Harmonic analysis. Part one

A course in the analysis of the chords and of the non-harmonic tones
to be found in music, classic and modern

Author: Benjamin Cutter
Year of publication: 1902
Source: B. Cutter. Harmonic analysis. A course in the analysis of the chords and of the non-harmonic tones to be found in music, classic and modern. - Boston, 1902.

Preface

This book is designed, primarily, for those who have studied Harmony and would apply it in their every-day musical life, – in other words, in their playing and in their teaching. It is planned – although no premium is hereby placed on superficiality – as much for those people who have made poor work of their harmony, so far as turning out a good-sounding product is concerned, as for those to whom the difficulties were as naught. Furthermore, it is planned for him who, living in some place inaccessible to the best performances, would fain review what he has learned, – if possible, in a manner other than that of laboriously writing exercises, – would broaden his musical horizon and thus increase the gift Heaven has given him, and appease, in a measure, that hunger for chords and for things harmonic which characterizes so strongly this present day.

By not a few observers it has often been thought that the ordinary course in harmony ceased before its rightful end, and that there was no connection, or not enough connection, made between harmony and playing; i.e., between harmony and practical musicianship. Harmonic Analysis, it has been held, would give the ordinary non-composing student an opportunity to make his harmony a live thing; and experience has justified this idea. The Course of Instruction in the New England Conservatory of Music, Boston, Massachusetts, from which this book has grown, was planned for forty class lessons. After canvassing the field, the following representative works were chosen, and have been held to with gratifying success: Schumann, Scenes from Childhood, Op. 68; Bizet, Suites, L'Arlesienne, Nos. 1 and 2 ; Chopin, Preludes ; Wagner, Selections from Tannhäuser, Lohengrin, Tristan; and, in some cases, Grieg, Humoresken, Op. 6. Absence of a text-book made imperative, however, from the beginning, not only the dictation of principles, but also constant discussion, time-robbing, and, because more or less was naturally forgotten, unsatisfactory. A text-book became necessary; a book comprehensive and up to date. On studying his subject for the recitation and for this book, the writer confesses that he was dismayed by its dimensions and by the very multitude of conditions he was forced to consider and to explain. The number of harmonic phenomena, of tone combinations, complications, which occur in modern music, although they admit of a reasonable classification, goes well-nigh into the infinite. More than this, not a few are extremely subtle in nature. Things which the composer absorbs unwittingly, and accounts for in an off-handed manner, if at all, may to the layman, when he comes to define them, present very considerable difficulties. Furthermore, these subtleties occur in every-day modern music. For these reasons a certain breadth of scope and entrance into detail has been observed from necessity in the general plan of this book.

As will be noted, examples have been drawn from the most varied sources. The page-limits of a work of this sort forbade, however, the use of many quotations which might have been used and which suggest themselves, no doubt, to the connoisseur. But it is thought that this presentation will be found ample enough for practical purposes, provided the student applies what he has acquired here.

It is believed by men generally, that the understanding of a thing heightens its enjoyment. We read "Hamlet" with care, that not a point of stage-craft may be lost, and our emotions are moved the more powerfully because of our knowledge. In the same way, it is believed that by a careful study of this book, one may learn not only to analyze and to understand anything in the way of harmony that he may chance to meet in musical literature, classical or modern, but – what is far more important – through his heightened powers of comprehension he may be enabled to hear with greater understanding, to read at sight with more facility, to play and to sing with more intelligence, and consequently may have his musical perceptions – those gifts of delight to man – quickened and made more responsive, whether he act as player, as singer, or as listener.

And, lastly, – and this is not the least consideration, