following additional chords of the seventh : The 1st and 4th degrees are fundamental to a chord of the seventh, consisting of a major triad and a major seventh (a). 24 THE CORRECT ORDER OF The 2d, 3d, and 6th degrees are fundamental to a chord of the seventh, consisting of a minor triad, and a minor seventh (i). The 7th degree is fundamental to a chord of the seventh, consisting of a diminished triad and a minor seventh (c). Primary Chord of the jth. {&) (») w -JL- The preparation of a seventh takes place most naturally when the fun- damental of the chord preceding occurs a filth higher (or a fourth lower) than that which contains the seventh. In this manner the third of the first remains as seventh of the second fundamental. All thirds can remain as seventh, except the third of the dominant. The resolution of the seventh is effected when the chord containing a seventh is followed by another whose fundamental occurs a fifth lower (or a fourth higher) than that which contains the seventh. The seventh of the first will then resolve into the third of the second fundamental. Primary Chord of the -Jtk. Resolution. Preparation. Chord of the 7th. Resolution. tS>- - lin^ &- -a~ From the chord of the seventh the following inversions are derived : 1st. — If the octave of the fundamental is transposed above its third, leaving the latter as bass, we have the chord of the fifth and sixth, consisting of a third, fifth and sixth. It is marked |, which includes the third. The fifth of this chord, being the original seventh of the chord of the seventh, must be prepared, and resolved one degree downward. All the other tones retain their qualities,' original and fundamental. This chord may occur without its third. *==f£ a -O.^ sSS 1 zzaL — s- 5&F IZ2." 3Z; Fund. D 2d. — If the octave and the third of the fundamental are transposed above the fifth, leaving the latter as bass, we have the chord of the third and fourth, FUNDAMENTAL HARMONIES. 25 consisting of a third, fourth and sixth, marked f , which includes the sixth. This chord can only occur when either its bass and third, or third and fourth are prepared. The third (original seventh 1 ) must resolve one degree down- ward, and the bass one degree downward or upward, or it may remain. «K 25 — — — «•-.-= ^ iz= — fO-^- _=3_ :=g= -c-l W- E - f ~£— — & — ' "- i =#-! Pk- ,*-|tm, 5?" s> & & iS> — — o g. ' At iV /V . =^ * — at 6 ^ -2 6 6 2 8 fV " !• i= _. <=■ ■ = The inversions of the dominant chord of the seventh may enter free and unprepared, provided no other rule is violated ; but that tone which takes the place of the seventh, must always be resolved one degree downward. SECTION XIII. The chord of the ninth consists of the third, fifth and ninth of the fundamental, and is marked 9, which includes the third and fifth. The fundamental, third, and fifth,, being consonances, may enter free and be doubled, provided the fifth is not dubious nor diminished. The ninth, being a dissonance, must be prepared, and resolve one degree downward, and cannot be doubled. (Sec. XL.) The ninth can never appear close to the octave of the fundamental, nor below, but must invariably occur as the actual ninth to that voice which sounds the fundamental. The dubious and diminished fifth must always occur in a 26 THE CORRECT ORDER OF higher voice than the ninth, as otherwise progressions of parallel fifths would be the result. (Sec. XXIII. ) Note. — The chord of the ninth should never be mistaken for the chord of the eeond. SECTION XIV. Here it should be mentioned, that the four-voiced harmony is the most usual, although harmonies which are more or fewer-voiced are admissible. The upper voice is termed Soprano, the next below, Alio, the next below, Tenor, and the lowest, Bass. The leading of the Bass must be effected according to ' the strictest rules, particularly in the case of the alternate employment of primary and inverted chords. To enable us to proceed correctly, it is absolutely necessary to bear the primaiy chords distinctly in mind, to which end we indicate the fundamental beneath the actual bass- note. In this way, having the fundamental bass ever in view, we obtain a clearer insight into the leading of the voices. SECTION XV. The position of a chord in four- voiced harmony, is determined (a) by the intervals involved in the disposition of the upper three voices, or {b), by the tone which the Soprano sounds at the beginning of a harmonic phrase. A. From the former consideration (a) we derive the following three positions : 1st — A chord appears in Close Position, whenever its upper three voices lie so near to each other, that no tone belonging to that chord can be added, without doubling one of them. 2d. — A chord appears in Open Position, whenever a space is left between each two of the upper three voices, which might be occupied by a tone belonging to the chord. 3d. — In the case of a still wider space between each two of the three upper voices, the position is said to be dispersed. B. The Soprano determines the following three positions : 1st. — A chord appears in the position of the octave {Octave position) if the soprano sounds the octave. 2d. — A chord appears in the position of the third {Third position) if the soprano sounds the third. 3d. — A chord appears in the position of the fifth {Fifth position) if the soprano sounds the fifth. FUNDAMENTAL HARMONIES. 27 Ad A. n (1) (2) (»)* ■* — — » ffTf — "f v-)' % C£ •V « ^ u KJ .4d B. (1) i ^ iiE^E (2) 3BI (3) i= :^r»r The position of the first chord of a harmonic phrase is a matter of choice, whilst that of the second and all other succeeding chords is subject to the laws of harmonic progression. SECTION XVI. The voices progress in the following modes : ist. — Two voices progress in parallel motion, when they rise or fall at the same time. 2d. — They progress in contrary motion, if one voice rises while the other falls. 3d. — The oblique motion arises, if one voice remains on the same tone, while another is progressing. (1) (2) fcSi^^P^g^Ppl^ r (3) % -jL — *- 5=^: i 1 l The parallel motion requires a thorough discussion. In the case of parallel motion of two voices, .proceeding by degrees, the best progressions are, by thirds or sixths (a). The parallel motion of three voices progressing by degrees, is best effected by letting the middle voice proceed by thirds and the upper by sixths, together with the lower voice (6) ; but it is of rather doubtful pro- priety when the middle voice proceeds by fourths and the upper by sixths to the lower voice (c). This lower voice in the four-voiced harmony, fur- thermore, cannot be the bass, as the latter in this case is obliged to move in contrary motion ( i -4- i — « — — * — i — » — i 4 i ■#- =f= i — # — i — # — -f- — * — i — # — .-\ — — f~ -r— —* — What has been said above concerning the progression by degrees, applies also to that by skips. . Other objectionable progressions in parallel motion are parallel primes, octaves and fifths. Two voices move in open primes, if starting from the same degree, they are found to be again perfect primes at the next progression, whether by degrees or by skips, e. g. ; jhpf^ ±=J= Two voices move in covered. primes, if, beginning with any interval, they are found in the next progression to be perfect primes, e. g. : M=H^-I FUNDAMENTAL HARMONIES. 29 Two voices progress in open octaves, if they begin and end with a perfect octave (a), or in covered octaves, if they begin with any interval, to terminate with a perfect octave (V). («) n J J -4- (») _<2- S -ffl rszr r *3 Two voices progress in open fifths, if they begin and end with a perfect fifth (a), or in covered fifths, if they begin with any interval, to terminate with a perfect fifth (d). (a) (J) d=,M 3T. w I^ffeiZ ^22= Two voices may progress in fifths, if the first fifth is perfect, and the sec- ond diminished (a), provided this does not occur in the outer voices (&). (a) Good. : - 4rH£ §g^§l Faulty. 19 t 9 I i g - H 9^i^ ^F p— fr- il (A) faa/y. fHHg ^ r f 1 Another objectionable progression occurs, when, of two tones forming a prime on the accented part of the measure, one skips away to form with the other a prime on the accented part of the next measure, e. g. : Si j£l zzbesr aac 7777: -OTO- F Note. — The above rule and example might at first sight seem to apply only to oblique, rather than to parallel motion, with which we are now concerned. But examples of this kind, in which the skips occur on the unaccented part of the meas- ure, belong more strictly to parallel than to obliq«e motion. The same remarks apply to a similar progression in octaves (a) and in fifths (5), e. g. : 30 (a) S THE CORRECT ORDER OF -jsr. "1 — r - zzn (») m M B r -<^& r esc -g-p I Of the above-named different kinds of motion, only the parallel motion in two or three voices can be applied by itself, due regard being had to the rules given above. But it oftener happens that a third or fourth voice pro- gresses in contrary motion, while at the same time still another remains on one degree, thereby producing the oblique motion. The best arrangement is, to let the bass ascend, while the other voices descend, and vice versa. SECTION XVII. During the continuance of the same fundamental, the component tones of its triad may exchange their positions. The most natural exchange is, when that voice which has the octave of the fundamental, skips to the third, while another, which has the third, skips to the octave, (a). The next best exchange is, when that voice which has the third, skips to the fifth, while, another, which has the fifth, skips to the third if). One octave may ascend to the third, while another descends to the fifth (c). Again, the third may descend to one octave, while the fifth ascends to another octave [d). One fifth may ascend to the octave, while another may descend to the third (e). The octave may descend to one fifth, while the third ascends to another fifth (/). One third may skip to the octave, while another skips to the fifth (g). The octave may ascend to one third, while the fifth descends to another third {h). The octave may ascend to the fifth, while the fifth descends to the octave (i), and lastly: The fifth may ascend to the octave, while the octave descends to the fifth (A). FUNDAMENTAL HARMONIES. 31 («) I 2k±Kfe5 (») r _J__«L to (d) r -.7S1 ^r- i i I f= —3: 1= ill -62 Ei=Ei: 1 4— -tf— (») T -P- f 1 'I r ■ r — * -% -^ e ■ » «• * — %- These exchanges may be used singly, or combined, e. g. : 5 -=5t r k^^kk -& 1= "Jt- 35t a Trrrrr^Tf SECTION XVIII. The exchanges may likewise be effected with the component , tones of the chord of the seventh of the same fundamental, with the exception of the seventh itself, which admits of no exchange, unless it be the dominant seventh, which alone has that privilege, e. g. : m dSi • f2 'HZ' 5=SE j2z T "T ^ ff~r ^m Fund. A . % Tf T m -(2 1- =^= Fund. G. A -J- A-A -f- 32 THE CORRECT ORDER OF SECTION XIX. In making the exchanges mentioned in Sections XVII and XVIII, in- termediate tones, not belonging to the chord, may be inserted, as follows: ist — If in the skip to the third the intermediate tone is inserted, thus effecting two melodic steps, instead of a skip, such inserted tone is called a passing tone, e. g. : Skip. Melodic connection. Skip. Melodic connection. i Passing tone. Passing tone. 2nd. — If in the skip to the fourth the two intermediate tones are inserted, thus effecting three melodic steps, instead of a skip, they are likewise called passing tones, e. g. : Skip. Melodic connection. Skip. Melodic connection. J _J L. m *> *> • 1 T TV _ ! m P r P j— i l b< r 1 1 r i r i J i 1 ' *• 1 1 -J r The following are examples of these passing tones in four-voicea harmony : i S u -S U^L4 •=■2= ^ ^ m *i- k i I'll 1 ' 4- ' I I .1 ' I \i ' 1 * i J<£^ at J j£j H r I v I | ' |i" y i i The exchanges of the component tones of a chord of the seventh of the same fundamental, may also lie effected by means of passing tones, if the rule laid down in Section XVIII is observed. FUNDAMENTAL HARMONIES. 33 Fund. A The application of passing tones during the continuance of the dominant * chord of the seventh, with the respective exchanges of the seventh itself, may be seen in the following examples ; Without exchange of the seventh. fe -§- Ii«; If -V t? r dtr-d r r r r > ~? f^TP "" /=•>/»-t- =f i=^-f» — &- 1 1" In place of Two .voices progressing by this returning motion must proceed by thirds or sixths in parallel motion, or, contrary motion must be applied, e. g. : A simultaneously returning motion of three voices may be effected as follows : 3=J -9=*^ fU In place of ^^^^ I I I I I I ffiV^ *> P'<« of tk^4 $0^=\ l» P^c of ^ ^ It is to be noticed in regard to these returning motions, that they pass to tones belonging to chords which bear a relation to the tonic triad, viz. : the chord of the seventh, and the chord of the seventh and ninth of the 5th degree.and to the triad of the 4th degree. FUNDAMENTAL HARMONIES. 35 SECTION XXI. During the continuance of the same fundamental, the octave in the triad rriay descend one degree, whereby a seventh, called a passing seventh, arises, while the remaining tones either maintain their position (a), or another octave skips to the third or fifth, the passing seventh requiring a resolution one degree downward (6) . $ MA J. i ®A £L^St f=E -j— JFfa The examples given at (5) are based on the same fundamental as at (a). If the fundamental appears to ascend, at first a third and then a second, only two fundamentals occur in reality, viz. : the first and last ; because the second bass note is obtained- by the skip of one octave to its third : and its fifth Is obtained by the step of another octave to the passing seventh, which is amply illustrated by the last example, and by the following ones : m -f- -4— Jl rffil t9- c i iiJiiJji^ A rrfyfr^ ^mmms^ r r E A... r G C SECTION XXII. Simultaneously with the step of one octave to the seventh, of the same fundamental, and the skip of another octave to the third, the third may also ■ skip to the octave, by way of exchange, which, however, can be done 36 THE CORRECT ORDE'R OF without reserve upon the dominant only, as the free and simultaneous en- trance of the fundamental and seventh of the other degrees would sound too harshly, e. g. : Without exchange : A _a_ ra:: i !=d= :22 ^= 3ZflC w 1£ *tzrj£i ^£r iac zar i With exchange =± 3St J=± g^^g-r-^: ¥^ iffii :sc ^ f=T^ a=E *»— r— < z - zaz 321 3t i f- • Eg V*== "^_^ ^ -«»- ngsc This harshness, however, may be easily avoided by allowing the entrance of the passing seventh to occur simultaneously with the passing tones of the exchanges, e. g. : $ -*UL J- (2 0- m j=± £ _^ZiI -r— r s_ ?z- & =i=p= -» zr P2ZI FUNDAMENTAL HARMONIES. 37 I E^E di J35-— &E=E^ik— sLrzf^=r |rj=~==fE=;j ii iip r— r* T ^ I I 9i=— #— ^ t — f J2I :a?z The ist, 5th, 7th, nth and 15th measures seem to contradict the prin- ciples laid down in Sec. XI ; but the correctness of the progression, never- theless, becomes apparent, when we consider that the harmony of the third quarter of each of these measures occurs only accidentally, not necessarily. If, simultaneously with the step of the octave to the passing seventh, and the entrance of the remaining passing tones, the fifth descends one de- gree, to return immediately, then care should be taken that the octave occur above the fiftl},as in the following examples, at (a) and (d), since otherwise consecutive fifths would occur in parallel motion, as at (c). (a) (*) m ^ (c) Fault). ^J>_ r l 1 t? :_ _ _ jz. ♦ T r Fund. Q. SECTION XXIII. Simultaneously with the step of the octave to the passing seventh, the tenth may step to ike ninth of the same fundamental. Special care should be tafcen, that the dubious fifth of the 2d, and the diminished fifth of the 7th degree occur above the ninth, since otherwise consecutive fifths would result (a). <•) % ¥ m T ■jsr -jajsi ^=-Jt- =f= TF isss: gus ^ m-. aczL 38 THE CORRECT ORDER OF The perfect fifth may appear below or above the ninth, because it can either ascend or descend (6). W £ H — r 3£ m ^ 3& -O- zz>z ■^ ~ig~ These progressions are usually effected in four-voiced harmony without the fundamental voice ; and in three-voiced harmony even without the. lower third, e. g. : i Fund. D. :sfc SECTION XXIV. If the fundamental descends a fifth, or ascends a fourth, the regular progressions of the tones take place in the following different ways : i st. — The third of the first ascends one degree to the octave of the sec- ond fundamental (a) ; it ascends a fourth, or descends a fifth to the third of the second fundamental (3), or it remains as seventh of the second fun- damental (c). (a) 0) . . (*) i s 3E ^ =3C ZZSBT raz 2d. — The octave may remain as fifth (a), or skip to the third (6), or to the octave, provided the bass remains, or progresses in contrary motion (f). (a) % W PPP _£>jy_ zzzz. IP- Fund. C F F 3d. — The fifth ascends one step to the third (a), or descends one step to the octave (6). The progression to or from the fifth should be by de- FUNDAMENTAL HARMONIES. (a) _ (J) t =2: c F m FutuL C F 4th. — The seventh descends one degree to the third, e.g. 39 SECTION XXV. Although progressions involving covered fifths and ortaves are usually prohibited, yet they cannot in all cases be avoided, unless the most natural progressions in harmony are also to be rejected. The following examples will elucidate the special exceptions to this rule : A. Covered fifths 'are admissible : (a). If the fundamental descends a fourth, while the third of the first descends one degree to the fifth of the second fundamental — (6). If the fundamental ascends a fifth, while the octave of the first ascends one degree to the fifth of the second fundamental — (c). If the third of the first fundamental ascends to the octave of the second, while the octave of the first fundamental ascends to the fifth of the second, e. g. : (a) (b) (c) zz=*z ^=r- --T- The above exceptions, however, are subject to the following limitations : 1 st. — They are not to be applied in two-voiced compositions, on account of the unsatisfactory effect produced by the absence of the third, but exclu- sively in compositions of three or more parts, affording the effect of full harmony. 2d. — The skipping voice must skip in reality, i. e., without insertion of passing tones, e. g. : («) • (*) J ! («) mtmMmm r rr 40 THE CORRECT ORDER OF (d). Covered fifths are admissible also, when the same chord is inverted into other positions, e. g. ; B. Covered octaves are admissible : (a). If the fundamental descends a fifth, while the fifth of the first de- scends one degree to the octave of the second fundamental. (6). If the fundamental ascends a fourth, while the third of the first ascends one degree to the octave of the second fundamental, e. g. : («) (») $ I T I i Provided, however, that this does not occur in two-voiced harmony, be- cause of the unsatisfactory effect resulting from the absence of the third, but in compositions of three or more parts, affording the effect of full harmony : bearing in mind, as before, that the skipping voice must skip in reality, without insertion of passing tones, e. g. : i^Ppip#N SECTION XXVI. Conditions which are indispensable to a good four-voiced harmony, are the following : ist. — Each voice must have a different progression from the others, which excludes octaves and fifths in parallel motion. 2d. — Whenever a primary chord is immediately succeeded by another, the two must be connected by a natural link, which with some chords pro- duces a closer connection than with others. This natural link is a perfect one, whenever the fifth is prepared, which may be effected by the descent of the fundamental by a fifth or third. The ascent of the fundamental by FUNDAMENTAL HARMONIES. 41 a fifth or third, affords the preparation of the octave, which, however, in some instances leads to results which are objectionable. 3d. — In every harmonic phrase one voice, at least, should progress by melodic steps. SECTION XXVII. The most important chord is the triad of the 1st degree, which degree is also called Tonic, or principal tone, because every harmonic combination must bear relation to it, and because it forms the proper conclusion, as it is also the most appropriate commencement, of a composition. The next important chord is the triad, or chord of the seventh, of the $th degree, which degree is also called Dominant, inasmuch as it leads back directly to the tonic, and accordingly occurs as the last chord but one. The triad of the t,th degree, which degree is called subdominant, is the third of the important chords, because it also affords the opportunity to return to the tonic triad. Generally speaking, the tonic triad bears a direct relation to the domi- nant triad, or chord of the seventh, and to the subdominant triad, e. g. : 7 7 Degrees : I V I V I IV I V I, expressed by notes. n -&&- ^&- ^^=2z;: A «=at is The next best progressions are those in which the fundamental descends by a fifth, or ascends by a fourth, (the resuh in both cases being the same), because the harmonic link is always perfect, and the change of harmony enters very distinctly ; and finally, because the dissonances may be better prepared and resolved, e. g. : I. Succession of triads formed upon fundamentals descending hy a fifth. &~ : Z2ZC *Ffe=I=l s~ II. Succession of chords of the seventh upon the same fundamental progression. W^^Sz E^3E7fr jO?Z. Sl&T- 42 THE CORRECT ORDER OF The expression, regular progression of fundamentals, implies a succession of fundamental descents by a fifth. The descent of the fundamental by a third 'affords two links, the fifth and third, and is much weaker in its effect, because it introduces but a single new tone, e. g. : 2^S: 3Efe=I= i*^fc js.— Hff«Z- I SECTION XXVIII. The progression from the chord of the seventh of the 5th to the triad of the 1st degree, is regarded as the most important. It serves as a model for many other progressions, and is called the closing cadence. It is effected in the following manner : 1st. — The voice having the fundamental tone of the chord of the seventh of the 5th degree, skips to the fundamental tone of the triad of the 1st de- gree, i. e., it either descends by a fifth, or ascends by a fourth. This is therefore called a. fundamental progression. 2d. — The voice having the seventh of the chord of the seventh of the 5th degree descends one degree to the third of the triad of the 1st degree. This is a melodic progression. 3d. — -The voice having the third of the chord of the seventh of the 5th degree, ascends one degree to the octave of the fundamental of the triad of the 1st degree : this being also a melodic progression. 4th. — The voice having the fifth of the chord of the seventh of the 5th degree, descends best one degree to the octave, or ascends one degree to the third of the triad of the 1st degree ; both progressions being melodic. 5th. — The voice having the octave of the fundamental of the chord of the seventh of the 5th degree, remains, to become the 5th of the triad of the 1st degree, thereby forming the harmonic link between both chords. The closing progression may also be made from the triad (instead of from the chord of the seventh) of the 5th to that of the 1st degree. This requires no other alteration than the omission of the seventh, and its sub- stitution by the octave, which then descends by a third, or, the fifth of the 5th degree ascends one degree, in order to reach the third of the 1st degree, e. g. : i -*2h &3~ gg ^ FUNDAMENTAL HARMONIES. 43 The same, with fewer voices : $ In case a perfect close should not be intended, the third of the 5th de- gree may (instead of progressing to the octave) skip to the third of the 1st degree, e. g : The same, ivith inversions : m J&z m^m i ~^& Z&r =Ez] Fund. G 75" C G =<=?£: er Fund. G Z&Z lf=± G T0&Z SECTION XXIX. The ascent of the fundamental by a fifth or a third, depends on the per- fection or independence of both chords, in so far, that is, as the first chord requires no resolution, and the second no preparation. Hence the follow- ing perfect progressions remain, viz. : A. The ascent of the fundamental by a fifth ; 1st. — From the 1st to the 5th degree ; 2d. — From the 4th to the 1st degree ; 3d. --From the 6th to the 3d degree. B. The ascent of the fundamental by a third : 1st. — From the 1st to the 3d degree ; 2d. — From the 4th to the 6th degree ; 3d. — From the 3d to the 5th degree ; 4th.— From the 6th to the 1st degree. The ascent of fundamentals by a fifth or third, may be effected either actually, or by a passing motion, which latter, in some cases, is preferable to the former, as allowing the second chord to be succeeded by a more marked harmony than would otherwise be the case. The means of effecting this have been already elaborately discussed in Sections XVII — X.XIII. Here follow a few examples : From the \st to the id degree. From the 6th -to the 3d degree. 1. By fundamental *. By passing *. By fundamentat 2. By passing progression. 1&- By passing , motion. . By fundamental progression. tilts Ippinipip^ Fund. C 1* c 44 THE CORRECT ORDER OF SECTION XXX. Whenever a succession of harmonies containing conditional tones * is in- tended, which cannot, according to rule, be prepared or resolved, the rule is that the fundamental of the second chord cannot follow, and that the pro- gression must be effected by means of passing motions on the fundametital of the first chord. Means for effecting passing motions are : isL— The exchanges (Sees. XVII, XVIII). 2d. — The passing tones of the exchanges (Sec. XIX). 3d. : — The reluming motions (Sec. XX). 4th The passing seventh (Sees. XXI, XXII), and 5th.— The passing ninth (Sec. XXIII). SECTION XXXI. Passing motions must be applied in the following instances : i st — If the triad of the 5th degree is to be succeeded by that of the 7th degree, which would be a fundamental ascent by a third : 2d. — If the triad of the 5th degree is to be succeeded by that of the 2d degree, which would be a fundamental ascent by a fifth : 3d. — If the triad of the 2d degree is to be succeeded by the harmony of the 4th degree, which is a fundamental ascent by a third ; and, 4th. — If the triad of the 2d degree is to be succeeded by the harmony of the 6th degree, which would be a fundamental ascent by a fifth. The triad of the 5th degree cannot be succeeded by the diminished triad of the 7th degree, by way of fundamental progression, because the diminished fifth is not prepared ; nevertheless, this succession may be effect- ed by means of a passing motion, upon the fundamental of the 5th degree, if one octave skips to the third or fifth, while another descends to the seventh, e. g. : $ p :g= 3C T Fund. G.. I ^m s r The triad of the 5th degree cannot be succeeded by the harmony of the 2d degree, by fundamental progression, because the dubious fifth of the 2d degree is not prepared, but this harmony may be obtained upon the fundamental of the 5th degree, if one octave descends to the seventh, at the same time that * Tones which should be prepared and resolved. FUNDAMENTAL HARMONIES. 45 another octave ascends one degree, to descend again simultaneously with the descent of a returning step of the third, e. g. : $ All - :*» —±s — i— 1 .Fawrf. G. I G.. a 3 ISasC The triad of the 2d degree cannot be succeeded by the harmony of the 4th degree, by fundamental progression, because the dubious fifth of the 2d degree would thereby lose its dubious character, by remaining as third of the 4th degree, in which character it might ascend. But this succession may be based upon the fundamental of the 2d degree, if one octave of the fundamental ascends to its third, while another descends to the seventh, e. g. : $ i.- 1 =&^ -&^ i n -4?=t ^^f: Sii -js&i G Fund. D. I The triad of the 2d degree cannot be succeeded by the harmony of the 6th degree, by way of fundamental progression, because the dubious fifth of the 2d degree would lose its dubious character by remaining as perfect octave of the 6th degree : still, this connection may be effected upon the fundamental of the 2d degree, if one octave descends to the seventh, sim- ultaneously with the descent of the tenth to the ninth, while at the same time another octave skips to the fifth, e. g. : A jru ^A^Jki mm A J. „ . r **tu£ D. 19- I rr SECTION XXXII. The diminished triad of the 7th degree cannot be succeeded by the har- mony of the 2d degree, by way of fundamental progression, because the diminished fifth of the 7th degree cannot become a consonance by remain- ing as third of the 2d degree, and because the dubious fifth of the 2d degree is not prepared. This succession, if attempted, must be made thus : b, d 46 THE CORRECT ORDER OF and f are treated as third, fifth and seventh of the 5th degree, the tone a being reached by returning motion of the third descending, while the other tones remain, e. g. : m -a- :q- 1 G m The entrance of the harmony of the 4th degree after the diminished triad of the 7th degree, must be effected in the same way ; here a and c are reached by returning motions, while f remains, e. g. : fej T * 7 Fund. G C By means of the passing motions mentioned in Sec. XXX, chords of the seventh, or their inversions, may be introduced, even when the strict rule regarding them cannot be observed. This, however, can be done only under the following restrictions : 1st. — That at- least one tone from the first chord (especially that which represents the fundamental of the new chord) remains in the new chord. 2d. — That all tones reached by skips be component tones of the funda- mental. 3 d. — That all tones foreign to the fundamental(especially that which rep- resents the seventh) be reached by steps. 4th. — That no free* seconds, sevenths or ninths occur, and 5th. — That every free fourth to the bass (whether in parallel or in contrary motion) be avoided. A ♦ £ -*- r<» — l — BO I ^ I — •%" I "£Z= Fund. G. I G.. r SECTION XXXIII. A direct, regular progression of fundamentals by degrees, does not exist, since their harmonies are without any natural link. But this progres- * Free intervals are such as are formed abruptly, t. e., when neither of their tones is prepared. FUNDAMENTAL HARMONIES. 47 sion may be effected by the introduction of a tacit fundamental, which should be a fifth above the second actual fundamental, thereby affording the desired natural link. Here follow examples of fundamentals ascending by degrees : € Futid. C A D More correctly thus -^r^—jzr "fb G -!»- -a- BG DB -& $m^P»i .&- r 5 "? 1 raz 3": It is obvious, that, with the seeming ascent of the fundamental by de- grees, the first harmony undergoes a change by the insertion of the tacit fun- damental. The first triad is changed into a chord of the seventh of a tacit fundamental, or, in other words, the triad of the first fundamental stands as substitute for a chord of the seventh, the resolution of which leads to the desired harmony of the second (actual) fundamental. SECTION XXXIV. Irregular progressions of the voices may occur under the following restrictions : ist. — The seventh may ascend one degree, a license which can never occur in the bass, and is admitted in the other voices only under the following conditions, viz. : (a) That the fundamental is not audible, so that the seventh does not appear as such : (6) That another voice (most generally the bass) effects the resolution of the seventh ; and (c) That such a position is chosen, as enables the ascending third of the first (tacit) fun- damental to stand above the irregular ascending seventh, so as to prevent any succession of fifths, which would otherwise result, e.g. : i sUrzri— i Faulty. 3E Fund. G r c r r r G C 48 THE CORRECT ORDER OF Under the above-named conditions, two sevenths, one descending, and the other ascending, may occur simultaneously, e. g. : 8 _J J- i .a- -3L Fund. C A C A T D EiH C A 2d. — The third, instead of ascending to the octave, remaining as seventh, or skipping to the third of the second fundamental (all of which are regular progressions), may skip to the fifth of the second fundamental, either by de- scending a third, or by ascending a sixth. This license also cannot be ex- tended to the bass, and is admitted in the other voices only on the following conditions : (a) the skip from the third of the first to the fifth of the second fundamental, if it descends a third, must occur in a lower voice than that in which the regular progression of the fundamental descent, or the regular progression of the fifth of the first to the octave of the second fundamental takes place, since otherwise consecutive fifths would arise, e. g. : $ S- z&z -ffl- Z2SZ =3S: zsz 1* Faulty. Fund. G G (6). For the same reason, namely, to avoid covered fifths, the skip from the third of the first to the fifth of the second fundamental, if it ascends a sixth, must appear in a lower voice than that in wMch the regu- lar ascending progression of the fundamental by a fourth, or the regular progression of the third of the first to the octave of the second fundamental occurs, e. g. : 19- jBT- nn 19- . Fund. C 111 ^ 3= c 3d. — The fifth, instead of ascending one degree to the third, or descend- ing one degree to the octave of the second fundamental, may ascend or descend to the fifth of the second fundamental. This license is also denied to the bass, and is granted to the other voices under the following restric- tions, viz. : (a). The skip of the fifth of the first, to the fifth of the second funda- mental, whenever it descends a fifth, must occur in a lower voice than that in which the regular fundamental descent by a fifth, or the regular progres- FUNDAMENTAL HARMONIES. sion of the fifth of the first to the octave of the second fundamental occurs. e. g. : Faulty. % jBZ Fund. G G -f-- G Hi G =f= (b. ) The skip from the fifth of the first to the fifth of the second fun- damental, if it ascends a fourth, must be made in a lower voice than that which effects the regular progression of the fundamental ascent by a fourth, and than that which effects the regular progression of the third to the oc- tave of the second fundamental, otherwise consecutive fifths occur, e. g. : Faulty. y — . a , a.- Fund. G G i-§z i mpii G SECTION XXXV. The descent of the fundamentals by degrees depends likewise upon the insertion of a tacit fundamental, which should form a fundamental descent, by a fifth to the second actual fundamental. The triad of the first {actual) fundamental . will then stand as subst'tute for a chord of the seventh and ninth, the resolution of which effects the desired harmony of the second (actual) fundamental (a). The position of the first triad must be so- chosen, that that fifth which is to be changed into a ninth, does not appear in the soprano, as consecutive fifths would arise (6), nor in the bass, as parallel fourths would occur against the bass (c ), This leaves as bass note of the first triad its fundamental only, which would stand as fifth (d), or its third, which would stand as seventh (- ^F EfE IfflU # % >2T mm ^_._„ Fund. F D G Fund. D F D G It should be observed, however, that after the triad of the tth degree, the dominant chord of the seventh enters abruptly, inasmuch as these two chords have no common tone to form a connecting link. m Instead of: Instead of: idzs?z iac :1 ISE^^^^SE^-^r J£g- TBgZ r- A € -a~ G Instead of: :zt at D G SECTION XXXVII. The tonic triad cannot be succeeded by the triad of the 7th degree in direct fundamental progression, because the diminished fifth of the latter is not prepared ; but the succession may be effected seemingly, if the triad of 52 THE CORRECT ORDER OF the 7th degree is introduced as substitute for the dominant chord of the seventh, that is, without fundamental, e. g. : i 32= -77-. ■zr G 0g- =S- 3Z= Fund. C G C C G C The diminished triad of the 7th degree may stand as substitute for the dominant chord of the seventh, not only after the tonic triad (a), but also after the triad of the 5th (6), and that of the 3d degree (c), which progres- sions are particularly adapted to three-voiced harmony, e. g. : i * Fund. C (*) -zr G G zsz G 1 ~P \ ' = ^~~P 'tt /*•««£ G. r % w =2fc 4 ii r ' ' Aou/. E G E G E G In free harmony the dominant ninth may enter unprepared, if it appears in the upper voice, e. g. ; I £ -fSO I 1 -zr Fund. C G G G This is all the more allowable if the dominant chord of the seventh and ninth appears without its fundamental, i. e., when its substitute (the chord of the seventh of the 7th degree, or one of its inversions) takes its place, e. g. : zz-fe I ffi^f J. j .jj j J j ;' j-jj ~3BZ Fund. C i A UM^ L J . . ^. 1 G J G I I r% a —o — Fund. G G EG EG Additional examples will be given with the suspensions. 3C 'I "ZT E ■zr ' G FUNDAMENTAL HARMONIES. 53 SECTION XXXVIII. In Section XXVIII, allusion was made to the progression from the dom- inant triad, or the chord of the seventh, to the tonic triad, which we termed the closing cadence (authentic close), and which we shall now fully discuss. The formation of a perfect closing cadence depends upon the contrac- tion of all the diatonic tones of the scale, and is effected by the succession of the triads of the 4th, 5th and 1st degrees, or by the succession of the triads, or chords of the seventh, of the 2d and 5th, and the triad of the 1st degree (a). Other cadences (still more contracted) arise, if the harmony of the 6th is succeeded by that of the 5th degree (6), and the harmony of the 3d by that of the 5th degree (c), but they have not the decisive character of the former ones, e. g. : & K=iEyEi= &z -^f^=^ 3& i'uiul. F D F D (*) -*- C («) sr G \% ~-W: E£ f: -gyg- Fund. A D G Fund. E G JZ1 -f-- The plagal close (succession of the triads of the 4th and 1st degrees), heldom stands alone, being preceded most generally by the authentic close. SECTION XXXIX. The succession of the harmonies of the 4th, 5th and 1st degrees, and of the 2d, 5th and 1st degrees, as explained in the foregoing Section, invest the third of the 5th degree with an unmistakable tendency to resolve itself into the octave of the 1st degree, in consequence of which, it is called leading tone. This tendency of the leading tone forbids its duplication (as involving octaves or primes), and also its skipping away to any other tone, SECTION XL. In contradistinction to the progression of the component tones of a chord simultaneously with the entrance of a new fundamental, it may hap- pen that one voice does not resolve, as expected, but continues as a foreign 54 THE CORRECT OR&ER OF tone in the new harmony, and resolves at a subsequent period. In such a case we have a suspension. A suspension may occur in any voice, and every tone of a triad, or chord of the seventh, may be suspended in the following manner : i st. — If the fundamental descends by a fifth, or ascends by a fourth, (which gives the same result), and the first chord is a chord of the seventh, the fifth and seventh must descend one degree. By their suspension at the entrance of the new fundamental, the seventh is changed into an eleventh, and the fifth into a ninth, both of which must afterwards resolve into the tenth and octave. Accordingly the eleventh stands as a suspension of the third from above, and is accompanied by the octave and fifth of the fundamental ; and the ninth as a suspension of the octave from above, and is accompanied by the third and fifth. The names eleventh and ninth are adopted for the reason that the eleventh must form a ninth above that voice which sounds the third, and the ninth a ninth above that voice which sounds the octave ; because a tone can never lie close to or above its suspension, e. g. : (a) (6) (a) {a) (b) X-X =f= *4 £ -*-& ■ Fund. G =JF 3«±=£ "ST G C G =^e sr I ■&• G C The examples show the disposition of the voices, (a) when the suspen- sions occur accompanied by their suspension-tones ; (6) when they occur alone. 2d. — If the fundamental descends by a fourth, or ascends by a fifth (which gives the same result), and both chords are triads, the octave and the third must descend one degree, but their suspension changes them, at the entrance of the new fundamental, the former into an eleventh, and the latter into a thirteenth, both of which must afterwards resolve into the tenth and twelfth. The thirteenth, therefore, stands as suspension of the fifth from above, and is accompanied by the third and octave. Here the term thir* teehth is adopted, for the reason that the thirteenth must form a ninth above that voice which sounds a fifth, e. g. : (a) i 3" .4,^ (»), _3£ The examples show the disposition of the voices, (a) when the suspen- sions occur unaccompanied by their suspension-tones; (6) when they occur accompanied by them. FUNDAMENTAL HARMONIES. 55 The following examples show the use of suspensions in inverted chords, and moreover, some faulty leading of the voices. . It should here also be noticed, that the ninth, eleventh and thirteenth, should not be mistaken for the second, fourth and sixth ; the former being tones foreign to the harmony, admitting of no duplication ; while the latter are component parts of the harmony, and may be doubled. It should be observed, furthermore, that the suspensions do not remove parallel fifths or octaves, as the following examples will show : Faulty; $ 5fc 4- I I :z£ -jag- -&-^&r ~zt M J j Fund. G Good. V m Fund. G Faulty. & SB- I L ^=F=g^^ :*32t ~A1 ZSlZ F Faulty. F t p ^ A—A EIE2 =a; i— r-r-^ ^e Fund. G C F •&■ F B £ B E A i st. — Suspensions of the Ninth. Faulty. Faulty. m -*&&- '=&- First inversion. =- L - L ** With octave of the fundamental. £ S Ei ./?«»<£ G C Second inversion. With octave of fundamental. '^ z&r. J=*t Fund. A ^F Good. Good. m -*—■#- Tg- £=M Without ot rave. Without octave. i -o- m rzt: G C A -fS<-. A 66 THE CORRECT ORDER OF If the suspension of the ninth occurs in the bass, the octave of the fun- damental cannot appear in a higher voice, for the reasons given above. Suspension of tie fcj t ninth in the bass. CEp i ^ ^p^--^ "^ f I eej -g»- i i Hrrf inversion. Second inversion. -?~3> P -^^ 32Z ■*? /'w** r C F ^ immmm FUNDAMENTAL HARMONIES. 59 Three suspensions may occur simultaneously, viz. : those of the eleventh, ninth, and ascending seventh, e. g. : \^m -&L ^£=A S£- -&r %a The first inversion of the foregoing suspensions can follow only after the chord of the third and fourth, e. g. : i zSX. Fuud. G Second inversion. Z ^L =§= SET. 3Z G "ST c m ^ Fund. G J*- c m ■7ZBT G * If the fundamental ascends a fourth, and the second chord is a chord of the seventh, the delay of the resolution of the fifth of the first chord changes the same into a ninth, whereby the sound of the third, fifth, sev- enth and ninth is obtained, which latter must then resolve into the octave, e. g. : Without suspension. With suspension. gupil $Hiilil It may be observed here, that the ninth possesses merely a fictitious in- dependence, since it is only a suspension of the octave, although the simul- taneous sound of the fundamental, third, fifth, seventh and ninth, is called a chord of the seventh and ninth. When the dominant chord of the seventh and ninth occurs without fun- damental, the chord of the seventh of the 7th degree appears to be succeeded by the first inversion of the dominant chord of the seventh ; hence an op- portunity is given for the descent of the fundamental by a third after every Chord of the seventh, by which descent the seventh, fifth; third and funda- mental are changed into the ninth, seventh, fifth and third of the new funda- mental. The subsequent resolution of the ninth effects then a new Chord of the seventh, e. g. : r&-^—& ■ S^ 60 THE CORRECT ORDER OF If the second chord of the last example occurs without fundamental, the harmony of the dominant chord of the seventh is succeeded by that of the chord of the seventh of the 3d degree. The progression of the fundamentals in this case may be termed artificial, because both chords cannot be pri- mary chords. SECTION XLI. The resolution of the ninth, instead of taking place during the contin- uance of the same fundamental, may be effected simultaneously with the entrance of the new fundamental, when it leads into the fifth. Two fundamental progressions are necessary : the first, to introduce the ninth as suspension of the octave ; and the second, to resolve the ninth (of the second) into the fifth (of the third fundamental). Without suspension. ^E Suspension of the ninth. Delayed resolution of the ninth. '^£-e-4- ^T ^ T^Tf f*^ In cases like those just exemplified, it is necessary that the dubious fifth of the 2d, and the diminished fifth of the 7th degree should stand above the ninth, since otherwise consecutive fifths would arise ; while a perfect fifth may stand above or below the ninth, because it can ascend or descend, according to the necessity of the case. J J ziz 1 l — — ■= *$-&z=&- F=^a Of «?■■" 5' 'if — ^ — 'S"" 1 **~ fB ■-<&- 7 7 L J. J ^C — iff— -6>- 7*"" ■Z7- 7 7^ L Jr J 5? — \ \ V r> - . . -&- 6 7 5> * a Fund. B E The harmonic connection- of the 7th and 2d degrees may be effected by means of the passing motions only (Sec. XXXII). «/9 ■ Y+ «ll f3 —&& — \ -/& " H 9- J — ' ' fl> - 4rJ 8 3 — — « L e -p h - fS -f* — »- r— m -*- -J " r ( 1 1 a ^ ,1 . ±=?± 1 -1 Fund G FUNDAMENTAL HARMONIES. 60 -,-» m E£ ^mm EEEfE ZJBT T- Fund. G. The diminished triad of the 7th degree cannot be succeeded by the chord of the seventh of the 2d degree (Sec. XII), nor by its first inversion (Sect. XXXII). The harmonic connection of the 7th and 4th degrees can be effected only by means of passing motion (Sec. XXXII). No. a. Fkttd. G C Remarks. —The progression in the 1st example is effected by means of the skip of the third (of the tacit fundamental) to the seventh, a license which is admitted only upon the fundamental of the dominant. The pro- gressions in the 2d and 3d examples are preferable to those in the 1st, because the succession is effected by means of the returning motion and melodic connection of the exchanges, and because the duplication of the seventh is avoided. The diminished triad of the 7th degree cannot be succeeded by the chord of the seventh of the 4th degree. The succession of the diminished triad of the 7th and harmony of the 5th degree must be effected by means of a passing motion, the former being treated as substitute for the dominant chord of the seventh. No. 1 No. No. 3. E J^U- -JJL No. 4. 3E m eI £ -&c 6 "A ne Fund. G. 70 THE CORRECT ORDER OF No. 5 . No. 6. No. 7. P EIr =fc ^=f -0- m t H 9 i ^ ^ ^ ^"k«-— fid •"-* *— ^ ' g- — -g ■ a,— *— zr^ 6 S „ 1 ^ ZJ» — & S2__ Ei ~* — \-0~\ Fund. B "ZT- 6 f J-- ^^=- ~e: 321 Fund. B FUNDAMENTAL HARMONIES. 71 VII. I. ^i^^im §=^12=111 Fund. B G C B G C The last example contains one license, viz. : the irregular progression from the fifth of the first to the fifth of the second fundamental. See Sec- tion XXXIV, 3. VII. VI. Fund. B E A D VII. II. — By means of passing motions. % t=i= m -G> -6>- jSZ ZIO&ZZ WsmmM Fund. G.. ^=^ ^Z ?=& IffiZ -*- m -s>- zaz ^ee£-. zvr Fund. G C G C VII. IV. — By means of passing motions. ^F m rnm^- -s> Z3- s a 6 -0 mi^ T-- zaz Fund. G.. (I -4=m zaaz m mz zzzznaz Fund. O. -„ -I zoz 72 THE CORRECT ORDER OF $ VII. V. — By means of passing motions. =±=± Z22ZZ ZSZ I3>- ^T ^r (sr s 3ZI f ^r— r is: .FW. G. &=%— -to— ■ ' — ' ig 6* ' e> !■■ ' g «> -SJ— i ffc^ : « i ' — g ' 1 S^d I bf— 1 '. t ^ 1 fV *> ■ )' n 1 1 -"' i a i /TJ » I * a 1 1 r i 1 1 i r ftiui. G. The Harmonic Successions of the Second Inversion of the Dimin- ished Triad of the 7th Degree. VII. fl III. V j- ^s> e> fcr^~ 5 V-\) ^> ~*% <&■ _e* 6 4 6 e t r ^? r ^-^ ^-^ 4 7 8 7 7 r\*^^^^g — & " «> «.j /* /^ » & a ^ The second inversion of the diminished triad of the 7th degree cannot be succeeded by an inversion either of the triad, or of the chord of the seventh of the 3d degree (Sec. XI). The second inversion of the diminished triad of the 7th degree cannot be succeeded by the harmony of the 6th degree (Sec. XI). VII. II. — By means of passing motions. i ^ -0- ?El % % £ =*2= Ptmd. G. FUNDAMENTAL HARMONIES. VII. IV By means of passing motions. (i E£ ±E± 78 ^ d^* =F % r i § 9- f ^^ ~tm Fund. G C G. VII. V. — By means of passing motions. -&B- r~r zsz gia £^' Fund. G. The Harmonic Successions of the Chord of the Seventh of the 7th Degree. VII. III. (&__„ I * r r r— , | g ^ — 19 ] 7 7^777 t f c\f _ (- ^ , la-. ffl a . \ -=£ s> &—\—0 O <9 — — - r :a: > — «- ^ S^-~ 8 6 S '-42=- ^ E^lE ^ r-f=^f 1 AWixi B.... G C. B... G C. G C... B G C.... 74 1HE CORRECT ORDER OF VII. IV. — By means of passing motions. i=s _S2_ rzzz -»- f=r i i ££ m -f --B --r- Fund. B G C B G C B G C The mark A implies that no other tones of the chord are required than those seen below it. VII. V. — By means of passing motions. I ( i&j^^j^ ^-t-: 3z: 122: 3ZI /»« !=*=*=• a isi^d- /5 e erg — 1 g 8 --% % f =g- 9-—f — p- rj "T — 5~ rj lS> j -^H F — — M—- f- i F.nrf. B G C. 76 THE cdRRECT ORDER OF The Harmonic Successions of the Second Inversion of the Chord of the Seventh of the 7th Degree. VII. III. IffiZ =^SZ?= ZeZl Fund. B VII. I. -73S- nb^^=li ~-fS^ Fund. B G C VII. II. — By means of passing motions. B G C. Up $_ =i J- zzr sfc 5* iff: is: 2C -1 •- ~ar Z&L- H2ZI ^ ^ .Faxrf. B. VII. IV. — By means of passing motions. ± — ! ~~~ i — ±1 — » h g i* — ~S~ =b!h=*=| t — 7 t — « l^-S a -— 1 — « »-^= - — 1 1 6 f 5 f /a- H — =-» «- =\ a "P 1 ^ \^=^ _ "I — - 1« <-l 1 ■ Fund. B G C. B-. VII. V. — By means of passing motions. i *m* -s>- -IB fimi. B ft 3& f-~ zataz ^ FUNDAMENTAL HARMONIES. 77 . The Harmonic Successions of the Third Inversion of the Chord ok the Seventh of the 7th Degree. VII. III. $^?m^m 2 6 1*LzztZZJS1 Fund. B VII. II. — By means of passing motions. J. -& — «■— ■= \^- a lS> ~-<&z zzzz %r ^i: 2 ZffiZ Fund. B E VII. IV. — By means of passing motions. Fund. B G C B VII. V. — By means of passing motiom. S^fe S^ ie: 2 7 ■jsz Fund. B VIL VI. ^& F*md B Efefei w=r zsz 78 THE CORRECT ORDER OF PART II HARMONIC PROGRESSIONS IN THE DIATONIC MINOR SCALE. INTRODUCTORY. It being taken for granted that all the general rules and licenses explained in Part I are perfectly understood, new rules and licenses, appertaining to the Minor Scale, will now be discussed. SECTION I. The minor scale, which is characterized by a minor third and a minor sixth, exhibits a three-fold variety of progressions. The first and most natural progression of its component tones is, from the ist to the 6th degree upward (from the ist to the 2d degree, one step; from the 2d to the 3d, one diatonic half-step ; from the 3d to the 4th, one step ; from the 4th to the 5th, one step ; and from the 5th to the 6th, one diatonic half-step), descending in the same order to the ist, followed by the leading-tone of the scale, and concluding with the ist degree. E. g., in A minor: 8345 66432171 i =p=s 1 — 1 — "— j ■ — • or 8 7 8 In the second mode, in which an ascent is made from the ist to the 8th degree, the 6th degree must be raised a chromatic half-step, in order to avoid the progression by an augmented second, in the step to the raised 7th degree (leading tone), which must be succeeded by the 8th degree. Kg., in A minor : 12345678 FUNDAMENTAL HARMONIES. 79 In the third mode, in which a descent is made from the 8th to the ist degree, the 7th degree must be lowered a chromatic half-step, in order to effect a melodic step to the natural 6th degree, since all degrees must be natural. E. g., in A minor : m ^ * : £ The ist, 2d, 3d, 4th and 5th degrees remain unchanged in each mode ; the only changeable degrees being the 6th and 7th, while the 8th degree remains unchanged, like the first. The changes are thus easily explained : ist. — The nature of the 6th degree after the 5th, depends upon the direct return to the 5th, or the ascent to the 7th degree. If the 6th degree is to be succeeded by the 5th, the former can be distant only a minor second, and must be natural. When the 6th degree is succeeded by the 7th, the former must be distant a major second, which is effected by raising it a chromatic half-step, in order to form a melodic step to the raised 7th degree (leading tone), which leads to the octave. 2d. — The nature of the 7th degree after the 8th, depends upon the direct return to the 8th, or the descent to the 6th degree. When the return to the 8th degree is intended, the 7th degree can be distant only a minor second, i. e, it must be raised. If the 7th degree is to be succeeded by the 6th, the former must be distant a major second, and therefore be natural, in order to form a melodic step to the natural 6th degree, which leads to the fifth. Hence the following four modes in A minor, which will represent all other minor scales : p ^33[3^t^^^r^=n SECTION II. The variable nature of the 6th and 7th degrees of the minor scale, gives rise to four new intervals, now to be discussed : The augmented second arises from the simultaneous sounding of the natural 6lh and raised 7th degrees : IpiSlipl so THE CORRECT ORDER OF The diminished fourth arises from the simultaneous sounding of the raised ftk and the loth degrees : fe^^ The augmented fifth arises from the simultaneous sounding of the 3d and raised "jth degrees : i =^^=E? The diminished seventh arises from the simultaneous sounding of the raised "jth and natural 13th degrees : i ¥ Tl— * t^i-' Z The tones forming these four intervals must, in general, be introduced and progress according to the rules given in Section I. SECTION III. The character of the minor scale depends upon the perfect tuning of the minor triads of the 1st and 4th degrees, and of the major triad of the "jth degree. A piano-forte tuned for the scale of C major, contains upon its 6th degree the perfect minor triad of A, which may be used for the tonic triad of A minor. In order to change the tuning from C major to A minor, it is necessary to tune the sub-dominant (D) — which in the major scale was too high by the ninth part of a step — so much lower, thereby restoring to the dubious minor third its right proportion. The minor triad of the 5th exists already upon the 3d degree in C major. Thus also the major triads of the 3d and natural 6th degrees, in A minor, exist upon the 1st and 4th degrees in C major. The raised 6th and 7th degrees have now to be added to the other (natural) degrees as major thirds of the 4th and 5th degrees. In conse- quence of the lowering of the 4th degree (D) by the ninth part of a step, the fifth of the natural 7th degree will, in like proportion, approach its fun- damental, and thereby exhibit the same dubious character in the minor scale, that the fifth of the 2d degree does in the major scale. The fifth of the 7th degree, accordingly, must be -prepared, and resolved one degree downward. FUNDAMENTAL HARMONIES. 81 Although the tuning of the piano-forte, according to the tempered system in vogue, does not exactly "correspond with the theory above ex- plained, still, the rules of the perfect system should be strictly adhered to. Tonic Triad. Dominant Triad. Sub-dominant Triad. >■ .. — Perfect Perfect Fifth. m ;-Vi Diminished Perfect '. Perfect Fifth. Fifth. Octave — -^ Major Third. — ~^~ ~- . -~ Minor Third. m Tonic Triad. Dubious Fifth. 3E Z? Perfect Octave. Major Thinct. Minor Third. SECTION IV. From the variable nature of the 6th and 7th degrees arises a greater - number of triads in the minor than in the major scale. The 1st degree is fundamental to a minor triad (a). The 2d degree is fundamental to both a diminished and a minor triad [b, c). The 3d degree is fundamental to both a major and an augmented triad ! (d, e). The 4th degree is fundamental to both a minor and a major triad (f, g). . The 5th degree is fundamental to both a minor and a major triad [h, i). . The natural 6th degree is fundamental to a major triad (/). The raised 6th degree is fundamental to a diminished triad (k). The natural 7th degree is fundamental to a major triad (/). Finally, . ' The raised 7th degree is fundamental to a diminished triad (m). i (a) (») («) ( « mmm FUNDAMENTAL HARMONIES. 83 The dominant chord of the seventh is the most harmonious. It may r alone of all chords of the seventh, enter free and unprepared. SECTION VI. The upward tendency of the raised 6th and 7th degrees forbids their standing as ninths. It also precludes their being fundamental to chords of the ninth, as their resolution would effect a duplication of the raised 6th and 7th degrees. In the minor scale, the dominant chord of the seventh and ninth, and its inversions, are allowed to enter free and unprepared. The same license is extended to the diminished chord of the seventh of the raised 7th degree and its inversions, when standing as substitutes for the dominant chord of the seventh and ninth. The resolution of the ninth may be effected accor- ding to Sections XL, XLI and XLII of Part I of this treatise. Here follow examples of the dominant chord of the seventh and ninth, and its inversions, with audible fundamental : Fund. E Examples of the diminished chord of the seventh of the raised 7th de- gree, and its inversions, as substitutes for the dominant chord of the seventh and ninth : ^^^^^^^^^^^^t Fund. E. F A E A 3Z?Z •rr 1 ^ Delayed resolution of the dominant ninth (resolution with entrance of the new fundamental) : IM ^a^EEffr zsr A E A E A Even the diminished triad of the 2d degree may stand as substitute for ■3Z. IB- Fund. E A E A 84 THE CORRECT ORDER OF the dominant chord of the seventh and ninth, being, as such, succeeded by the first inversion of the tonic triad, e. g. : The same examples more complete. Fund. B E A -f-- B E E 2?" E SECTION VII. In all the following examples, the chord ' of the seventh of the raised 7th degree and all its inversions, may enter freely, when they stand as sub- stitutes for the dominant chord of the seventh and ninth. This case occurs : 1 st. —After the tonic triad and its inversions. 2d. — After the dominant triad and its inversions. Even the diminished triad of the 2d degree may enter freely, if it stands as substitute for the dom- inant chord of the seventh and ninth. [ad j.) ppf §*§EiSiEgijlE3? as= Fund. A * E ( ad 2.) A -27- E A, iifeil rs: Hi Fund. A E E A E A.... T &- E E. A 122. az: 22: w± Fund. A E E A E EiE r r ^r j2gz & ee; A.... A E E A ZZ2L ±=± 4- i3|E:g ^=r aa ;€&: 3* 2ZZ .Fkk<£ E E A E -* 1 - 7Sr -3T E A E "f-^ E A r E. fr fcgj Jz^ :22Z Fund. E A.. E ~9r. -sr\ FUNDAMENTAL HARMONIES. 86 3d. — The seventh of the raised 7th degree may appear in another voice than that in which it is prepared ; but this occurs most appropriately after the diminished triad of the 2d, and after the minor triad of the 4th degree, and their inversions, e. g. .* -O- \JB- 3^ Jg-_ z m JO. -<2r B >- v — ■ — « -&-'- gz Fund. A lf- B E jSZ.- - JB. T ■A=*. J&1 -j&I T^rj 32= HiL- D B E. m Fw;d. A F B E. J2- I A sssigiii "wm r~r 3= /"u»rf. A D B E. ^S-^-iSt- hzr is- A A D GfE A or: A D B E A D B E SECTION VIII. During the continuance of the same fundamental, the component tones of the chord may exchange their positions, as explained in Sections XVII and XVIII of Part I. The said exchanges may also take place in any chord containing the natural or raised 6th or 7th degrees, and be linked together by means of passing tones, e. g. ; Exchange. $ m f^ji ~£i mm Fund. D Exchange. % ¥ '32.- a_ W- 2EEf=*m ^ Fund. D. With passing tones. ^m 3E EEEEEE D D. With passing tones. ~JS1 ^ ~Z= m w-- ^ Exchange. THE CORR ECr , ORDER OF With passing tones. % i=£z =f= r-r ^ ( B-rVrVfH i&EEg ^r a? f^^r=FT=i /Vow/. E Exchange. With passing tones. fe rn *= I iliillll^ fe«=^^ ^tH't N ^ Fund. Exchanges during the continuance of the chord of the seventh of the natural 6th degree. P^ ^% ^ ^ ^E -*>- ¥^F /;««/. A F B. The same tuith passing tones. g^_ — ^H^ S m FT^ =± f ¥^ jSSC ¥=^ ^=^ -+-T-P- fci -s Fund. A. Exchanges during the continuance of the chord of the seventh of the natural ■jth degree. m -^ 3g=*=g &L—~ JGZ 3^ I E& fc Aim/. A F FUNDAMENTAL HARMONIES. 87 The same 'with passing tones. - A i Fund. A C FJ wmmmwm B But this same triad may be preceded also by the second or first inver- sion of the harmony of the raised 7th degree, and be succeeded by th,e triad of the 1st degree, by means of which, however, no close is yet reached, since the latter can be obtained only by regular fundamental progressions. E. g. : m --& $m Fund. A Gt A D B w 5th. — That the minor triad of the 4th' degree can be preceded by that of the 1 st degree, has been shown ; but it may also be preceded by the triad of the natural 6th degree. It is succeeded by the diminished triad of the 2d degree, etc., whenever a close is intended : if not, it may be followed by the triad of the 1st degree, e. g. : sr 6th. — The major triad of the 4th degree can, strictly speaking, be preceded only by the triad of the i9t degree. In Section XIV, 4, it was succeeded by the minor triad of the 2d degree ; though it may also be followed by the first inversion of the triad of the raised 7th degree, after which a close may be reached by the successions of the dominant harmony and of the triad of the 1st degree, e. g. : i^l^^l&ii^iP Fund. A Gt A 94 THE CORRECT ORDER OF 7th. — That the major triad of the 5th degree may be preceded by the triad of the 1st, or by the diminished or minor triad of the 2d degree, we have seen repeatedly. It has also been shown that this triad of the 5th degree ma)' be succeeded by that of the 1st, or by the augmented triad of the 3d degree. 8th. — That the minor triad of the 5th degree may be preceded by that of the 1st degree, has been exemplified. But it may also be preceded by the major triad of the 3d, or by that of the natural 7th degree, although without exhibiting any characteristic of the minor scale. It cannot follow after the diminished triad of the 2d degree, because the nature of the minor scale demands the characteristic succession of the major triad of the 5th degree. The minor triad of the 5th degree, may be succeeded by the major triad of the 3d degree. 9th. — That the triad of the natural 6th degree may be preceded by the triad of the 1st, or the major triad of the 3d degree, we have already seen ; yet it may also follow after the minor triad of the 4th degree. This triad of the 6th degree may be succeeded by the diminished triad of the 2d, or by the minor triad of the 4th degree. Moreover, this triad may be followed by that of the 1st degree, but without forming a close, as this must be reached in the usual manner, e. g. : m • IB- -JSr. --#-- rer V-£> =f=: JS. M mm 10th. — The triad of the raised 6th degree is preceded most generally by that of the 1st, or by the augmented triad of the 3d degree, as in both cases the diminished fifth of the raised 6th degree is prepared. This diminished fifth will be resolved by the succession of the minor triad of the 2d degree, or one of its inversions, e. g. : Ff B E A nth. — The triad of the natural 7th degree needs a preparation on account of its fifth, that is, it must be preceded by the minor triad of the 4th degree. The resolution of this fifth is obtained by the succession of the major triad of the 3d degree, etc. E. g. : n "Z?I JS. -- -W- ~<3f9~ ^bsso : M 1 TJSZ sr Fund. A GJt • IS" c sr - B ^PSf^PlPlHIit^ Fund. A GJ F A ^S^- w fefef^^Q^l Fund. A sr- — --J9- B Gf B A §^^ Fund. A FJ B Gff Si=ii3ii3^ c lip zai SECTION XVI. THE PREPARATION OF THE CHORDS OF THE SEVENTH. 1st. — The chord of the seventh of the 5th degree, with major third, may enter freely, though it may be prepared by the triad or chord of the seventh of the 2d degree with diminished fifth ; by the minor triad of the 2d, or by the diminished triad of the raised 7th degree. E. g, : m z *&~- W- *#3 sazz ^^ VT-l .-^sr_ 11 - \ U fff J — Ufc£_.^*_ffi£ 1 — #<•? L & — "*~TI» — * — ^ — ^ — Fund. A -zr GJ 96 THE CORRECT ORDER OF If it is preceded by the minor triad of the 4th degree, the latter must be treated as substitute for the chord of the seventh of the 2d degree, with- out fundamental, as is seen in the following examples : ^^mm If it is preceded by the major triad of the 4th degree, the latter must occur without fifth, since this seeming triad of the 4th degree stands but as substitute for the chord of the seventh of the 2d degree, as is seen in the following examples : More correctly : jtg- 3g= l_^gE w< ^F^ 2d. — The chord of the seventh of the 5th degree, with a minor third, is the most unusual. It is best introduced, by means of passing motion after the minor triad of the same degree, and should be succeeded by the chord of the seventh of the 1st degree, which may be followed by the minor triad of. the 4th degree, etc. , e. g. : 3d. — The chord of the seventh of the 2d degree always contains a diminished fifth, and is prepared by the minor triad of the 4th, or by the triad of the natural 6th degree. It should be succeeded by the major triad, or the corresponding chord of the seventh, of the 5th degree, etc., e. g. : If this chord of the seventh is preceded by the tonic triad, the latter stands as substitute for the chord of the seventh of the natural 6th degree, without fundamental, e. g. : More correctly : FUNDAMENTAL HARMONIES. 97 4th. -s— The chord of the. seventh of the natural 6th degree may be pre- pared by means of the tonic triad, or the major triad of the 3d degree. It resolyes most naturally into the diminished triad, or the corresponding chord of the seventh, of the 2d degree, etc., e. g. : m^m vg^tt & ^m ^m '>-%- m -TBT -zr If this chord of the seventh is preceded by the minor triad of the 5th; degree, the latter must be treated as substitute for the chord of the seventhi of the 3d degree, with perfect fifth, but without fundamental, as is seen in; the following example : More correctly : «0- -»e- I ST -w- I -&■ > If this chord of the seventh is preceded by the major triad of the 5th degree, the latter stands but as substitute for the chord of the seventh of the- 3d degree, with an augmented fifth, e. g. : l§£E|l^ibi= js*z =£z=\ r-r~ ^^^M^m^m 5th. — The chord of the seventh of the raised 6th degree, is best pre- ceded by the tonic triad, or the augmented triad of the 3d degree^ .to resolvev into the 2d inversion of the minor triad of the 2d "degree, e.g..- ^ ;eie Fund. A %S- ZZ~ #*- n B A ^mm^^^^m Fund. A F* B 98 THE CORRECT ORDER OF Whenever this chord of the seventh is preceded by the major triad of ithe 5th degree, it stands as substitute for the chord of the seventh of the, 3d degree, with an augmented fifth, e. g. : , Fund. A 6th. — The chord of the seventh of the 3d degree, with a perfect fifth, may be preceded by the minor triad of the 5th, or the triad of the natural 7th degree. This chord of the seventh may be succeeded by the triad, or the chord of the seventh, of the natural 6th degree, etc: , e. g. : £ ^=2z za= -+ — a~ -JST- 35¥ ^ i & — ^feJI^M* S?- zez If this chord of the seventh occurs without fifth, after the diminished triad of the 2d degree, this triad stands as a substitute for the chord of the seventh of the natural, or raised 7th degree, e. g. : ^m %^~-%-x-=% IS-' -h-- More correctly : Fp It* .fe w^wmmi ST Or* i t r- -j "S3 — J - ^~^ -2^— — ^ — % — ' — ^ Z7SZ <* Tfr ^ — - A 3£~ #- F« ■5T B 8th. — The chord of the seventh of the natural 7th degree, may be pre- ceded by the minor triad, or the corresponding chord of the seventh, of the 4th degree, or, seemingly by the diminished triad of the 2d degree, to resolve into the major triad, or the corresponding chord of the seventh, of the 3d degree, etc, e. g. : % — ?3&P7Z ~rs&- ~<&- *3^s t=z??= ■27" S 4^H ^f---^-,^— T _-§L_ T _ ( J@- Eg=: ^m M-es ««= 6>— = *—S~ m m Fund. A D B ^zsr:: -%~^ h. -& ZJS1 100 THE CORRECT ORDER OF oth. — The chord of the seventh of the raised 7th degree, may also be preceded by the minor triad, or the corresponding chord of the seventh, of the 4th degree, to resolve into the second inversion of the augmented triad, or corresponding chord of the seventh, of the 3d" degree, etc, e. g. : ■^ —r " ir— — — — -rr tt —t F GJ ' C -a- B E A Fund. A iK^' 37 " GJ C & zaz IS? IS :53c ■zr B H^ 3«E c FJ 10th. — The chord of the seventh of the 4th degree, with minor third, may be preceded by the triad, or chord of the seventh, of the 1st degree, or by the triad of the natural 6th degree, to resolve only into the chord of the seventh of the natural, or raised 7th degree, or its second inversion, etc, e. g. : ^g^gPl z&z Fund. A E D ' GJ C F B E 232=^- ~jBSB1 s^+=f= 321 C^2= GJ sr _ B ST E A If this chord of the seventh occurs without fifth, after the major triad of the 3d degree, it stands as substitute for the chord of the seventh of the 1st degree, e. g. : More correctly : ^ S^ etc. fe35 g zso etc. FUNDAMENTAL HARMONIES. 101 nth. — The chord of the seventh of the 4th degree, with major third, may be preceded by the triad of the 1st, or the first inversion of the triad of the raised 6th degree, to resolve into the second inversion of the triad of the raised 7th degree, e. g. : pf^^l^i^iiS Gf C F B Fmid. A FJ D $ip^§iiliis&:ip Gf 1 2th. — The chord of the seventh of the 1st degree may be preceded by the minor triad, or by the corresponding chord of the seventh of the 5th, or the major triad of the 3d degree, to resolve into the minor triad, or corresponding chord of the seventh, of the 4th degree, e. g. : **=hM m 3S: -~£& z a* m --£-- -BS- ei c. i|E|i^i|^jii§t=i r SECTION XVII. Whenever, in the progression from the major triad of the 4th to the chord of the seventh of the ^lh degree with a major third, the former occurs with its fifth, it stands as substitute for the chord of the seventh of the 2d degree with perfect fifth, which was rejected. Considering that this fifth of the 2d degree must ascend to the leading tone, whithei the seventh also tends, and that a duplication of the leading tone is forbidden, it remains only that the seventh should ascend, which, in this case, is quite proper, in- asmuch as its fundamental is inaudible. This progression of the ascending 102 THE CORRECT ORDER OF seventh is allowable only when the third occurs above it, as otherwise consecutive fifths would arise. Hence, the major triad of the 4th degree with its fifth, should be succeeded, not by the dominant chord of the seyenth itself, but by its third inversion, the chord of the second. The case is the same, if the first inversion of the major triad of the 4th degree is succeeded by the first inversion of the dominant chord of the seventh. K g. . fc^g: D B ZS2I -^ -7 9t i~ eg ■ ■t - : «£ c 3E zaai More strictly thus : HS= Js= =1 g§315ll3 I2fc Z3C naaL—^JS. E Fund. A The directions just given for the above progressions, may also be applied to the following analogous cases : (a) when the harmony of the chord of the seventh of the natural jth degree is preceded by the triad of the natural 6th degree ; (6) when the harmony of the chord of the seventh of the raised ith degree is preceded by the triad of the natural 6th degree ; and (f) when the harmony of the chord of the seventh of the 1st degree is preceded by the triad of the natural jth degree. F D $ E^5 zzzmz ?fr -M Fund. A (ft) =^?=rJfa gg fe S zar ^Vuu£ A F D GJ /W<*. A F D G£ FS w. 1p FUNDJMEN7 iL HARMONIES. -** — l— — — J— e>— \— &>— \—a&- Jt 103 1221 Three-voiced. D iJEjE^lipg^ ^ $ G E A -ffl.- ^IeS 3=1* .F*k&*- 1S>3r ^1 3s=s ZsZZ ^- -»b Fund. A -a- --§."--4= etc. GJ 4th. — If the major triad of the 3d is followed by the minor triad of the 4th degree, the former stands as substitute for the chord of the seventh of 1st degree. E. g : Xf= is- C A E 3H 5th. — When the minor triad of the 4th is followed by the major triad of the 5th degree, the former stands as substitute for the chord of the seventh, of the 2d degree, with diminished fifth. E. g. : More correctly : m^^m 5Tf M FUNDAMENTAL HARMONIES. 105 N. B. — Since the succession of the minor triad of the 4th and the major triad of the 5th degree, can be effected only by means of the real or seeming chord of the seventh of the 2d degree, it is evident that the former cannot be followed by the minor triad of the 5th degree. 6th. — Whenever the minor triad of the 5th is succeeded by the triad of the natural 6th degree, the former stands as substitute for the chord of the seventh of the 3d degree, with a perfect fifth. K g. : 1$ ■&L JS2Z mm More correctly : dgr =f= i zsr. jSSZ r Tr 7th. — If the major triad of the 5th is succeeded by the triad of the natural 6th degree, the former stands as substitute for the chord of the seventh of the 3d degree, with augmented fifth. E. g. : m fe z%z f*5?*Z More correctly : m etc. Fund. A E C F 8th. — If the triad of the raised 6th is to be succeeded by the triad of the raised 7th degree (a progression which may be allowed for the sake of practice, although it is far from being euphonious, for the reason that the diminished fifth of the raised 7th degree enters free), it follows, that the former stands as substitute for the first inversion of the chord of the seventh, of the 4th degree, with major third. E. g. : More correctly t J. Fund. A F|D Gf E Fg D Gl 106 THE CORRECT ORDER OF 9th . — Whenever the triad of, the raised 7th is succeeded by that of the 1 st degree, the former must be considered as substitute for the first inversion of the chord of the seventh of the 5th degree, with a major third. .£. _g. : More correctly : i Fund. A -# GJE A 5=? I 12212 S -iS>- A =i= 3=~ 1221 ** G« A SECTION XIX. The formation of suspensions in the minor scale must be effected in the same manner as in the major scale, with this modification however, that the raised 6th and 7th degrees cannot be used as suspensions from above, while the natural 6th and 7th degrees cannot serve as suspensions.// - *™ below. The ascending seventh occurs upon the 1st, 3d and natural 6th degrees. SUPPLEMENTARY REMARKS. Since the chords appertaining to the minor scale contain a greater num- ber of dissonances, and other tones, viz : the raised 6th and the natural 7th degrees, which latter, strictly speaking, are foreign to the nature of the minor scale, care must be taken during the study of the exercises relating to them, that their entrances and progressions be effected in exact conformity with the rules laid down. The descent of the fundamental by a fifth (ascent by a fourth,, or by a third, requires no additional explanation, but the ascent of the fundamental by a fifth (descent by a fourth), or by a third, involving progressions other than those which in Section XI were pronounced unexceptionable, must be effected in accordance with the directions given in Sections XXX, XXXI, XXXII, XXXVII, and XLII— XLIV of Part I of this treatise, and be perfected by means of exchanges, passing motions, and suspensions. Practice in connecting the harmonies of the minor scale, may be best obtained by writing exercises, following the order laid down in the subjoined table, and transposing them into the different minor scales. The supple- mentary remarks at the end of Part I will here be found of practical application. I: FUNDAMENTAL HARMONIES. 107 TABLE. $ 8 , * 1, 6, f, 3, 4, 4, fr, 2, 2. V : 1, 3, 6, 4, 7, 2. IV: 7, |, 2, 6, 5, 5, 3, 1. IV: i, % J3, 3, 1, 5. ' * i II: 5, 5, 1, 3, 4, 7, fr, 6. II: 5, .4, 3, 1, i f VI: 2, 4, 1, 3, 3, 5, 7, f. » , 8 ' « *- 8VI : 2, f, 5, 4, 1, 3. Ill : 6, 1, 5, 7, 4, 2. * * , 8,8 III: 2, 2, 6, t, 1, 4, 4, fr, 5. VII : 3, 5, 1, 6, 2, 4. » i # * 8 8VII : 3, 6, f, 5, 1, 2, 2, 4, 4. 8 8 • $ $ I: 4, 4, 5, 5, 6, f, 3, 3, 2, 2, 7, V. By way of explanation of the above table, we would remark that the Roman numbers indicate the harmony of that degree which is considered as a starting-point. The student should set himself the problem of joining to this harmony, as far as is practicable, the harmonies of the respective degrees indicated by Arabic numbers. The sharps placed above the ciphers indicate that the harmony of the respective degrees must contain a major third ; the figure five with a dash ( $ ) signifies, with the 2d degree, the harmony with the perfect — , and with the 3d degree, the harmony with the augmented fifth. The numbers and f, with a dash, indicate raised degrees, as do also the Roman numbers preceded by a sharp ($VI and $VII). In all other cases, the harmony of the respective degrees must be composed of natural tones. 108 THE CORRECT ORDER OF PART III DIATONIC MODULATION. INTRODUCTORY. Modulation may be defined, a change of tonic, or iey-t. e., a transferring of the character of tonic to a tone which did not possess it before. Diatonic modulation from one scale or key into another, is effected only by means of their diatonic tones, in so far, that is, as the same chord which stands in the first scale upon the ist degree, is found on another degree in the second. Now, in order to show how we may modulate in all cases by means of diatonic progression, it is necessary to make use of every scale from d? to Cfi major, and from AJ? to A$ minor. Those who hold that it is one and the same thing whether they modulate into the scales of Ftt or <3v major, into B or Q> major, into C§ or Db major ; or into those of A$ or B^ minor, G$ or AJ? minor, and Dtt or E? minor, may, by means of this treatise, be easily convinced of the contrary. If the modulations by diatonic means are found to be more extensive, and to require more chords, they have, on the other hand, the advantage of being more positive and precise, than the chromatic and enharmonic modulations. SECTION I. A triad is readily recognized as a tonic triad, if it is preceded by its dominant harmony and its subdominant triad/ e. g. : I IV I V I, or: I IV I V L It has been seen, that the succession of the subdominant triad and dominant harmony should be connected by the actual or tacit harmony of the 2d degree, which would give the following fundamental progressions : I IV II V I, or: I IV II V I, or : I IV II V I. FUNDAMENTAL HARMONIES. 109 But since the chord of the seventh of the 2d degree contains all the component parts of the triad of the 4th degree, the latter may be omitted at discretion, whenever the former is preceded by the triad or chord of the seventh of the 6th degree, e. g. : I VI II V I, or: I VI II V I, or: I VI II V I. When this succession is to be extended, the harmony of the 6th may be preceded by that of the 3d degree, e. g. : I III VI II V I, or: I III VI II V I, or: I III VI II V I. If the subdominant triad is succeeded by the harmony of the 7th de- gree, the harmonic passage will be still more extended, if the latter is fol- lowed by the harmony of the 3d degree, etc. , e. g. : I IV VII III VI II V I, or: I IV VII III VI II V I. All these progressions may be effected by means of primary chords, or their inversions. SECTION II. A. The major triads occur, in the major scale, upon the 1st, 4th and 5th degrees, and in the minor scale upon the 3d, 4th and 5th and the natural 6th and 7th degrees. Hence, each of the above named degrees may be fundamental to the same major triad. E. g. : The major triad of C occurs upon the 1st degree of the scale of C major, upon the 4th degree of G major, and upon the 5th degree of F major ; it occurs also upon the 3d degree of A minor, upon the 4th degree of G minor, upon the 5th degree of F minor, upon the natural 6th degree of E minor, and upon the natural 7th degree of D minor. B. The minor triads occur : in the major scale upon the 2d, 3d and 6th, and in the minor scale upon the 1st, 2d, 4th and 5th degrees. Hence, each of the above named degrees may be fundamental to the same minor triad. E. g. : The minor triad of A occurs upon the 2d degree of G major, upon the 3d degree of F major, and upon the 6th. degree of C major, and also upon the 1st degree of A minor, upon the 2d degree of G minor, upon the 4th degree of E minor, and upon the 5th degree of D minor. Only major or minor triads can become tonic triads. The following Section contains the modulations from C major into its relative scales, and also the returns of those modulations into their original no THE CORRECT ORDER OF scale. The examples will be the better understood by means of the added fundamentals, and the Roman ciphers, which mark the degrees in each particular modulation. SECTION III. From C major into G major : is- * c Degrees in G major ; IV From G major into C major: Degrees in C major From C major into F major J£ zaz Tff*- :§= F D IV II G V j^fgLj: Degrees in F major .' V S- From F major into C major : Degrees in C major . IV From C major into A minor : Degrees in A minor : III F VI B II E V From A minor into C major: ^ ~Ol: =SE Degrees in C major: VI From C major into E minor : D II G V c T ■leiii^ Degrees in E minor : C VI n ii From E minor into C major : i :2= wk ~^^ Degrees in C major : III A VI D II -s>- G V c I FUNDAMENTAL HARMONIES. Ill From C major into D minor: $ -—a<2i S>- S ~%2*z z^EzEE Degrees in D minor: VII From D minor into C F III VI E II A V D I C major : p fe: D Degrees in C major : 1 1 From C major into G minor G V ISI c I 'i^pp *i*iiFfcg Degrees in G minor : IV A II D V G I In order to effect a return of the modulation from G minor into C major, a mediation is necessary, because the minor triad of G is not contained in the scale of C major. This mediation, or intermediate scale, may be that of F major or D minor, both of which contain the minor triad of G. After having effected the modulation into F major, we treat its tonic triad as standing upon the 4th degree of C major. If a modulation is made into D minor, its tonic triad is treated as standing upon the 2d degree of C ma- jor, from which degree, as well as from the 4th, the major triad of C is easily reached. E. g. : From G minor into F major : Degrees in F major : From F major into' C major : zb> s '—. :: Degrees in C major : F IV D II BX?Z G V —fW 1 From G minor into D minor Degrees in D minor : IV From D minor into C major G inor: IV 1=118 E II A V D I Degrees in C major : II From C major into F minor : ^ G V =i=J ^m igcr Degrees in F minor I V IV G II C V F I H2 THE CORRECT ORDER OF The modulation from F minor back into C major, must again be effected by means of an intermediate scale, because the minor triad of F is not contained in the scale of C major. To do this, we treat the minor triad of F as standing upon the 4th degree of C minor, which is then suc- ceeded, first, by the harmony of the 2d degree, and next, by the dominant harmony of that scale. The homogeneous nature of the dominant har- mony in the major and minor scales, offers then the best opportunity to resolve into the major rather than into the minor scale. From F minor into C minor, resolving into C major : Degrees in C minor: SECTION IV. Modulations from A minor into its relative scales, with returning modu- lations into A minor. From A minor into G major: g •<- Idfc -«§^ Degrees in G major From G major into A minor: Degrees in A minor: VII From A minoF into F major = $^^Sitl^ ^ r : Degrees in P major : III From F major into A minor: Degrees in A minor : From A minor into C major: D VI G II C V F I a .-k — "=&2- -Jr% qis>- ZOBSL m Degrees in C major : VI Ftom C major into A minor: Degrees in A minor : III D II G V F VI B II E V A I FUNDAMENTAL HARMONIES. US From A minor into E minor: 1 /m l Degrees in E minor : A IV '3ST- **£. m FJ II B V From E minor into A minor: ~i a — -X.-&9— T 3L r- Degrees in A minor : V C III F VI B II Zjgf- S- A i From A minor into D minor : f^l za_ "rsz Degrees in D minor : V F III VI E II A V -0- D I Fiom D minor into . A minor: ^~ g~>^ gg^=|=g D B E A Degrees in A minor : IV II V I 1 u 5? + ft . i Tl 25 -^T-^a l k„ Ji L J c ^ A D G. Degrees in G minor : II V I The modulation from G minor tack into A minor,: must be effected by- means of an intermediate scale, as the minor triad o£ G is not contained in the scale of A minor. This intermediate scale may be that of D minor or of F major. From the modulation into F major we obtain the triad of the natural 6th degree, and from that into D minor, the minor triad of the 4th degree of A minor, from either of which degrees a close may be reached by the regular fundamental progressions. E. g. : From G minor into F major : lp=5|; Degrees in F major : G. II ^ C. V From F major into A minor : :(fty- JpiEgj|sig'~•'' !ggl^Efz~gag "| Degrees in A minor: F Vt B' IIi, Ej Vv A. I 114 THE CORRECT ORDER OF From G minor into D minor : Degrees in D minor : IV From D minor into A minor: Degrees in A minor : IV B II zsz «= £ V n A I SECTION V. The following table will materially assist us in obtaining a g&rieral sum- mary of the scales more closely or more distantly related to C major, which stands in the centre. From its right, extend the scales as they increase in the number of sharps, and from its left, as they increase in flats. Imme- diately below the scale of C major stands that of A minor, and similarly, under every other major scale, stands that minor scale with which it has a common signature. The major scales are marked by capitals, and the minor scales by small letters. Cfc>, G>, Dfc Al?, E[>, Btv F, C, G, D, A, E, B, F#, C#. a> eb, bV, f, c, g, d, a, e, b, f#, c#, gjf, d#, a£ Now, just as the scale of C major is akin to the scales of G and F major, which stand nearest to it on both sides, and since it is also related to those of A minor, E minor, and D minor, which stand right below the above named major scales, and, moreover, to the three minor scales of G, C, and F, standing still more towards the left, it follows that every major scale bears a relation to both major scales on its sides, to the three minor scales stand- ing below these three major scales, and finally, to the three minor scales standing still farther towards the left. Thus : F$ major is related to CJf and to B major, to D$, A$, and Gjt minor, and also to C$, Fjf, and B minor. SECTION VI. The modulation into any scale outside the circle of relationship, must be effected by means of one or more intermediate scales. i st. — The modulation into a scale which is but one step outside of the relation, would be too abrupt to be effected without abridgment Hence, FUNDAMENTAL HARMONIES. 115 the modulation, e.g., from C major into D major proceeds most naturally by means of the intermediate scale of G major, or of E minor, and from thence into the scale of D major. FronvG major into D major, by way of G major : ^kzsEz l£5EE — g Z5 j^rz rMr*. ^e= Degrees in G major: IV Degrees in D maj.or : I IV I From C major into D*major, by way .of E minor : Ss*i 7^821 a 3&= =^ : Degrees in S minor : VI " Dmxlor: I II ;gsi- A similar course must always be pursued if a modulation is to be effected from any major scale into the second upon its right. The modu- lation liom C major into Bb major, proceeds most naturally, first, to the scale of F major, or of D or G minor, and from thence to that of.Bb major. From C major into Bt> major, by way of F major: Sn Degrees in F" -major : V 9&~ ziz: r-T-/a»g"-r ^g J $g=i~dec£. IV II V IV II V ST I From C major into B[? major, by way of G minor : ^B:ji $z Degrees in G minor ; IV " " iflj major : V m m JBZ v%r ~W i VI A like method must always be pursued, if a modulation is to be made from any major scale into the second on its left. 2d. — The modulation from any major scale into the third on its right, admits of one abridgment, i. e., we proceed first to that minor scale which possesses the same first degree with the desired major scale, and is always found immediately below the major scale from which we started. Thus : the modulation from C major into A major is effected by means of A minor, which contains the major triad of C upon its 3d degree. This triad is to be succeeded by the regular fundamental progressions, until the dominant 116 THE CORRECT ORDER OF harmony is reached, which affords the best opportunity to close in A major, instead of A minor. From C major into A major, by way of A minor: $ -r0f=^ 3E _agj^s ^FT^c Degrees in A minor: III " "A major: VI II V V The modulation from any major scale into the third on its left finds its abridgment in the treatment of the .first chord as* dominant triad of the minor scale standing on the left of the desired major scale, the tonic triad of which is obtained by the regular fundamental progressions. Thus : in the modulation from C into Eb major, the major triad of C is treated as dominant triad of F minor, and succeeded by the tonic triad of that scale. This triad is again treated as that of the 2d degree of Eb major, whose tonic is reached by the regular fundamental progressions. From C major into Eb major, by way of F minor : Degrees in F minor : V " u Eft major: 3d. — The modulation from any major scale into the fourth on its right, may be abridged by treating the first major triad as that of the natural 6th degree of the minor scale next at its right, to be succeeded by the regulai fundamental progressions until the dominant harmony is reached, which, being identical in major and minor, affords the best opportunity to conclude with the desired major triad. Thus : in the modulation from C major into E major, the major triad of C is treated as that of the natural 6th degree of E minor, to be succeeded by the regular fundamental progressions, until the dominant harmony of that scale is reached, which may be followed by the tonic triad of E major. From C major into E major, by way of E minor: fe^^^^P degrees in E minor : VI II tt ** E major : V V The modulation from any major scale into the fourth on its left may be abridged by treating the first major triad as dominant triad of the minor scale standing immediately below the desired major scale, to be succeeded FUNDAMENTAL HARMONIES. 117 by its tonic triad. This latter stands upon the 6th degree of the desired major scale, whose tonic triad is reached by the regular fundamental pro- gressions. Thus : in the modulation from C major into At> major, trie triad of C major is treated as dominant triad of F minor, to be succeeded by the tonic triad of that scale. The triad of F minor is then treated as that of the 6th degree of Ab major, whose tonic is reached in the usual manner. From C major into Aj? major, by way of F minor: m ^§ aar Degrees in F minor ." V '* u A\) ma/or: Eg*:zpt2Sif - - fog— * *°— e> — -b a ^ —o — ifcfcC I VI 4th. — The modulation from any major scale into the fifth on its right, needs two abridgments. It leads at first into the major or minor scale on its right ; after that, into the minor scale on the right of the latter. The dominant harmony of this last scale being reached, may be succeeded by the major triad of the desired tone. Thus : in the modulation from C major into B major, the major triad of C is treated as that of the 4th de- gree of G major, or of the natural 6th degree of E minor. After reaching the tonic triad of either of these scales, the modulation leads into B minor, whose dominant harmony may be succeeded by the tonic triad of B major. From C major into B major, by way of G major and B minor: Dtgrees in G major : IV " " B minor : " " B major : The modulation from any major scale into the fifth on its left, may be effected by treating the first major triad as dominant triad of the fourth minor scale on its left, leading to the tonic triad, which is then treated as triad of the 3d degree of the required major scale, whose tonic is reached as usual. Thus : the modulation from C major into Dt* major leads first into the scale of F minor, whose tonic triad, being treated as triad of the 3d degree of Db major, is succeeded by the usual fundamental progressions, until a close is reached. From C major into D£? major, by way of F minor : Degrees in F minor : V ■* "Db major: I III VI II 118 THE CORRECT ORDER OF 5th. — The modulation from any major scale into the sixth on its right, requires two abridgments, viz. : the first major triad is treated as that of the 3d degree of the minor scale standing immediately below, to be succeeded by the usual fundamental progressions, up to its dominant harmony, lead- ing into the major scale of the same name. The tonic triad of this scale is again treated as triad of the 3d degree of the minor scale standing imme- diately below, to be succeeded by the usual fundamental progressions up to its dominant, leading into the major scale of the same name. Thus : the modulation from C major into Fjt .major may be effected by treating the major triad of C as that of the 3d degree of A minor, whose dominant may be succeeded by the major triad of A. This triad being treated as that of the third degree of Fit minor, is followed by the usual progressions up to the dominant, leading into the major triad of the required scale. From C major into Fjj major, by way of A minor and F$ minor: Degrees in A minor: III " " A major: " " F % minor : " " F$ major : The modulation from any major scale into the sixth on its left is also abridged twice. The first major triad is treated as dominant of the fourth minor scale on its left, to be succeeded by its tonic. This minor triad is treated as that of the 2d degree of the second minor scale on its left. The tonic triad of this last minor scale is treated as triad of the 6th degree of the desired major scale, whose tonic is reached as usual. Thus : in modulat- ing from C major into Gb major, we begin by treating the major -triad of C as dominant of F minor. The minor triad of F stands upon the id degree of Eb minor, whose tonic is reached as usual. The minor triad of EJ> stands upon the 6th degree of Gb major, whose tonic is also reached in the usual manner. From C major into Gb major, by way of F minor and Eb minor : Degrees in F minor : V ** " Sjff minor: " " Gfr major : 6th. — The modulation from any major scale into the seventh on its right, is abridged twice. The first major triad is treated as that of the natural 6th degree of the minor scale on the right ; the dominant of this FUNDAMENTAL HARMONIES. 119 minor scale, having been reached in the usual way, may be succeeded by the major triad of the same name, which occurs in the table as the fourth one on the right. The new major triad being treated as triad of the 3d de- gree of the minor scale immediately below, is followed by the usual funda- mental progressions, leading into the tonic of the desired major scale. Thus : the modulation from C major into Cti major begins by treating the major triad of C as that of the natural 6lh degree of E minor, whose dominant being reached, is succeeded by the major triad of the same name. The major triad of E, which stands upon the 3d degree of Cjf mi- nor, is followed by the usual fundamental progressions up to the dominant, leading into the major triad of C$. From C major into Ctt major, by way of E minor and Ctt minor: Degrees in E minor : VI II V '* '* E major : V I 44 " C$ minor : III " " C$ major: m^^rn^^ VI V V The modulation from any major scale into the seventh on its left, is also abridged twice. The first major triad is treated as dominant triad of the fourth minor scale on the left; the tonic triad of this minor scale is then treated as triad of the 2d degree of the 2d minor scale on its left, whose tonic triad stands upon the 3d degree of the required scale. Thus : the modulation from C major to C? major begins by treating the major triad of C as dominant of F minor. The minor triad of F is treated as triad of the 2d degree of EP minor, the tonic of which stands upon the 3d degree of Ct* major, whose tonic is reached in the usual manner. From C major into C|? major, by way of F minor and E|? minor: Degrees in F minor : V " "E\, minor: ** " C\/ major: I III VI II SECTION VII. 1st — The modulation from any minor scale into the second on its right, admits of no abridgment, because these two scales are not related. In other respects, this modulation must be effected in the same way as in the concluding examples of Section IV. 120 THE CORRECT ORDER OF The modulation from any minor scale into the second upon its left, is easily effected, as has been already shown in Section IV. 2d. — To modulate from any minor scale into the third on its right, the tonic triad of the first scale is succeeded by its dominant, which must be treated as triad of the natural 7th degree of the desired minor scale, the tonic of which is reached in the usual manner. Thus : to modulate from A minor into F$ minor, the minor triad of A is succeeded by its dominant triad, standing in Ftt minor upon the natural 7th degree, to be followed by the regular fundamental progressions, until a close is reached. From A minor into F«f minor : -*- §ni Degrees in A mhwr: I " M F% minor : V VII ■JF in ~ =£= -=snf. ^js~: :z&jE3£%^z ~M S^fe 1F w VI The dominant triad of A minor may also be considered as dominant of A major, without, however, altering the result, as the tonic triad of A major stands in all cases upon the 3d degree of Ftt minor. The modulation from any minor scale into the third on its left, is effected by treating the first minor triad as triad of the 2d degree of the major scale on the right, to be followed by its dominant and tonic, which fetter is then treated as dominant of the desired scale. Thus : we modulate from A minor to C minor, by treating the minor triad of A as triad of the zd degree of G major, to be followed by the dominant and tonic of this scale ; the major triad of G is then treated as dominant, to be succeeded by the minor triad of C. From A minor into C minor: fct: ZJSZ Degrees in G major : II " " C minor : V ST I V HI 3d. — To modulate from any minor scale into the fourth on its right, we treat the first minor triad as triad of the 4th degree of the next minor scale on the right, to be succeeded by the regular fundamental progressions up to, the dominant, leading into the major triad of that name. This major triad, occurring upon the 3d degree of the desired minor scale, is followed by the regular fundamental progressions, until its tonic is reached. Thus : in modulating from A minor into C$ minor, we treat the minor triad of A as that of the 4th degree of E minor, which is then succeeded by the usual FUNDAMENTAL HARMONIES. 121 progressions, leading into E major. The major triad of E, being treated as that of the 3d degree of Cjf minor, is followed by the regular fundamental progressions, until its tonic is reached. From A minor into Ctt minor, by way of E minor and E major: r ret s in E minor : IV II V a " £ major ." V I 11 " C% minor: III Ir VI > To modulate from any minor scale into the fourth on its left, we treat the first minor triad as that of the 6th degree of the major scale standing immediately above. The tonic triad of this scale, having been reached as usual, is treated as dominant of the desired scale, to be succeeded by its tonic. Thus : in modulating from A minor into F minor, we treat the minor triad of A as that of the 6th degree of C major, whose tonic triad is reached as usual, and treated as dominant of F minor, after which the mi- nor triad of F may follow. From A minor into F minor, by way of C major : ZS.Z Degrees in C major : " " F minor : VI II V I v 4th. — In modulating from any minor scale into the fifth on its right, we treat the first-minor triad as that of the 4th degree of the next minor scale. on the right, which, being succeeded by the regular fundamental progres- sions up to its dominant, leads into the major scale of that name. The tonic triad of this scale is treated as triad of the natural 6th degree of the desired minor scale, whose tonic is reached as usual. Thus : in modulat- ing from A minor into G$ minor, we treat the minor triad of A as that of the 4th degree of E minor, to be succeeded by the regular fundamental progressions, leading into E major. The major triad of E is then treated as that of the natural 6th degree of G$ minor, whose tonic is reached as usual. From A minor into Git minor, by way of E minor and E major: rfe Jet m^mmm^ Degrees in E minor : IV II V " U E major: V 1 " " OS minor: VI F 122 THE CORRECT ORDER OF In modulating from any minor scale into the fifth on its left, we treat the first minor triad as that of the 3d degree of the major scale on the left, whose tonic triad, being reached in the usual manner, is treated as domi- nant of the desired minor scale. Thus : in modulating from A minor into B[? minor, we treat the minor triad of A as that of the 3d degree of F ma- jor, whose tonic is reached as usual, and treated as dominant of Bl> minor, to be succeeded by the tonic of the desired scale. From A minor into Bt> minor, by way of F major : Degrees in F major '. Ill '* " -Sly minor: 5th. — To modulate from any minor scale into the sixth on its right, we treat the first minor triad as that of the 4th degree of the next minor scale on the right, which, being followed by the regular fundamental progressions, leads into the major scale of that name. The major triad of this scale may be treated as triad either of the 4th degree of the major scale on its right, or of the natural 6th degree of the minor scale on its right. From the tonic of either of these scales, the desired scale is easily reached. Thus : in modulating from A minor into D$ minor, we treat the minor triad of A as that of the 4th degree of E minor, to be succeeded by the regular fundamental progressions, leading into E major. The major triad of E is then treated as that of the 4th degree of B major, or as that of the natural 6th degree of G$ minor, the first standing upon the natural 6th, and the 'latter upon the 4th degree of Djf minor, whose tonic is reached in the usual manner. From A minor into Do minor, by means of E minor and B major : Degrees in E minor : IV II V " " E major : V I " " B major'. IV (i " D% minor : II I VI To modulate from any minor scale into the sixth on its left, we treat the first minor triad as that of the 6th degree of the major scale standing imme- diately above, to be followed by the regular fundamental progressions up to its tonic. The tonic triad of this scale is then treated as dominant of the fourth minor scale on its left, to-be followed by its tonic, which is treated as triad of the 2d degree of the desired scale, whose tonic is reached in the usual manner. Thus : in modulating from A minor into El? minor, we FUNDAMENTAL HARMONIES. 123 treat the minor triad of A as that of the 6th degree of C major, whose tonic is reached as usual, to be treated as dominant of F minor. The minor triad of F is treated as that of the 2d degree of Et> minor, whose tonic is reached as usual. From A minor into Ef minor, by way of C major and F minor: -& -a<2Z ifea: 1 far Degrees in C major : V I " " F minor : u '' E)y minor : I II 6th. — In modulating from any minor scale into the seventh on its right, we treat the first minor triad as that of the 4th degree of the next minor scale on the right, which, being succeeded by the regular fundamental progres- sions, leads into the major scale of that name. The tonic triad of this scale is treated as triad of the 3d degree of the minor scale immediately below, to be followed by the regular fundamental progressions, leading into the major scale of that name. The tonic triad of this scale is then treated as triad of the 3d degree of the desiied minor scale, whose tonic is reached as usual. Thus : in modulating from A minor into A§ minor, we treat the minor triad of A as that of the 4th degree of E minor, to be succeeded by the regular fundamental progressions, leading into E major. The major triad of E is then treated as triad of the 3d degree of C$ mimor, to be fol- lowed by the regular fundamental progressions, leading into C$ major. The major triad of Cft is treated as triad of the 3d degree of Ajf minor, whose tonic is reached as usual. From A minor into Aft minor, by way of E minor and Ctt minor : JF 57 " **" ^—^ S= Deg. in E minor : IV ** '* E major : " " C% minor : ** ** C% major : * u A% minor: V V III VI V V III VI II In modulating from any minor scale into the seventh on its left, we treat the first minor triad as that of the 3d degree of the next major scale on the left, to be succeeded by the regular fundamental progressions until its tonic is reached. The tonic triad of this scale is then treated as dominant triad of the fourth minor scale at its left, to be followed by its tonic. This last minor triad is treated as that of the 2d degree of the desired scale, whose 124 THE CORRECT ORDER OF tonic is reached as usual. Thus : in modulating from A minor into Ab minor, we treat the minor triad of A as that of the 3d degree of F major, to be succeeded by the regular fundamental progressions, till the tonic is reached. The major triad of F is then treated as dominant triad of Bb minor, to be followed by the tonic of that scale. The minor triad of Bb is treated as that of the 2d degree of Ab minor, whose tonic is reached as usual. From A minor into Ab minor, by way of F major and B7 minor : m -2&l :sz^ :g^: =£ & <6>lS-^ ^- [7; i»-^ -jsz 1-r Degrees in F inajor : III " " £k minor : " u A\j minor: VI II I V I II SECTION VIII. In the table, in Section V, we notice that the scale of C$ major has no step to its right, and that the scale of Cb major has none to the left. The scale of F$ major has but one step to its right,' and that of Gb major, but one step tQ its left, etc. The modulation from Cb major into Cjf major may be effected in four progressions, viz. : 1st, from Cb major into Eb major, by means of Eb mi- nor ; 2d, from Eb major into C major, by means of C minor ; 3d, from C major into E major, by means of E minor ; and 4th, from E major into C$ major, by means of CS minor. The modulation from C$ major into Cb major may be effected in six (shorter) progressions, viz.: 1st, from C$ major into F$ minor; 2d, from F# minor into E minor ; 3d, from E minor into C major ; 4th, from C major into F minor ; 5th, from F minor into Eb minor ; and 6th, from Eb minor into Cb major : or, 1st, from C$ major into Fjf minor ; 2d, from F$ minor into D major ; 3d, from D major into G minor ; 4th, from G mi- nor into Eb major ; 5th, from Eb major into Ab minor ; and 6th, from AP minor into Cb major. The modulation from Ab minor into Ajj minor may be effected in five progressions, viz. : 1st, from Ab minor into Eb major, by means of Eb minor; 2d, from Eb major into G major, by means of G minor ; 3d, from G major into E major, by means of E minor ; 4th, from E major into C$ major, by means of Cjf minor ; and 5th, from Cjf major into A$ minor. The modulation from A$ minor into Ab minor may be effected in six progressions, viz. : 1st, from A$ minor into F$ major ; 2d, from F$ major into B minor ; 3d, from B minor into G major ; 4th, from G major into FUNDAMENTAL HARMONIES. 125 C minor ; 5th, from C minor into Et> major ; and 6th, from ~& major into At>. minor. All these modulations should be practised with primary chords, as well as with their inversions. SECTION IX. The close to which a modulation tends, which, for our present purpose, we will call its resolution, may be accelerated by contraction (use of sub- stitute chords), of which examples have been already given, in connection with the discussion of the major and minor scales. The following exam- ples of contraction are recommended as the most. practical. 1st. — The triad of the 4th degree requires the succession of the harmony of the 2d degree, if the dominant harmony is desired. But the triad of the 4th degree may be succeeded directly by the dominant harmony, if the former is treated as substitute for the chord of the seventh of the 2d degree. m razz -O- Fundamentals : F D Degrees in C major: IV II G V n =522= Fundamentals : D B Degrees in A minor: IV II E V A I 2d. — The triad of the 6th degree must also be followed by the har- mony of the 2d degree, if the dominant harmony is to be reached. But the triad of the 6th degree (or rather its first inversion) may be succeeded directly by the dominant harmony, if it is treated as substitute for the chord of the seventh and ninth of the 2d degree. r- 1 1 '_, I i ft. . ei k-g I!- u_ Fundamenta Degrees in C 's: m A D zjor: VI 11 G V C I p ^0^m^E^ Fundamentals: F B E Degrees in A minor : VI II V A I 3d. — The triad of the 3d degree of the major scale may be succeeded directly by the dominant harmony. Fundamentals : £ Degrees in C major : III 126 THE CORRECT ORDER OF 4th. — The major triad of the 3d degree of the minor scale may be suc- ceeded directly by the diminished harmony of the 2d degree, if the former is treated as substitute for the chord of the seventh and ninth of the natural 6th degree. Fundamentals : Degrees in A minor 5th. — The triad of the natural 7th degree of the minor scale may be considered as dependent, if it occurs without octave. In this case, that tone which otherwise would be fundamental, may be treated as thirteenth of the of the 2d degree ; as such it may resolve one degree downward, while the other tones remain, thereby producing the harmony of the 2d degree, which may be succeeded by that of the dominant. J2. Fundamentals : B Degrees in A minor : II SECTION X. The resolution of a modulation may also be delayed, by progressions of the following kind : 1st. — The dominant harmony may be followed by the tonic ; the succes- sion of the harmony of the 3d degree delays the resolution. 2d. — The harmony of the 2d may be followed by that of the 5th degree ; the succession of that of the y/h degree delays the resolution. 3d. — The harmony of the 6th may be followed by that of the 2d degree ; the succession of the harmony of the t,th, of the is/, or of the 3d degree, delays the resolution. 4th. — The harmony of the 3d is followed most naturally by that of the 6th degree ; the succession of the harmony of the is/ degree effects no close, as this must be reached by the usual fundamental progressions. 5th. — The triad of the 4th is followed by the harmony of the 2d degree, if a direct close is intended ; but this close will be delayed, if the triad of the 4th is followed by that of the 6/A or is/, degree, or by the harmony of the 7M degree. FUNDAMENTAL HARMONIES. 127 SECTION XI. If, of two periods, one closes with one tonic, while the succeeding one begins with another, the passage is best effected in accordance with the laws of natural relationship, as shown in the following scheme : a. If the first period ends with a major triad, the new period properly begins with the major triad of its perfect upper or under-fifth, or with the minor triad of its major upper-third, or of its minor under-third, or of its perfect under-fifth. In other words : the concluding triad of the first period must stand upon the 4th or 5th degree of the new major scale, or upon the 3d, or 5th, or 6th degree of the new minor scale. For example : the first period having closed with the major triad of C, the second may begin with the major triad of G or F, or with the minor triad of A, E, or F. b. If the first period closes with a minor triad, the new period begins most naturally with the minor triad of its perfect upper or under-fifth, or with the major triad of its minor upper-third, or of its major under-third, )r of its perfect upper-fifth. In other words : the concluding triad of the first period must stand upon the 4th or 5th degree of the new minor scale, or upon the 6th or 3d degree of the new major scale. The last triad occurs only upon the 5th degree of the first minor scale. For example : the first period having closed with the minor triad of A, the second period may begin with the minor triad of D or E, or with the major triad of C, F, or E. Hence we conclude, that the succession of two triads, each of which is to represent a tonic, must be regarded as a regular fundamental pro- gression. What course is to be pursued in the case of two periods, respectively closing and beginning with a major and minor triad on the same degree, will be self-evident when the explanation of chromatic progressions is fully com- prehended. SECTION XII. We have had, in this part of our treatise, but one object in view, viz. : to show how much may be effected by means of diatonic modulation alone, and to prove that this species is not merely an apparent, but a real modula- tion, enabling us to recognize every tonic triad as such, at any time, and under all circumstances. We conclude with the remark, that the species of modulation explained in this part, is founded on the tempered system — a circumstance which, however, places no difficulties in the way of effecting the harmonic succes- sions in exact conformity with the rules laid down in Parts I and II of this treatise. • 128 THE CORRECT ORDER OF PART IV. CHROMATIC PROGRESSIONS IN THE SCALE OF C MAJOR. SECTION I. If, between two tones of a diatonic scale which are distant a major second, a tone of a relative key is inserted, we have a chromatic progression. The keys related to C major, our normal scale, are : G major, F major, A minor, E minor, D minor, G minor, C minor, and F minor. The insertions in the ascending scale take place as follows : Between c and d we place ci, from the scale of D minor, which also con- tains c and d. Between d and e we place di, from the scale of E minor, which also contains d and e. Between f and g we place f§, from the scale of either G major or G minor, which latter also contains ^"and g. Between g and a we place g§, from the scale of A minor, which also contains g and a. Between a and b we place b\>, from the scale of either F major, D minor, or C minor, the last two of which also contain a and b. Descending. — Between b and a we place bV, from the scales of F major, D minor and C minor, the latter two of which also contain b and a. Between a and g we place a?, from the scale of C minor, which also contains a and g. Between g and f we place f\, from the scale of G major or G minor, which latter also contains g and f. Between e and d we place e\>, from the scale of G minor, which also contains e and d. Between d and c we place db, from the scale of F minor, which also contains d and c. Remark. — We distinguish, moreover, periods artificially related, in which the insertions, descending, are effected thus : g§ is placed between a and g ; d $ between e and d, and ci bdtween c and d. FUNDAMENTAL HARMONIES. 129 SECTION II. A careful examination of Section I will show that the relative minor scales furnish most of the means for chromatic progressions, owing to the fact that the 6th and 7th degrees of the diatonic minor scale succeed each other raised, in ascending, and natural, in descending ; whilst, in chromatic progression, each of these degrees may successively appear in both forms. In the diatonic scale of A minor, the ascending degrees, 5, 6, 7 and 8 ' are, e, f§, g§, a ; those descending from the 8th to the 5th degree are, a ' g> f> e - I n chromatic progressions they are intermingled in the follow- - ing manner : e, f, f.§, g, g$, a ; descending in reversed order. In the diatonic scale of E minor, the ascending degrees from the 5th to ■ the 8th, are, b, cjf, d§, e ; those descending from the 8th to the 5th are; e, d, c, b. The intermingling of these will give the chromatic progressions, , b, c, c$, d, djk, e ; descending in reversed order. In the diatonic scale of D minor, the ascending degrees from the 5th to the 8th are, a, b, c\, d ; those descending from the 8th to the 5th are, d,' c, b)r, a. The intermingling of these gives the chromatic progressions, a, bfy, b§, c, ci, d ; descending in reversed order. The examination of these three minor scales enables us to understand : the nature of the ascending chromatic scale of C major, since it is of their- component tones that this scale is made up, viz. : c, c§, d, d%, e, f, f\, g, g\, a, bV, 6% c. The diatonic scale of C minor ascends from the 5th to the 8th degree : g, a, b, c, and descends from the 8th to the 5th : c, bfy, a\>, g. These - being intermingled give the chromatic progressions : g, afc», a % ^?, &% c, descending in reversed order. The diatonic scale of G minor ascends from the 5th to the 8th degree : <^> e , f% g ; an d descends from the 8th to the 5th : g, f, e\>, d. These • being intermingled give the chromatic progressions ; d, efy, et\, f, f\, gj descending in reversed order." The diatonic scale of F minor ascends from the 5th to the 8th degree : c, d, e, f, and descends from the 8th to the 5th : f, e\>, dv, c. These being intermingled give the chromatic progressions : c, dfy, dfy. efy, e\ r f>\ descending in reversed order. The examination of the last three minor scales enables us to understand the nature of the descending chromatic scale of C major, since it is of their component tones that this scale is made up,> viz,-: f, b>- bb, a, a^ g, ffo /% e, eb, d, d\f, c. 130 THE CORRECT ORDER OF SECTION III. The use of the chromatic tones cannot be extended to fundamentals. The chromatic scale of C major, therefore, has no other fundamentals than the diatonic, -viz. : the diatonic degrees, C, D, E, F, G, A, and B, which may be treated for a short time as degrees of a relative scale, thus : I. The tones, C, F, D, G, may be treated as the same degrees in C major and C minor, that is, in both as the ist, 4th, 2d and 5th degrees.. II. The tones, D, G, E, A, 2d, 5th, 3d and 6th degrees in C major; may be treated as ist, 4th, 2d and 5th degrees in D minor. III. The tones, E, A, B, 3d, 6th and 7th degTees in C major, may be treated as ist, 4th and 5th degrees in E minor. IV. The tones, F, G, C, 4th, 5th and ist degrees in C major, may be treated as ist, 2d and 5th degrees in F major, or F minor. V. The tones, G, C, A, D, 5th, ist, 6th and 2d degrees in .C major, may be treated as ist, 4th, 2d and 5th degrees in G major, or G minor, VI. The tones, A, D, B, E, 6th, 2d, 7th and 3d degrees in C major, may be treated as ist, 4th, 2d and 5th degrees in A minor. SECTION IV. I. The triad of the ist degree of the scale of C major may be altered into that of the ist degree of C minor by changing its major third, e, into a minor third, el*. It needs no alteration to stand as triad of the 5th degree of F major, or F minor, unless the minor triad of the latter is required. It needs no alteration to stand as triad of the 4th degree of G major, but its major third must be altered into a minor third, if it is to stand as characteristic triad of the 4th degree of G minor. Its change into the characteristic triad of the 3d degree of A minor is effected by the alteration of its perfect fifth, g, into an augmented fifth, g%.. It needs no alteration to stand as triad of the natural 6th degree of E minor, or (in the tempered system) of the natural 7th degree of D minor. The triad of the ist degree of C major cannot by right be altered into that cf the 2d degree of Bl? major, ncr into that of the 6th degree of Et? major, because bb and ev are not diatonic tones in C major, and therefore cannot stand as fundamentals of that scale. II. The triad of the 2d degree of C major (in the tempered system) needs -no alteration to stand as triad of the ist degree of D minor. Its change into the triad of the 5th degree of G major or G minor depends upon the alteration of its minor third, f, into a major third, _/$. It needs no .alteration (in the tempered system) to stand as minor triad of the 4th FUNDAMENTAL HARMONIES. 131 degree of A minor, or of the 6th degree of F major. Its alteration into tire characteristic triad of the 2d degree of C minor is effected by the change of its {dnbious) fifth,- a, into a diminished fifth, a\f. III. The triad of the 3d degree of C major needs no alteration to stand as triad of the 1st degree of E minor, or as that of the 6th degree of G rflajor. Its alteration into the characteristic triad of the 5th degree of A mi- nor, is effected by the change of its minor third, g, into a major third, g$. Its alteration into the characteristic triad of the 2d degree of D minor, or of the 7th degree of F major, is effected by the change of the perfect fifth, b, into a diminished fifth, b\>. IV. The triad of the 4th degree of C major needs no alteration to stand as triad of the 1st degree of F major, or of the natural 6th degree of A mi- nor. Its change into the triad of the 1st degree of F minor, or into the characteristic triad of the 4th degree of C minor, is effected by altering its major third, a, into a minor third, «|?. Its change into the characteristic triad of the 3d degree of D minor, is effected by the alteration of its perfect fifth, e, into an augmented fifth, c Jf. It needs no alteration (in the tempered system) to stand as triad of the natural 7th degree of G minor. It cannot be treated as the triad of the 5th degree of Bl> major, nor be altered into the t*i*d of the 2d degree of Eb major, or of the 6th degree of At> major, because bY, eV and a\r are not diatonic tones of C major, and therefore, cannot occur as fundamentals of that scale. V. The triad of the 5th degree of C major needs no alteration to be- come the triad of the 1st degree of G major, nor of the 5th degree of C minor, unless the minor triad of the 5th degree of C minor is required, which is obtained by the alteration of the major third, 6, into a minor third, ik Its change into the triad of the 1st degree of G minor, or of the 2d degree of F major, or of the 4th degree of D minor, is also effected by the alteration of the major third, b, into a minor third, bb. Its alteration into the characteristic triad of the 2d degree of F minor is effected by the change of the major third into a minor third, and by that of the perfect fifth, d, into a diminished fifth, db. Its change into the characteristic triad of the 3d degree of E minor, is effected by the alteration of its perfect fifth, d, into an augmented fifth, d$. VI. The triad of the 6th degree of C major needs no change to stand as triad of the 1st degree of A minor, or of the 3d degree of F major, or Of the 4th degree of E minor. It also needs no alteration (in the tem- pered system) to stand as triad of the 2d degree of G major. Its change into the characteristic triad of the 2d degree of G minor is effected by altering its perfect fifth, e, into a diminished fifth, e\f. Its alteration into the characteristic triad of the 5th degree of D minor, is' effected by the change of its minor third, c, into a major third, <:$. 132 THE CORRECT ORDER OF VII. The triad of the 7th degree of C major cannot become a triad pf the 1 st, 5th, 4th or 3d degree of a relative scale, because its diminished fifth cannot be altered into a perfect fifth. Hence it can be treated only as triad of the 2d degree of A minor, as triad of the raised 6th degree of D minor, and as triad of the raised 7th degree of C minor. These observations sufficiently explain the mode of alteration also of the chords of the seventh, and of the seventh and ninth, of each degree in the scale of C major, with special reference to their occurrence in a relative scale. We will only add, that major sevenths or ninths may be altered into minor sevenths -or ninths, but not vice versa. In the chromatic alterations of chords of the seventh, and of the seventh and ninth, when in the former the third and seventh, and in the latter the fifth and ninth are to be lowered, it is better that the change effecting a nearer, should precede that effecting a more distant relationship. Whenever, in a chromatically altered chord of the seventh, a simulta- neous resolution of the minor third and minor seventh is intended [e. g., by means of the ascent of the fundamentals by degrees), care must be taken to place the minor third above the seventh ; otherwise, consecutive fifths would result For the same reason, i. e. , to avoid consecutive fifths, the diminished fifth of a chromatically altered chord of the seventh and ninth must stand above the ninth, whenever their simultaneous resolution is intended. SECTION V. The case of different triads upon the same fundamental, is best explained by the consideration of the several minor scales. Both the major and the minor triad of C stand upon the 4th degree of G minor, and upon the 5th of F minor. Both the major and the augmented triad of C stand upon the 3d degree of A minor. Both the minor and the major triad of D stand upon the 4th degree of A minor, and upon the 5th of G minor. Both the minor and the diminished triad of D stand upon the 2d degree of C minor. Both the minor and the major triad of E stand upon the 5th degree of A minor. Both the minor and the diminished triad of E stand upon the 2d degree of D minor. Both the major and the minor triad of F stand upon the 4th degree of C minor. FUNDAMENTAL HARMONIES. 133 Both the major and the augmented triad of F stand upon the 3d degree of D minor. Both the major and the minor triad of G stand upon the 4th degree of D minor, and upon the 5th of C minor. . Both the minor and the diminished triad of G stand upon the 2d degree of F minor. Both the major and the augmented triad of G stand upon the 3d degree of E minor. Both the minor and the major triad of A stand upon the 4th degree of of E minor, and upon the 5th of D minor. Both the minor and the diminished triad of A stand upon the 2d degree of G minor. SECTION VI. Alterations of diatonic into chromatic tones, and vice versa, must be made in the same voice, Otherwise, harsh and faulty progressions will result, e. g. : $ Faulty : ^ =&>-- FfeE« T 4- r? ir- A faulty progression of this nature is termed false relation, and is avoided in the following way : I I On: $ r * -> ai* t-f- js=te 1 i ^£ If it be asked, whether this false relation still exists, if the chromatic alteration takes place with an exchange of tones of the same fundamental, such exchange being connected by the intermediate passing tones, e. g. : ^ i=± i 'j- J_l *t =8 £ ■*&- zz£i *?=*? r r t r r -r t r r we maintain, that even in this case the false relation still exists. But there are other progressions, seemingly involving the false relation, yet perfectly legitimate, viz. : 134 THE CORRECT ORDER OF ist. — Progressions which occur during the continuance of the same fundamental, by means of chromatic changing notes, e. g. : Diatonic : Chromatic : More correctly ptk£?h $^s=k^k fa^f dsi Diatonic Chromatic : I More correctly : ^j=JEgg^f=g f^ fff i 2d. — Progressions which occur, if two voices sound the same tone of a chord ; in which case, both voices may progress chromatically in contrary motion, e. g. : 3e T *± * ^es« E^E =F^= -^ 3d. — Progressions which occur if the chromatic alteration takes place at the end and at the beginning of a passage ; provided the altered tone caus- ing the false relation does not occur in the two upper or in the two outer voices, e. g. : Good. i -0 ' FT — 1 P — fl . ' * » r T^^~^f^~f Faulty s i I22Z r=r ^=22=^ =221 Efc zaz SECTION VII. Every chromatic harmony should be founded upon a diatonic harmony. How chromatic progressions arise from a diatonic passage, will be exem- plified hereafter. FUNDAMENTAL HARMONIES. 136 It should also be mentioned in this place, that in those examples in which the fundamentals are found in the lowest voice, the usual indications of the fundamentals will be omitted ; and that in those examples in which inversions occur, the fundamentals will be indicated by capital letters. SECTION VIII. The returning motion of each component tone of any triad of the dia- tonic majoK scale, is effected by means of chromatic steps, i. e. , we raise or lower that tone which is to be reached by the returning motion ; provided the tone thus altered is a diatonic tone of some relative scale. . The following examples will illustrate the returning motions of the com- ponent tones of the tonic triad of C major. Diatonic, (a) Chromatic. {5) Diatonic, (r) Chromatic, (d) ^^PS^ ^S^=g The progression at (d) is preferable to that at (c), as /%. more naturally t ret,urns to g than/ft. Diatonic, (e) Chromatic . (_/") Diatonic, {g) Chromatic. (A) it £p T<9—>- **¥- a I -»\ T '¥■ ^i The progression at (A) is not good, because it is more natural for f\ to ascend to g, than to return to e. Chromatic . Chromatic. mm^m^ m^m^ .In the above two examples, the chromatic alterations indicate the dominant harmony of F minor and F major. Diatonic. Chromatic. Diatonic. Chromatic. $^pi%*S $E^?=^i|ip ■* V* -sr m^^m^m 136 THE CORRECT ORDER OF SECTION IX. 1st. — The chromatic steps arising from the descent of the fundamental by a fifth, or its ascent by a fourth : Diatonic. Chromatic. Chromatic. M, -*. i* ffegEl 1 ■-J-' g-H i Diatonic. ^3^ Chromatic . i Diatonic. Chro-maiic. ^s: za. zsr 1SSL IE =~2?- C hro malic . mmw^m^ An exemplification of the chromatic progression from the dimin- ished triad of the 7th to the minor triad of the 3d degree, is involved in Section XVI. 2d. — The chromatic steps arising from the ascent of the fundamental by a fifth, or its descent by a fourth : Diatonic. Chromatic. Diatonic. i -&. I25L M: m ti^m w== Chromatic, ^ -JSL J?* 3d. — The chromatic steps arising from the descent of the fundamental by a third : Diatonic. Chromatic. Chromatic. i^psf^p ^^p^# i z2z Chromatic . mmm =#= The last examples may also occur with passing tones, and returning motions, e. g. : FUNDAMENTAL HARMONIES. 137 m if-=z* H raz /Tmf. C. i^ -iS- A fei 4th. — When the fundamental ascends by a third, chromatic steps are not conceivable. 5th. — With regard to the seeming ascent of the fundamental by degrees, it may be observed, that if the first chord contains two minor thirds of the tacit fundamental, one of them may step to the major third, and the other to the minor ninth, eg.: ^ms^mm D C f ^ ° -f r r Fund. CAD Cjf and Wy are component parts of the dominant chord of the seventh and ninth of D minor. Diatonic. Chromatic. n - and dfy are component parts of the dominant harmony of F minor. Diatonic. Chromatic. i Fund. A D G Eifc T D G #t*# ^~ /"jf from G minor is reached by returning motion. The following examples will illustrate other chromatic steps arising from the seeming ascent of fundamentals by degrees : Diatonic. Chromatic. i Al -£ Fund. C A Diatonic. D pte?y IffiZ £ C Chromatic. -AM-- A D za= EIE Fund. E C Diatonic. E C F Chromatic. $ ~sz 4l & — r— zr f»W. G E A G £ A In the succession of the chord of the seventh of the 6th degree and the triad of the 7th degree, of the scale of C major, the former must be treated as substitute for the chord of the seventh and ninth of the 4th degree. This progression can be chromatically altered only by a license, which we may explain thus: The triad of the 4th degree cannot become a triad of the 2d degree (see Section IV, rv), whilst the triad of the 7th degree cannot become one of the 5th degree (See Section IV, vii). But since the above-mentioned succession introduces the 4th degree only as tacit fundamental, the license of treating _/$ as such, instead of fk, may be made use of, and f\ may be treated as standing upon the 2d degree of E minor. The succeeding triad of b may then stand as dominant triad of E minor, leading into the tonic triad of that scale, e. g. : Diatonic. Chromatic. w ^ Fund. A. -Z7- B 3E E F 3 % ISE ai n sr B FUNDAMENTAL HARMONIES. 139 Diatonic, Chromatic. mmmmm Fund. G. 6th. — The chromatic steps arising from the seeming descent of the funda- mental bv degrees : Diatonic. Chromatic- i Ei£* ->s>- Fund. C ' Diatonic. =3^ G zetz r^T- =3F c. Chromatic. i ^ ipc =3g.: Fu7td. D G Diatonic* % ^ D G C Chromatic. =§= \e!%^\ i Diatonic, a . Fund. £ A V Chromatic. £. 19 m zz~o~ rzrar ^ -£l 3& Fund. G F G. Diatonic. C F Chromatic. % =3fc 3. fe= -t- =3§r /«k£ A D The diatonic harmonies of the 4th and 3d degrees, and of the 7th and 6th degrees of the major scale, cannot be connected by chromatic progres- sions. SECTION X. When any harmony (triad, chord of the seventh, or chord of the seventh and ninth) is followed by a triad whose fundamental is a fifth lower, such harmony may be treated as standing upon the 5th, and the triad as stand- ing upon the 1st degree of a relative scale; provided the component tones of the former are altered accordingly, and that the latter consists of a major or minor third, and perfect fifth of diatonic tones. 140 THE CORRECT ORDER OF A dominant harmony succeeded by a major third of diatonic tones, may also be considered as arising from a minor scale, if its major ninth is altered into a minor ninth, thus : ist. — The harmony of the ist degree of C major may be treated as dominant harmony of F major, or F minor, and the succeeding funda- mental as tonic of F major : « X-a- S J&L & za_ Fmid. C. § Chromatic. a. i*r—4*- -j?r7 ^ feafe % z=&± Ti- Fund. C. F F 2d. — The harmony of the 2d degree of C major may be treated (in the tempered system) as dominant harmony of G major, or G minor, and the succeeding triad of the 5th degree of C major as tonic triad of G major : Diatonic. 3d. — The harmony of the 3d degree of C major may be treated as dominant harmony of A minor, and the succeeding triad of the 6th degree of C major, as tonic triad of A minor : Fund. E Chromatic. % ^ d=k 3 ^gy=q^^^ ^ HI Fund. E. FUNDAMENTAL P4RM0NIES. 141 4th. — The harmony of the 4th degree of C major cannot be treated as a dominant harmony (vide Section IV, iv). 5th. — The harmony of the 5th degree of C major may be treated as dominant harmony of C minor, if its major ninth is altered into a minor ninth ; but it must be succeeded by the major triad of C : Diatonic. i J Chromatic. z3&.\ '1 j>n~ is/Ssl. 6th. — The harmony of the 6th degree of C major may be treated as dominant harmony of D minor, if it is succeeded by the triad of the 2d degree of C major. Diatonic- m ~s>~ -TO- =3g: rsc 1ST : F ^m Fund, A. Chromatic* J *L fr^TfTr^ -&- ZT3Z ~r<~ ^ -g= r Fund. A.. SECTION XI. In any cadence-like succession (involving fundamental descents by fifths) of two chords of the seventh and a triad, the first chord of the seventh may be treated as that of the 2d degree, the second as that of the 5th degree, and the triad as that of the 1st degree of a relative scale, provided this triad consists of a minor or major third and a perfect fifth, and contains no other but diatonic tones of the first major scale. Thus : 1st. — The chords of the seventh of the 2d and 5th degrees in C majoi, may be treated as those of the same degrees in C minor, and be resolved into C major : Diatonic. Chromatic. % Zjgl -& 3E Pip 142 THE CORRECT ORDER OF ■2d.-— The chords of the seventh of the 6th and 2d degrees in C major, may > be treated as those of the 2d and 5th degrees of G major, or G minor, and be resolved into G major : H Diatonic 3 9r-z cr-6>~. ..a ■ Chromatic. , *3 W-^-teBr-l rS-% - ^ r~a :i £=#5 \. S7 ~s- :=g^ -^rH -S— !! ^==fe =5*2 * ~%- A -5M-'- f J « r s> J — ir ~ ■ — __ a 3d. — The chords of the seventh of the 3d and 6th degrees in C major, may be treated as those of the 2d and 5th degrees of D minor, and be resolved into D minor : Diatonic. a— Chromatic. -CH2- — rr ^» — - j 4th. — The chords of the seventh of the 7th and 3d degrees in C major, may be treated as those of the 2d and 5th degrees of A minor, and be re- solved into A minor : Diatonic. Chromatic. t f- ^&&~ \-£z^z -££&. :ac 5th. — The chords of the seventh of the 5th and 1st degrees in C major, may be treated as those of the 2d and 5th degrees of F major or F minor, and be resolved into F major : Diatonic. •&• -09 1 ilH i Chromatic. -<&-. =*=£$ -6&- 73% Or % -60L- 3£ l^sz. — — y nr ■■ The following examples will show that the above explained chromatic progressions are related to the scale of C major : FUNDAMENTAL HARMONIES. VitU 3 and i. — Diatonic.. g_, n , rs- ■■ .'-l, ,9. fE^§=;S3EE=i^^i -»- -«?■- EIE Chromatic. 143 ^iyjp 4 and z. — Diatonic. 1222; zgrffc cf.. ■•LJ- ^ ISI Chromatic. i ^ ~ g^~ 3S2I ^ ^1, s ****=* 3F s Vide s- — Diatonic. $ stE&fe^gsS-SSESSsE! ^^^^ Chromatic. SECTION XII. Whenever a chord of the seventh (or triad, or chord of the seventh and ninth) is treated as standing upon the 2d degree of a relative scale, it should be succeeded by the dominant harmony of the same scale, to be resolved into its tonic triad, as explained in the last Section. But if, instead of a triad, a chord of the seventh follows (on the same degree), the preceding dominant chord of the seventh should be altered into a chord of the seventh of the 2d degree, by changing the major into a minor third, or by altering the perfect into a diminished fifth. The next chord of the seventh must then be altered into a dominant chord of the seventh, after which it is suc- ceeded by that tonic triad which is the natural resolution of the last chord of the seventh. 144 THE CORRECT ORDER OF The succession of three chords of the seventh and a triad in cadence-like order, may be explained thus : ist. — If, in the succession of the chords of the seventh of the 6th, 2d and 5th degrees, and the triad of the ist degree, of C major, we treat the first chord of the seventh as that of the 2d degree in G major, or if we alter it into that of the 2d degree in G minor; the succeeding (second) chord of the seventh should be made the dominant harmony of these scales ; but since this last chord of the seventh is not succeeded by the tonic triad of G, but by another chord of the seventh, it follows that the dominant harmony of G major should be so altered as to stand for a chord of the seventh of the 2d degree of C major, or C minor. The last (third) chord of the seventh must become a dominant chord of the seventh, and resolve into the tonic triad of C major : Diatonic. pmmmm- ^E Chromatic. Or: 2d. — The chords of the seventh of the 3d, 6th and 2d, and the triad of the 5th degree, of C major, may be treated as those of the 6th, 2d, 5th and ist degrees in G minor and G major : Diatonic. Ckromatic. 3d. — The chords of the seventh of the 7th, 3d and 6th, and the triad of bhe 2d degree, of C major, may be treated as those of the 6th, 2d, 5th and ist degrees in D minor : Diatonic. FUNDAMENTAL HARMONIES. 145 4th. — The chords of the seventh of the 2d, 5th and 1st, and the triad of the 4th degree, of C major, may be treated as those of the 6th, 2d, 5th and 1st degrees in F minor, or F major : Diatonic. Chromatic. - <5r5r""§! — 1 — w — zxr~ & — 1 r _ o — -jggu -<£ f-^pf^F ^m The following succession of the chords of the seventh will sufficiently show the relationship of all these chromatic steps to the scale of C major : Diatonic, SECTION XIII. Since the dominant chords of the seventh may enter free and unprepared,' it is not necessary to alter any of them into chords of the seventh of the 2d degree. Consequently, the chords of the seventh of the 3d, 6th, 2d, 5th, and 1st degrees, may be altered into the dominant chords of A minor, D minor, G major or G minor, C major and F major. In this manner, all minor sevenths may enter freely together with major thirds, e. g. : E§1 -- &Z & 1 »*&T * Fund: C. Or: C D D Chromatic. This last example shows that the dominant chord of the seventh in F major is treated as the chord of the seventh and ninth of the 5th degree in D minor, the minor third of which becomes a major third, simultaneously with the resolution of the ninth into the octave. The treatment will be similar in analogous cases, e. g., in C major, if the harmony of the chord of the seventh of the 4th is followed by that of the 2d degree — F and D being treated as natural 7th and 5th degrees in G mi- nor and G major : and again, if the harmony of the chord of the seventh of the 5th is succeeded by that of the 3d degree — G and E being treated as natural 7th and 5th degrees in A minor. These cases may be illustrated thus: Fuml: C. Or: C D D G... G E. A A F.... F D.. G G n C c The same example, with chromatic alterations : 148 THE CORRECT ORDER OF SECTION XV. The progressions given in the last Section may be treated as occurring upon the 4th and 2d degrees of a relative scale. Thus, e. g, in C major, if the harmony of the chord of the seventh of the 1st is followed by that of the 6th degree, they may be treated as occurring, the former, upon the 4th, and the latter, upon the 2d degree, in G major, to be succeeded by the chord of the seventh of the 5th, and the triad of the 1st degree. Chromatic. The following chromatic alterations may be made in the above progres- sion, with a view to indicate a relationship to the scale of G minor: Chromatic. Z22± g sHS ^3zr ^-f^?^^®rt$3sr-^& Fund. C. c D D G G c. c D D G G The treatment will be similar in the following analogous cases, e. g. , in C major; 1st. — If the harmony of the chord of the seventh of the 2d is succeeded by that of the 7th degree, they may be treated as occurring, the former upon the 4th, and the latter upon the 2d degree, in A minor, to be followed by the dominant and tonic triads, respectively. 2d. — If the harmony of the chord of the seventh of the 4th is suc- ceeded by that of the 2d degree, both may be treated as occurring on the same degrees in C minor, to be followed by the dominant and tonic triads of C major. 3d. — If the harmony of the chord of the seventh of the 5th is suc- ceeded by that of the 3d degree, they may be treated as occurring, the for- mer, upon the 4th, and the latter upon the 2d degree of D minor, to be followed by the dominant and tonic triads of that scale. The following examples illustrate the relationship of all these chromatic progressions to the scale of C major : FUNDAMENTAL HARMONIES. 149 Fund: C Or: C liiSi V.... D # F D The same, with chromatic alterations : stSsiii^ifllEpii ^s^ SECTION XVI. A chord of the seventh (or seventh and ninth), treated as that of the 2d degree of a relative minor scale, must be composed of a minor third, a diminished fifth, and a minor seventh (minor ninth). By altering the minor into a major third, without altering the diminished fifth, a chord is formed whose major third occurs in another scale than that which contains the diminished fifth, because the major third places the chord upon the 5th, while the diminished fifth places it upon the 2d degree. It is therefore a real chromatic chord, not to be found in any diatonic scale. Thus : the chord of the seventh, or seventh and ninth, of the 7th degree in C major, occurs in A minor upon the 2d degree, and has in both cases a diminished fifth. If we alter the minor into a major third, the chord will seemingly belong to the scale of E minor, or Y, major ; but in reality it must be treated as stand- ing upon the 2d degree in A minor, and, accordingly, be succeeded by the dominant harmony, resolving into the tonic triad of that scale. Diatonic. S Chromatic. =£jg^e0l Ep_J__ffl -pT^ ^jpl Dtereet in C Major .VII III VI Deg. in A minor: II 150 THE CORRECT ORDER OF This last example contains an interval hitherto not explained, resulting from the simultaneous sound of the major third and diminished fifth. If the major third occurs in the higher voice, an augmented sixth arises ; but if the diminished fifth occurs in the higher voice, we have a diminished third. The treatment will be similar in the following analogous cases : ist. — The chord of the seventh, or seventh and ninth, of the 3d degree in C major, may be chromatically altered into that of the 2d degree in D minor, by lowering its perfect into a diminished fifth, and by raising its minor to a major third; to be followed by the dominant and tonic triads ,of the same scale. 2d. — The same course may be pursued with the chord of the seventh and ninth of the 6th degree in C major, with the additional alteration of the major into the minor ninth ; this chord may then be treated as standing upon the 2d degree of G minor. 3d. — The same method may be observed with the chord of the seventh and ninth of the 2d degree in C major, when it may be treated as standing upon the 2d degree in C minor. 4th. — We may proceed in like manner with the chord of the seventh and ninth of the 5th degree in C major, when it may be treated as standing upon the 2d degree in F minor. The following examples illustrate the relationship of all these chromatic progressions to the scale of C major : I. Diatonic. Chromatic. J-M ^^ gj^ FUNDAMENTAL HARMONIES. 151 II. Diatonic. UP& Chromatic* ^^^--ri^i-M •s>- £** 3=5^3=^ i^HS J-Jte- — -^«- -^ i , i ->w — -j — £ P§e| ~2?- r The harshness of these chromatic chords of the seventh and ninth is easily avoided by the omission of the fundamental tone. This omission causes the chromatic chords of the seventh and ninth to sound (in the tem- pered system) like dominant chords of the seventh, the only difference being, that they are differently resolved. SECTION XVII. The resolution of these chromatic chords of the seventh and ninth, without fundamental, into the harmony of the major triad of the next funda- mental, depends (if the ninth resolves simultaneously with the entrance of the next fundamental) upon the following position of the component tones, viz : the seventh of the (tacit) fundamental must be in the lowest voice ; the ninth next above ; above this, the diminished fifth, and the major third in the highest voice, as in the following example : Diatonic. m i =sb ~-g-_-^_-g: g# &-%>- rsc 152 THE CORRECT ORDER OF I a& -»- ?=* jSSSZ F Chromatic. :z£ |d=^i i :sfc:#sfc -P-f— r #- r -»- -br : P— fp "rz^^f*^ 3E i 3k *=*=£ ■p - " *f *■ — -p- ^ *n ■w EEg ffi = j?^ fe^ -fStffl The delay of the resolution of the ninth into the fifth of the following fundamental, is best accompanied by the delay of the resolution of the seventh into the third ; both remaining suspended as thirteenth and eleventh of the next fundamental, until they resolve according to rule. Here the position is immaterial, as the fundamentals of the chords of the seventh and ninth are generally inaudible. Diatonic. FUNDAMENTAL HARMONIES A 153 «^ feS =^g r o&~Z F B ^m =t=^=t= ^^fp^?^F*r* : c SECTION XVIII. The perfect fifth of a chromatic chord of the seventh and ninth may be lowered to a diminished fifth, before the major third is changed into a mi- nor third. In the following example, the chromatic alterations, as at (b), will be found amply justified by the diatonic progressions, as at (a). Diatonic, (a) e — -a — t-> - £ -A --Z2T- ZZZ1 ZSZ =r~ - -f — s; I ' ~J- §i .eL. i ~ °- — & — | — - & i i s= ^ gt F 15 ^ CAD. Chromatic. (£) iHi ! I =St S I h I- p=§i I2Z: ISC 22Z =r=ie -&-*- -f- =ii Jfc*4 ^-w— J^i-U' 154 THE CORRECT ORDER OF * S m ZStL ~z>- 1U— 3^^# ~b ^~ SECTION XIX. Another license in chromatic progression is, that instead of a triad, capable of becoming a dominant, a chord of the seventh, or of the seventh and ninth, may enter free, provided it is followed by its relative tonic. ist. — The triad of the ist degree of C major cannot be succeeded diaton- ically by a chord of the seventh, or of the seventh and ninth, of the 3d degree, but chromatically, if C, E and A are treated as 3d, 5th and ist degrees in A minor. Diatonic. Chromatic. Or: EJ pJ^ I=j=jjl ^ --CZL : S=F _e/o_ O— S 7SZ 2d. — The triad of the 4th degree of C major cannot be succeeded diaton- ically by a chord of the seventh, or of the seventh and ninth, of the 6th degree, but chromatically, if F, A and D are treated as 3d, 5th and ist degrees in D minor. Diatonic. Chromatic. iMsksm^msm 3d. — The triad of the 6th degree of C major cannot be succeeded diatonically by a chord of the seventh, or of the seventh and ninth, of the ist degree, but chromatically, if A, C and F are treated as 3d, 5th and ist degrees in F major. Diatonic. Chromatic. _g_, _g_ 1 ^ FUNDAMENTAL HARMONIES. 155 4th. — The triad of the 7th degree cannot, as we have seen, be altered into a dominant triad. The triad of the 5th degree, however, may be succeeded by the major triad of the 7th degree, provided that this latter is followed by the triad of the 3d degree. Such a case would be, e. g., in C major, if G, B, and E were treated as 3d, 5th and 1st degrees in E minor. But this progression must be regarded only as a license, being foreign to the nature of the scale of C major. Diatonic. Chromatic. 5th. — In the following chromatic progression, sxN.B., the diminished triad of the 7th degree seems to be followed by the harmony of the chord of the seventh of the 5th degree of G major; upon examination of the fundamentals it will be found, however, that the seeming fundamental of the triad of the 7th degree is but a passing tone arising from the fundamen- tal D. Chromatic. J, N.B. 'iff" -27 " [ ' 1 rr ^ Fund. D. The following example shows the relationship of all these chromatic progressions to the scale of C major. Diatonic. Chromatic. i ^El =^= ^ llEIIlElBill 156 THE CORRECT ORDER OF SECTION XX. The chromatic progressions remaining, being of a very complicated nature, can be explained only by showing how one gives rise to another. In the following diatonic progression, containing the most important funda- mentals of the scale of C major in this order: C, F, D, G, C, — C and F may be treated as dominant and tonic of F minor — F, D, and G, as 4th, 2d and 5th degrees of C minor, D as dominant of G minor, resolving into G major, to be treated as dominant of C major, ultimately to resolve into the tonic triad of that scale. Diatonic. **=^^^ & Chromatic. Another variation arises, if the dominant ninth in F minor remains as thirteenth of F, and if the minor triad of F stands for the chord of the seventh of the 2d degree of C minor, to be succeeded by the dominant chord of the seventh and ninth of G minor ; and if the seventh of this chord remains suspended as eleventh of G, resolving into the tonic triad of G major, to close with that of C major. f i^iN^ i^fa •zt \=& 3C ^ ^ m FUNDAMENTAL HARMONIES. 157 Still another variation arises, if the thirteenth of F resolves, simultaneously with the passage of the other voices, into the harmony of D, which may be treated at once as dominant of G minor, thereby shortening the progression by one measure. r-fc (i zfc S See® * iffii =s= EE -S£ EE re: isr Now, the thirteenth of F may remain suspended, even until the entrance of the harmony of D (dominant of G minor) is effected, thereby introduc- ing tones belonging to two different scales not related. i at m A- ?pee1E :)f» r ^ Bfc -JZL This last example contains an interval which has not yet been mentioned, viz : the diminished sixth, occurring here between _/$ and d\> (/and dfy form- ing already a minor sixth). The inversion (dY and f$) gives an augmented third, since dfy and y form already a major third. The last progression may be varied again, if, with the thirteenth of F, the octave remains suspended after the entrance of the major third of D, giving the simultaneous sound of f\ and f (a diminished octave, whose in- version forms an augmented prime). m i £c J=±=»L=taJ £ rszz =*»= ^ is??: SECTION XXI. The nature of chromatic progression enables every degree but the seventh to stand as a tonic, provided the fundamentals succeed each, other in such order that each is independent of the other. The following succes- 158 THE CORRECT ORDER OF sionof fundamentals, with their triads, will be found of most practical use, viz : C, G, E, A, F, D, G, C. This succession may be justified thus : ist. — C, having stood in the outset as tonic, may be treated as sub- dominant of G major, to be followed by the tonic of that scale. 2d. — G, having stood as tonic, may be treated as 3d degree of E minor, to be followed by the tonic of that scale. 3d. — E having stood as tonic, the A following may be treated as 4th degree of the same scale, unless we treat E • as 5th degree of A minor, and A as tonic of that scale. 4th.— A, having stood as tonic, may be treated as 3d degree of F major, to be followed by the tonic of that scale. 5th. — F, having stood as tonic, may be treated as 3d degree of D mi- nor, to be followed by the tonic of that scale. 6th. — D, having stood as tonic, may be treated as 2d degree of C major, to be followed by G, as dominant of that scale. 7th. — G, having stood as dominant of C, may now be treated as tonic of G major. 8th. — G, having stood as tonic, may now be treated as dominant of C Oiajor, to be followed by the tonic of that scale. The recognition of a triad as tonic triad, depends upon its connection with its dominant or sub-dominant harmony, or with both. By way of illustration of the foregoing, we give the following succession of diatonic triads in C major, forming, as it were, a theme for chromatic variations. Theme., |=&=iSi=E:^^ii^^ &= -jsr. -jsz First variation : Each triad may become a tonic, during the continuance of the same measure, if followed by its dominant, which being a secondary harmony, the progression cannot be considered a real fundamental one in C major. 1 1 1 ip^ns^ ■e fHffi 32= A.. r-rt- FUNDAMENTAL HARMONIES. 159 Second variation : Each triad may become a tonic, during the continu- ance of the same measure, if followed by its sub-dominant, the latter, as secondary harmony, arising also outside of the real fundamental progression. ^H i r p r f -mi • ■ • i i r nr EEZE ^ziE? 21 r TJiird variation : Each tonic.(primary or secondary) may be succeeded by the (secondary) harmonies of its 4th, 2d and 5th degrees, which latter three occur again outside of the real fundamental progression : 4- i 4- 4= 4=4 SSi^ iZ35t rr =g A r— r- ' 1 1 1 r *=t £ -1 >«5!- m -u -1—4- I I "P — l p-tt 5> — ^=; rr- 1' 1 9t ^*£ is! tfF^ c. Not to give all the other possible variations, we may state, once for all, that the first bass-note of each measure may be treated as primary fundamen- tal, to be succeeded by all its secondary fundamentals. iPiS^iliifpi^ wmm^m^m Ife 160 THE CORRECT ORDER OF ^^^^ m ^sMz-^tm^ m *-£tr -• W - Each, triad of the diatonic theme may also be treated as standing upon the 5th degree of a relative minor scale, and, accordingly, be succeeded by the tonic of such scale. tJ^^M^^^ ki 1 1 =22= ^P- ^^m^^^, j — j — b ??=i £ : 122= SECTION XXII. The organ-point (tasto solo) is a preparation of the close of a more ex- tended composition, and consists in the sounding of the dominant or tonic of the scale by the lowest voice, while the upper voices form a regular har- monic progression. FUNDAMENTAL HARMONIES. Itfl The organ-point must conclude with the same harmony with which it commenced, yiz : with that of the dominant, if it commenced with the dominant, or, with that of the tonic, if it commenced with the tonic. This may at once be tried with the theme given in the last Section, in which the tone C, as being the leading fundamental, may continue throughout the eight measures, while the other voices" make the regular progressions. It should be observed, that, if the same progressions are to be employed as an organ-point upon the dominant, at least one measure should precede, in which that harmony is clearly heard. However, in all the variations we have suggested, the leading fundamentals (indicated by capitals) may always re- main audible throughout the measure. SECTION XXIII. Of all the scales related to C major, E minor is, so to speak, the least congenial to it, and the occurrence of its characteristic harmonies in con- junction with those of C major, has in Section XXI been to some extent justified, on the ground that the major harmony of B, as dominant of E minor, occurs simply as secondary harmony. The student should now attempt to apply the chromatic steps explained in this part, to the several exercises suggested in Part First. These exercises should be worked out in the different major scales : care being taken to apply the chromatic progressions as well to the inverted as to the primary chords. But in order to show how these secondary scales may be again trans- formed into chromatic scales, we must first treat of the chromatic progres- sions in A minor. 182 THE C0RREC1 ORDER Of PART V. CHROMATIC PROGRESSIONS IN THE SCALE OF A MINOR. SECTION I. The explanation of the chromatic progressions in C major, is parriy adaptable to those in A minor. The chromatic scale of A minor ascends as follows : a, bfy, bS, c, c J|, d, di, e, f, f%, g, g§ and a; hence it is identical with the chromatic scale ascending of C major, and is made up of the tones of the minor scales of A, E and D. The descending chromatic scale of A minor is composed most usually of the same tones as the ascending scale. The case may occur, however, when it should descend as follows: a, at», g, f\, /§, e, e\f, d, d\f, c, b, bV and a,' in so far as it enters into relation with the scales of C, F and G major, and of C, F and G minor. And since the succession of the fundamentals : A,' D, G, C, F, B, E, A (in the tempered system) remains the same, whether occurring in A minor, or in C major, it is natural, that the chro- matic progression descending should remain the same in both A minor and C major, and become distinct only at the close. SECTION II. The principal difference between the chromatic progressions of C major and those of A minor, lies in the tones /tt and g§, which are diatonic tones in A minor, and may be used (under restrictions) as fundamentals ; thereby affording a closer relationship to E minor and E major, and to D and A major, than could have been attained as long as C major was the leading scale. Far, in the same .way as the fundamental, F§ (raised 6th degree in A FUNDAMENTAL HARMONIES. 183 nflnor% may be treated as the 2d degree in E minor, just so may (in the tempered system) the fundamental, B (2d degree in A minoT), be treated as the 5th degree in E minor, or E major. It follows from this, that E, which in A minor is fundamental to a minor, as well as to a major triad; may be treated as 1st degree iri E minor, or E major. Since the 6th and 7th degrees are variable in the diatonic minor scale, if fclfows, that these two degrees may be altered, even in the chromatic pro- gression, if they occur as fundamentals, — hence another difference between the chromatic progressions of the minor and those of the major scale. The tones, f\ and g%, from A minor, establish also a relationship to A major and D major. The chromatic scale of A minor, therefore, embraces the scales of A, E, and D minor ; C, G and F major, A, D and E major, and C, G and F minor. SECTION III. In order to avoid repetitions of the explanations given in the treatise on the chromatic progressions inC major (including the allusions bearing upon the tempered system), and yet to elucidate clearly the mode of effecting chromatic progressions in the scale of A minor, we append the following : 1st — The triad of the 1st degree of A minor (A-c-e) may be used un- altered for the scales of C major, F major, G major, and E minor ; but its minor third must be raised to a major third, if it is to be treated as occur- ring in A major, or as dominant triad of D minor, or D major. Its altera- tion into the characteristic triad of the 2d degree of G minor is effected by the lowering of its perfect, to a diminished, fifth. 2d. — The diminished triad of the 2d degree of A minor (B-d-f) needs iw alteration, to be adaptable to the scales of C major, C minor, and D minor. 3d. — The minor triad of the 2d degree of A minor {B-d-f§) may be used unaltered for the scales of G, D, and A major ; but its alteration into the dominant triad of E minor, or E major, is effected by the change of its minor, into a major, third. 4th. — The major triad of the 3d degree of A minor (C-e-g) may be employed without alteration for the scales of C, G, and F major, and those of E,. D, and F minor ; its alteration into the characteristic triad of the 4th degree of G minor, and its adaptation for C minor, is effected by the lower- ing of its major, to a minor, third. 5th. — The augmented triad of the 3d degree of A minor (C-e-g§) is a special characteristic of this scale. 164 THE CORRECT ORDER OF '6th. — The minor triad of the 4th degree of A minor (D-f-a) may. bg, employed unaltered for the scales of C major, F major, and D minor j its . alteration into the characteristic triad of the 2d degree of C minor is. .effected by the change of its perfect, into a diminished, fifth. yth. — The major triad of the 4th degree of A minor {D-fty-a) needs, no alteration to be adaptable for the scales of A, D, and.G major, and those of G and E minor ; but it may be altered (by license) into that of the 3d. degree of B minor, by changing its perfect, into an augmented, fifth. • 1 8th. — The minor triad of the 5th degree of A minor {E-g-i) may be. used unaltered for the scales of C major, G major, D major, and E minor ; , but its alteration for the scale of F major or F minor, or into the character- istic triad of the 2d degree of D minor, must be effected by the change of its perfect, into a diminished, fifth. , . 9th. — The major triad of the 5th degree of A minor (-E-gifc-b) may be employed unaltered for the scales of A and E major. 10th. — The triad of the natural 6th degree of A minor (F-a-c) may be used unaltered for the scales of C and F major ; its change into the charac- teristic triad of the 3d degree of D minor is effected by the alteration of its perfect, into an augmented, fifth ; and its recognizable change for the scales of C and F minor is effected by the alteration of its major, into a minor, third. nth. — The triad of the raised 6th degree of A minor (Efy-a-c) may' be employed unaltered for the scales of G major, G minor and E minor. ' It cannot be adapted for the scales of E major; D major and A major, because its diminished„fifth cannot be changed into a perfect fifth. 1 2th. — The triad of the natural 7th degree of A minor {G-b-d) may be used unaltered for the scales of C major, ; G major, D major and C minor. Its adaptation for the scales of F major and D minor must be effected by the change of its major, into a minor, third,. while its adaptation for the scale of F minor requires the additional change. of its perfect, into a dimin- ished, fifth. Its adaptation for the scale of E minor may be effected by.the change of its perfect, into an augmented, fifth ; but this is a license, since' the fifth of the 7th degree should resolve one degree downward. 13th. — The triad of the raised 7th degree of A minor (G§-6-d) needs no alteration for the scale of A major. The chromatic alteration of the chords of the seventh , and of the sev- enth and ninth, of A minor, into chords of the seventh, and of the seventh and ninth, of relative scales, may be easily made, if the directions given for' the chromatic alteration of the triads are adhered to. It should not be forgotten, however, that no raised degree can ever become a seventh or a ninth. PUNDAMENTAL HARMONIES. Igf SECTION IV. The following table will show, that the diatonic tones of the scale of A minor, are fully adequate to indicate the most important fundamentals of its relative scales. The tones D, B, E, A, are, in A minor, the degrees 4, 2, 5, 1. <„.:.. in A major, the same 4, 2, 5, 1. The tones A, F$, B, E, are, in A minor, the degrees 1, 0, 2, 5. ,; ,' in E minor, and E major, the degrees 4, 2, 5, 1. The tones G, E, A, D, are, in A minor, the degrees 7, 5, 1, 4. in D minor, and D major, the degrees 4, 2, 5, 1. . The tones F, D, G, C, are,. in A minor, the degrees 6, 4, 7, 3. in C major, and C minor, the degrees 4, 2, 5, 1. The tones C, A, D, G, are, in A minor, the degrees 3, 1, 4, 7. ' : in G major, and G minor, the degrees 4, 2, 5, 1. V T'ne- tones D, G, C, F, are, in A minor, the degrees 4, 7, 3, 6. in F major, the degrees 6, 2, 5, .1. '' The- tones G, C, F, are, in A minor, the degrees 7. 3, 6. in F minor, the degrees 2, 5, 1. ' .The tonic, dominant, and subdominant are the most important funda- mentals : the triad of the last of these may be represented by the chord Of the seventh of the 2d degree, whence the formula (closing cadence/ IV, V, I, or II, V, I, arises. It will be better, of course, if all four fun- damentals are represented. ■ •• It should be mentioned here, that the several relative scales contain even a still greater number of tones in common with A minor than the above named, viz. : The tones : a, g, /, e, d, c, b, a, form in A minor, the degrees . . 8, 7, 6, 5, 4, 3, 2, 1. - and in C major 13, 12, 11, 10, 9, 8, 7, 6. or: 6, 5, 4, 3, 2 . '. 7. 6. The tones : a, b, c, d, e, /$, g, form in A minor, the degrees i, 2, 3, 4, 5, $, 7. and in G major 2, 3, 4, 5, 6, 7, 8. The tones : c, d, e, /, g, a, form in A minor, the degrees 3. 4» 5, 6, 7, 8. and in F major 5, 6, 7, 8, 9, 10. w THE CORRECT ORDER OF - r The tones :~e, /%., g, a, b, c, form in A minor, the degrees . ' 5, $, 7, 8, 9, to. and in E minor 1, 2, 3, 4, 5, 6. The tones : d, e, /, g, a, form in A minor, the degrees 4, 5, 6, 7, 8. and in D minor i, 2, 3, 4, 5. The tones : d, e, f§, g§, a, b, form in A minor, the degrees 4, 5, 0, ?, 8, 9. and in A major, the same 4, 5, 6, 7, 8, 9. The tones : d, e, f§, g, a, b, form in A minor, the degrees 4, 5, 0, 7, 8, 9. and in D major i> 2, 3, 4, 5, 6. The tones : e, /$, g§, a, b, form in A minor, the degrees 5, $, t, 8, 9 and in E major i> 2 f 3, 4, 5, The tones : /, g, a, b, c, d, form in A minor, the degrees 6, 7, 8, 9, 10, 1 1 and in C minor 4, 5, 0, t, 8, 9, The tones : c, — I — _*-_3^ — . — ^3 168 THE CORRECT ORDER OF ,. 1 3th. --From the chord of the seventh of the 2d degree to the dominant chord of the seventh : 14th. — From the minor triad, or chord of the seventh, of the 4th, to the chord of the seventh of the raised 7th degree : §=#=§§ -** TZL : ^m ¥■ $~ 15th. — From the harmony of the chord of the seventh of the raised 7th, to that of the augmented triad, or chord of the seventh, of the 3d degree, resolving into the harmony of the natural 6th degree : $ zsz i^^EEf gjgEgg^^ Fund. GJ ST c C F 16th. — From the major triad of the 3d to that of the natural 6th degree : -a — ., &- 17th. — From the triad of the 1st, to the major triad of the 3d degree r^| 1 8th. — From the minor triad of the 4th, to the triad of the natural 6th degree : 19th. —From the seeming step of the diminished triad of the 2d, to the minor triad of the 4th degree, when the latter stands for the chord of the seventh of the 2d degree : FUNDAMENTAL HARMONIES. 169 20th. — From the seeming step of the minor triad of the 5th, to the triad of the natural 7th degree, when the latter stands for the minor chord of the seventh of the 5th degree : & -tfZ- SECTION VI. The most simple chromatic progressions in A minor are the following ; Diatonic. Chromatic, Diatonic. Chromatic, ,.-L IV. \ ^^E^Z ■jsz. ^Z$F= ~ 2g ■ 1 " I ■ =E —g ■ Degrees in A minor : I " " D minor : 1.-11. v I Degrees in A 'minor ; II ** " E minor 1 ** u E major'. Diatonic. Chromatic* ) VI. 11. I^ip Degrees in A minor : VI Degrees in D minor : III 5 VI Chromatic. i VI. III. -f-- ZSL Degrees in A minor: VI III " " C major: IV I « •■ F •' I V 5.— III. VI. : Diatonic, Degrees in C minor: IV " "Fminor: I Chromatic. m fp =2: • — E^E Ha Degrees tn A minor: III VI * U C major: I IV Degrees in F major I 170 THE CORRECT ORDER OF Diatonic. 6.— IV. VII. Degress in A minor : IV " " C major: II Chromatic. g minor: IV i ^mpmmssm m Degrees in F minor : " '* F major: V V 4th. — From the chord of the seventh of the 1st, to that of the 4th, thence to that of the natural 7th, and thence to the triad of the 3d degree. Diatonic. Degrees in A minor i I " " C major: VI Chromatic . Degrees in G major: II ' ' " C " VI Degrees in G major: II " " G minor: m " C minor: " " major: VI 173 THE CORRECT ORDER OF 5th. — From the triad of the ist, to the minor triad of the 5th, thence to the chord of the seventh of the ist, and thence to the minor triad of the 4th degree. Chromatic. Degrees in A minor : I " " D " V II This example becomes more chromatic, if the last fundamental exhibits first a major, and then a minor triad. 6th. — From the chord of the seventh of the raised 6th, to the minor triad of the 2d degree, thence to the dominant chord of the seventh, and thence to the tonic triad. Diatonic. Chromatic. Degrees in A miner z JfVI II I Degrees in E minor : II V " m A « JVI 7th. — From the dominant chord of the seventh, to the tonic triad, and thence to the dominant triad. Diatonic. Chromatic. •i ^&=-- m BE zaz t2Zzz l ^^ r m Degrees in A minor: V V Degree* in A major: V I " " B minor : IV "A minor: V 8th. — From the tonic triad to the minor triad of the 5th, thence to the chord of the seventh of the 3d, and thence to the triad of the natural 6th degree. FUNDAMENTAL HARMONIES. 173 Diatonic. •49- Chrotnatic. mmm Degrees in A minor : I III VI Degrees in A minor : I V " u D minor: u ** F major: VI II VII III VII V I SECTION VIII. The following connected examples will show, that the chromatic pro- gressions explained in Sections VI and VII, are related to the scale of A minor. I. Diatonic. i^|Si= -Aa J "«- =3= HE ST- Fund. A D G C F.. Chromatic, with the same fundamental progression. B -t9- A m m -j-- j- p is -s>- -4-90- ESE JSH ?2^ II. Diatonic. % r±SE± si -a> — — • tt^r — »- "zr feg: Chromatic. ^^f^^M^ 5 ^^ III. Diatonic* w i§= 121 EfE ^2^53 E^ :ezc 1221 _ffil * Chromatic. •&•■ — ■•*!• — g — r ' ^E dt*-Uk±& £ E^N 174 THE CORRECT ORDER OF TV. Diatonic. m i£ a X^ I — g 1 -^- H «> — » i^^S Jbz ; ~-W z ¥ 3E3E zfe s -*- III. Diatonic, in which the second and fourth fundamentals admit only a triad : ^P ^iT& m l3B= =S= :?£ ^~ 7SHZ Chromatic, in which the second and fourth fundamentals admit the free entrance of a dominant chord of the seventh : ^ Chromatic, in which the same fundamentals admit the free entrance of a chord of the seventh and ninth ; this being best effected by omitting the fundamental tone. 51 gz= m Sz m j. zst && f^E -W 51 iSC Fund. A D BE. SECTION XI. The formation of chromatic chords of the seventh, and of the seventh and ninth, seemingly standing upon the 5th degree, because of their major third, but in reality standing upon the zd degree of a relative minor scale FUNDAMENTAL HARMONIES. 1TO because of their diminished fifth (Section XVI, Part IV), will now be ex- emplified in the scale of A minor. • , I. From the chord of the seventh of the 2d degree, to the dominant triad, closing with the tonic triad. Diatonic. Chromatic. When the chord of the seventh of the natural 6th degree precedes, the chord of the seventh and ninth of the 2d degree may follow, and the seventh of the 6th degree may resolve into.the major, instead of the minor third of the 2d degree. This major third of the 2d degree must be lowered to a minor third, if the dominant chord of the seventh is to follow, in order that the seventh may be prepared. Chromatic. IP zz=% g-fas In this last example, the ninth of the 2d degree resolves during the con- tinuance of the same fundamental. If it is to resolve into the fifth of the following fundamental, it must be placed in the most favorable position, as in the following example : Diatonic. Chromatic. ^ ZBZ ~a~ --B1 Fund. A A F The same favorable position is necessary, if the chord of the seventh and ninth of the 2d degree is to be followed by the dominant triad, 180 THE CORRECT ORDER Of Diatonic. * jBZ. o -J- ~st J=i * Fund. A F Chromatic. F :zt E§E 2?r *: I2JI B A Should the position of the voices be unfavorable for the progression al- luded to, the only way to avoid faulty progressions would be to suspend the seventh and ninth of the 2d degree, as eleventh and thirteenth of the nejrt fundamental. Diatonic. ^3 2B2I 33^2 3^= £ f^ Ami A F Chromatic A ^^m^mm^ Fund. A II. From the dominant chord of the seventh, to the tonic triad, and thence to the subdominant triad : Diatonic. Degrees in A minor : V IV Degrees in D minor : II If the first chord is a chord of the seventh and ninth, it may become a chromatic chord of the 2d degree in D minor, by the alteration of its per- fect into a diminished fifth : Diatonic. Chromatic. i ■Jb=t i m Xfe J r - a T 'S — j g — ■ FUNDAMENTAL HARMONIES. 181 The following examples illustrate the resolution of the ninth into the fifth of the following fundamental, in the favorable position, (a), and the suspension of the seventh and ninth, as eleventh and thirteenth, in the un- favorable position (6). m Chromatic . e> 9s- E Chromatic. T E. III. From the chord of the seventh of the ist, to the minor triad of the 4th degree, thence to the chord of the seventh of the natural 7th, closing with the major triad of the 3d degree : -&- _gg-L Degrees in A minor l I ** ** C major'. VI C Aromatic. IV II VII III V I Dfreeh in C major: VI ! '■** " G minor: " " C minor: If the first chord is a chord of the seventh and ninth, it may become a chromatic chord of the 2d degree in G minor, by the alteration, of its major into a minor ninth, its minor into a major third, and of its perfect into & diminished fifth. Diatonic. AjzJ^l iiili^P sz 182 THE CORRECT ORDER OF Chromatic, I 1 $ ^=j* =t=p: fct :a« P jSZ. 31 The resolution of the ninth into the fifth of the following fundamental, in the favorable position, and the suspension of the seventh and ninth as eleventh and thirteenth, in the unfavorable position, require no farther il- lustration. IV. From the minor chord of the seventh of the 4th degree, to the triad of the natural 7th degree (by way of returning motion), thence to the chord of the seventh of the 3d, closing with fhe triad of the natural 6th degree : Diatonic, Degrees in A minor : ** " O major : Degrees in F major', VI *' " G minor: 14 ** C minor : 41 " F minor I If the first chord is a chord of the seventh and ninth, it may become a chromatic chord of the 2d degree in C minor, by the alteration of its major into a minor ninth, of its minor into a major third, and of its perfect into a diminished fifth. Diatonic. J^-J. Chromatic. FUNDAMENTAL HARMONIES. 183 V. From the chord of the seventh of the natural 7th degree, to the major triad of the 3d, thence to the triad of the natural 6th degree: Diatonic. JO. -&■ :z>: J3S(. 1 Chromatic. ■« — ~9i ~Ez*^ -■S"-T-f- Eg ^ >l- «/ -!#- ■&- Tegrees in A minor : VII Ill VI Degrees in F major ; V " " C major: V I IV " " ^ ««»««■ . II V , If the first chord is a chord of the seventh and ninth, it may become a chromatic chord of the 2d degree in F minor, by the alteration of the major into a minor ninth, and of the perfect into a diminished fifth. Diatonic. ^ O^&Jr^ Chromatic. 1^ r^P 3rf -<*• SECTION XII. The following connected examples will show that all the chromatic pro- gressions explained in the last Section are related to the scale of A minor. 1 st. — The resolution of the ninth into the fifth of the succeeding funda- mental. A jfe m 5b A vl W zer 3a: l!i J- fl j J—^J- 184 THE CORRECT ORDER OF 2d. — The suspension of the seventh and ninth, as eleventh and thir- .eenth of the succeeding fundamental : SECTION XIII. The theme introducing the fundamentals in such order, that their triads may be treated as tonic triads, is not the same in the minor as in the major scale, since the triad of the 2d degree in the former cannot become a tonic. On the other hand, the triad of the natural 7th degree (in the tempered system) may become a tonic. The following two orders of fundamentals may be made use of : First order: A, E, C, F, D, B, E, A. Second order: A, D, G, C, F, B, E, A. Justification of the first order : 1st — The fundamental, A, having stood as tonic, may also be treated as sub-dominant of E minor, to be followed by the tonic of that scale. 2d. — E, having stood as tonic, may also be treated as 3d degree of C major, to be followed by the tonic of that scale. 3d. — C, having stood as tonic, may also be treated as dominant of F major, to be followed by the tonic of that scale. 4th. — F, having stood as tonic, may also be treated as 3d degree of D minor, to be followed by the tonic of that scale. FUNDAMENTAL HARMONIES. 185 5th. — D, having stood as tonic, may also be. treated as sub-dominant of A minor, to be followed by the 2d degree of that scale (B). 6th. — Since B cannot become a tonic, on account of its diminished fifth, it must be followed at once by E, as dominant of A minor, involving, accordingly, a major third. 7th. — The major triad of E, having stood as dominant of A minor, may also be treated as tonic of E major. 8th. — E, having stood as tonic, may now be treated as dominant of A minor, to be followed by A, as leading tonic. The following succession of diatonic triads in A minor, will serve as an illustration of the above order. Theme. i m m Z&E&L %r 9i: *=? ~sn First variation, introducing each triad of the theme as tonic for one measure, dvjring which it is followed by its own dominant harmony, which latter, as secondary harmony, forms no part of the fundamental progr.es- ^ni^ f^^ 1 1 Mr i^^! ISC B^pp^g^ *— ~ — 1 -H F=J—h H= Cvj J— r j -j— 4 * — 1 1 — r — 1 2. -*— #W * J_ 1 r t 1 — . u -^ —g> Second variation, in which each triad is followed by its sub-dominant harmony, this being also foreign to the regular fundamental progression. 186 THE CORRECT ORDER Of B ± ±±m 5 :£= -p*- f * G>- -f F 1 i I r r r fc. M L J^ ^^ -»l Fg z ^fe IT *? r~Tf =P=^ Third variation, in which, during one measure, every tonic is followed by the harmonies of its sub-dominant, 2d degree, and dominant, which also, as secondary, form no part of the regular fundamental progression. J-J-j-^ F±gH=i =zst FfT fc£=£ ■»-; P r -#te BS=^ ^ -?*- i 3=rf ^ it *— was — S» — «i — ^J z=liz=i H^£ IM— * h=S2= |4rir '*— *- *i T f -#a^!« 3£ PP=t Fourth variation, in which each measure contains a leading fundamen- tal, followed by all its secondary, fundamentals. FUNDAMENTAL HARMONIES. 187 §E ^-j— fc S Hi pP^#Sf|^|=f^ fc ^ |i|S sna- res *» The capitals everywhere indicate the leading fundamentals, which are sounded at the beginning and end of each measure, the intermediate chords being regarded as the harmonies of secondary fundamentals, and, as such, forming no part of the regular fundamental progression. SECTION XIV. Justification of the second order of fundamentals. ist. — A, having stood as tonic, is followed by D, which may be treated either as sub-dominant of A minor, or as tonic. 2d. — D, having stood as tonic, may also be treated as 2d degree of C major, and be followed by its dominant, G. 3d. — G, having stood as dominant, may also be treated, first, as tonic of G major, afterwards, as dominant of C major, to be followed by the tonic of that scale. 4th. — C, having stood as tonic, may be treated as dominant of F major. 5th. — F, having stood as tonic, may be treated as 6th degree of A mi- nor, to be followed by the 2d degree (B). 6th. — Since B cannot become a tonic, because of its diminished fifth, it mHst be followed at once by E, as dominant of A minor, involving, ac- cordingly, a major third. 7th. — E, having stood as dominant of A minor, may now be treated as tonic of E major. 8th. — E, having stood as tonic of E major, may now be treated as domi- nant of A minor, to be followed by A, as leading tonic. 188 THE CORRECT ORDER OF The following succession of diatonic triads will serve as an illustration of the second order : Theme. W^^f- 3= r r -8s- •*- ^= 32= zsz ZBZ First variation, introducing each triad of the theme as tonic during one measure, followed by its respective sub-dominant and dominant harmonies, which, however, must be treated as secondary harmonies only : m^mm^^ i s= 2C E =P #- =P^ **=I=F l^^i^ifP^ ^^^f^^^ff r-^ tf^g^ ^ ?EE335 B E. Second variation, in which the tonic of each measure is followed by the harmonies of its 6th, 2d and 5th degrees : FUNDAMENTAL .HARMONIES. 189 ^: lb: m &=r- --¥- -•—*- -^ ^ B E. SECTION XV. The formation of an organ-point in A minor will present no difficulty, if Section XXIII of Part IV is carefully perused, and then applied to the scale of A minor. Here we need only to exemplify how the secondary scales may again be- come chromatic. For this purpose, it will suffice, to give a chromatic form to the last example but one of the preceding Section, with such altera- tions of the intermediate chords as are involved in the rhythmical require- ments of the measure. g^ffcidiii piy^P^p^ ■ia_ala m 190 THE CORRECT ORDER OF PART VI OF ENHARMONIC CHANGES. SECTION I. We have shown, when speaking of the strict system of tuning, that a scale tuned according to this system differs materially from the same scale tuned according to the modern equal temperament, as it is called. Hence it is only in the latter system that two different scales — e. g., C$ and Dt» — can sound exactly alike, and differ only in their notation. Since, however, the equal temperament is now generally adopted, we may, from motives of con- venience,* especially in compositions for keyed-instrumerrts, substitute one scale for another having the same sound, but a different notation, — and in this consists what is called the enharmonic change. The most practical en- harmonic changes are the following, viz: cfy, for b; fjf, for d V ; d§, for e\f; fit, for gb ; g§, for afy; a§, for bV, — and vice versa. For the application of enharmonic changes, it will be found convenient to arrange the different scales in musical circles, like the following : i st — Beginning with the scale of C major, followed by the others as they increase in the number of sharps ; the 5th degree of each scale forming the 1st of the next, viz : C major, G major, D major, A major, E major, B major, F$ major. The next scale in this order would be C$ major. But we exchange Fft major for G\> major, and continue in that order in which the scales decrease in the number of flats, viz : G\> major, D|? major, Al» major, El? major, B[> major, F major, and C major, — which completes the circle. * Among these may be mentioned, greater facility in reading, e. g. in Afr, rather than in GJ major, with a signature of eight sharps. Again, a correctly applied enharmonic change will often clearly fore- shadow a harmony whose sudden appearance, without it, could hardly be comprehended, as far as its nota- tion is concerned. FUNDAMENTAL HARMONIES. 191 We may also begin with A minor, proceeding as before, viz. : E minor, B minor, F$ minor, Cjf minor, G$ minor, D$ minor. We then exchange Dfl minor for ~E\> minor, and continue in the same order, as follows : Bj> minor, F minor, C minor, G minor, D minor and A minor, which con- cludes the circle. 2d. — We may begin again with C major, introducing the scales as they increase in flats, the ist degree of each scale forming the fifth of the next, viz. : C major, F major, Bt> major, EP major, Ap major, Dp major, G\> major. We then exchange Gp major for F$ major, continuing in the same order, as follows : B major, E major, A major, D major, G major and C, thereby closing the circle. We may also begin with A minor, proceeding in like manner, as fol- lows; D minor, G minor, C minor, F minor, Bp minor, Ep minor. We then exchange Ep minor for D$ minor, and continue as before : Gjf minor, Cfl minor, F$ minor, B minor, E minor and A minor, thereby closing the circle. The following examples will serve for better illustration : Modulations from C major, proceeding in the order given under I, until C major is again reached. « i ±=± -OL. mi £ -i =*= ^ at ^m =f=f= mm Degrees. I, IV II V I, IV II V I, IV II V I, etc i ^=* T Wz f =# tr- i^^^igE^^^g^F^Ei^ iifci rzfc \? a §» ~H *c r :tez ?a =^ 192 THE CORRECT ORDER OF Modulations from A minor, proceeding in the same order, until A mi- nor is again reached. J. liJ^J ^m^^^m^mm fcEfeEtE * S 3= ¥ *= ~r~ =P -j- 0-h--^- #= Ep #= -s* — H &Sn 122: ?S=^ ~ff~ '■» LS -t y& i i r r ^PTT a $ =F =? I^Z Modulations from C major, proceeding in the order given under II, nntil C major is again reached. v-^ -v b^ h^JtdV^ ^ -Jat t*= & ±-±ti- -I — i»- p»^«i. i, v i rvnv i, v i rvnv i, v i ivnv i, v i, etc -k^J-^ ' ] i If.B. ' ' ^S^^p +5« i r i i * i -y FUNDAMENTAL HARMONIES. 198 il^iflp^iipp^^fe 2=tdfcdE gpg§i^i^s f=rzi=i q=iJ=±±|: Modulations from A minor, proceeding in the same order, until A mi- nor is again reached. 4—4 fMm^m^^m^ g= gg=a=g g Sc Dignn. I, V I IV II V I. V I IV II V I, V I IV II V Si Igs^iPN ^ at itafc -tf — 6S taMi^Sli^liiil I, V I IV II V h*te*± s^iPlS 3P*«s^5e i — r ^^g^=fpg mnm^wn^ feg|=i^^ S^iS^P^i^iriJ=l «=i=i=£F The mark N.D. always indicates the enharmonic changes, in these, as well as in the subsequent examples. 194 THE CORRECT ORDER OF SECTION II. If we wish to shorten the circle, we may modulate in the following manner : I. We begin with C major, modulate into A minor, and conclude in A major ; we then modulate into Fjf minor, concluding in Fi major, After having exchanged F$ major for G[> major, we modulate into E[» minor, and conclude in E(> major, from whence we modulate into C minor, concluding in C major. II. We begin with C major, modulate into E minor, and conclude in E major ; we then modulate into G4 minor, concluding in Gjf major; after having exchanged G$ major for A[? major, we modulate into C minor, and conclude in C major. The following examples may serve as illustrations : feE£ ife* M f^*£ "-£= 4ft Degrees. I, III VI I, III VI II B fe^ jj^S^P tgJ£=i R *^m uBH *= I, III VI II I, III VI II Degrea. I, VI II V I, VI II V I, VI II V By the aid of these examples, the student will be enabled to begin and end the circle with any other scale. FUNDAMENTAL HARMONIES. 195 The following peculiar progression will be found of service for the formation of a short circle of the minor scales : We begin with A minor, modulating by means of the dominant into G minor; thenqe by the same means into F minor; thence into E|* minor. After exchanging E^ minor for D$ minor, we modulate by means of the .dominant into Q§ minor ; thence intq B minor, thence into A minor, there- by closing the circle. i j=»t j=sLU^ £ a » >- Dtgrm. I, II I, II I, II Sp I^Ei mmm^&zmm W $* §S&E ^^ M i. ii i, a i. H We have a similar circle, if we begin and end with D minor, thus intro- ducing the other minor scales. SECTION III. By means of independent triads only, we may form a short circle in the following manner : I. We begin with the major triad of C, descending by a minor third to the minor triad of A, which latter we may treat at first as the 6th degree of C major, and then -as ist degree of A minor, when it may be followed by its dominant, the major triad of E. This triad may be followed by the mi- nor triad of Cjf, which latter we treat at first as that of the 6th degree of E major, and then as that of the ist degree of C§ minor, to be followed by its dominant, the major triad of Gjf. After substituting for the latter chord, the major triad of Ab, we may treat this latter as tonic triad of Al» major, to be followed by the minor triad of F; this again may be treated, at first as the triad of the 6th degree of Ap major, and then as tonic triad of F mi- 196 THE CORRECT ORDER OF nor, to be followed by its dominant, the major triad of C, thereby conclud- ing the circle. II. The triad of C major may be followed by that of C minor, and th« latter be treated as the triad of the 6th degree of EP major, to be followed by the major triad of A(?, as that of the 4th, and the major triad of Et? as that of the 1 st degree, of Eb major. • After substituting for the minor triad of E(? that of D#, we may treat the latter as the triad of the 6th degree of F$ major, to be followed by the major triad of B, as that of the 4th, and by the major triad of Fjf as that of the 1st degree, of Fjf major, to be altered into the minor triad of Fjf. This triad may now be treated as that of the 6th degree of A major, to be followed by the major triad of D, as that of the 4th, and then by the major triad of A, as that of the 1st degree, of A major, to be altered into the minor triad of A. . This latter we treat as the triad of the 6th degree of C major, to be followed at first by the major triad of F, as that of the 4th, and then by the major triad of C, as that of the 1st degree, of C major, thereby closing the circle. ir.B. aEJ^j ^^^ ^^^ ^-3- -I$E -*£= 1 *= 3SC Degrees. I I, VI IV I, VI IV fe^h**#^#^^^ I. vi IV I. VI IV The student should now begin and end the circle with other scales, a. g. : ■arith F major, G major, Bb major, etc. FUNDAMENTAL HARMONIES. 107 SECTION IV. Enharmonic modulations are effected chiefly by means of the dominant chords of the seventh and* ninth of inaudible fundamentals. In this con- nection it will be well to recall once more the table given in Section V of, Part III, viz: Major scales: Cb, Gb, Db, Ab, Eb, Bb, F, C. G, D, A, E, B, Ffc Cf Minor scales: ab, eb, bb, f, c, g, d, a, e, b, f% c%, g%, d%, ag ' ' It must be mentioned here, that at the third step to the right, from any minor, scaje, the major scale of the same name is found, both possessing the same dominant harmony, which differs only in the ninth. ■». . Hence arise the three following analogous series of relative scales : . • L Ab minor and Ab major have the same dominant, and the latter has the same tones with F minor. F minor and F major have the same domi- nant,, and the latter has the same tones with D minor. D minor and D major have the same dominant, and the latter has the same tones with B minor. B minor and B major have the same dominant, and the latter has the same tones with GJf minor. II. Eb minor and Eb major have the same dominant, and the latter has the same tones with C minor. C minor and C major have the same dominant, and the latter has the same tones with A minor. A minor and A major have the same dominant, and the latter has the same tones' with FJt minor. F# minor and Y% major have the same dominant, and the latter has the same tones with DJf minor. III. Bb minor and Bb major have the same dominant, and the latter has the same tones with G minor. G minor and G major have the same dominant, and the latter has the same tones with E minor. E minor and E major have the same dominant, and the. latter has the same tones with Cj| ' minor. CJJ minor and Cjf major have the same dominant, and the latter has the same tones with A$ minor. 1st — The first series is involved in the construction of the following mi- nor scales: Ab, F, D, B, and G$. The consideration of the dominant chords of the seventh and ninth (of inaudible fundamentals) of each of the above named scales, will show their influence upon the modulation. Thus : the dominant of Ab minor is Eb, the major third of which is g, its perfect fifth, bb, its minor seventh, d\>, and its minor ninth, fv. Consequentry, the dominant chord of Ab minor, omitting the fundamental, is : g-b b- -db-fb. The dominant of F minor is C, the major third, perfect fifth, minor seventh and minor ninth of which are: e, g, bb, and dv ; omitting the fundamental, we have e-g-bb-db. Th'e dominant of D minor is A t THE CORRECT ORDER OF the major third, perfect fifth, minor seventh and minor ninth of whicb are: c$, e, g and b\> ; omitting the fundamental, we have c%-e-g-bv. The dominant of B minor is F%, the major third, perfect fifth, minor seventh and minor ninth of which are : ajf, c%, e and g ; omitting" the funda- mental, we have a$-c $-e-g. The dominant of GJf minor is .Djf, the major third, perfect fifth, minor seventh and minor ninth of which are : fx, a% c Jf, and e ; omitting the fundamental, we have fX-a Jf-c J-e. If we sound successively on the pianoforte these five chords, vml: g-i \>-d b-/b ; e-g-6 \>-d\>; c %-e-g-b b ; a %-<%-t-g ; and fx-a %-c%-e ; the ear will discover no 'difference between them : they are distinguished only by a different notation. . 2d. — The explication of the above series answers also for the second series, in its application to the minor scales of Eb, C, A, FJf, and Djf. 3d. — The same explanation holds good for the third series, in its appli- cation to the minor scales of Bb, G, E, Cjf and AJf. SECTION V. Those chords of the seventh which are treated as substitutes for a domi- nant chord of the seventh and ninth of a tacit fundamental, are termed en- harmonic chords, on account of their polygenous nature, as shown in the' last Section. They may enter free and unprepared, and are well adapted for modulations into remote scales. Our tempered system presents to the ear only three enharmonic chords of the seventh, although they can be applied to different scales, and are distinguishable only by their different notation. In order to show how to reach any scale by means of these three enhar- monic chords, we begin with that of A minor. Its immediate relatives are E minor and D minor. The triad of A minor may be followed by its own dominant chord of the seventh and ninth, without fundamental, or by that of E minor or D minor, without presenting any difficulty. The enharmonic chord of A minor being identical in sound with that of C minor, FJf minor, D# minor and Eb minor, nothing prevents their being exchanged for each other. The enharmonic chord of E minor being identical in sound with that of G minor, Bb minor, CJf minor, and A# minor; these chords, again, may be interchanged. The enharmonic chord of D minor sounds the same as that of F minor Ab minor, B minor and Gjf minor ; and, accordingly, may be exchanged for any of these chords. FUNDAMENTAL HARMONIES. 109 A wider range of modulation is afforded by the circumstance that the dominant harmony of a minor scale may follow after, and lead into, a major triad. Here, however, the succession of the fundamentals is niore seeming than real, since only the dominant and tonic bear some relation to each other. The following examples show the enharmonic modulations from A mi-- nor into all the other minor scales. N.B. The fundamental tone must in every case be omitted. A H) Efe A AJ D* A BE ^^i pqp f^ggfegEjagm; E«.. AJ A Eb Ab A Fjf B A D$ • GJ The above examples may all begin with, and again resolve into^a major - triad. SECTION VI. Another opportunity is afforded for modulating by- means of these en* harmonic chords, if the ninth of the tacit fundamental remains suspended after the entrance of the tonic, so that the seeming chord which ■, arises- -from:, the suspension, is accepted as satisfactory. 200 THE CORRECT ORDER OF Instead of: Instead of : ^^ ^^^^ ^^ E A.. Instead of '. G C. Instead oft ^ppiife^iisgipa^ig^i Eb Eb CS FJ.. Instead oft AJ DJ From these examples are derived the following returning motions : The student should modulate with the two other enharmonic chords, after the manner just exemplified. SECTION VII. The chromatic chords of the seventh and ninth consisting of a major third, diminished fifth, minor seventh and minor ninth, which, without fundamental, sound like the harmony of the dominant seventh, furnish further means for enharmonic modulation. It has been shown in explaining the chromatic progressions, that each chromatic chord of the seventh and ninth stands upon the 2d degree of a minor scale, and may be followed by the harmony of the major triad of the 5th degree, when it occurs in the most favorable position ; otherwise, the seyenth and ninth may remain suspended as eleventh and thirteenth. In exemplifying this in A minor, we must recollect that the chromatic chord of the seventh and ninth of the 2d degree (of a tacit fundamental), contains only the tones, f, a, c, and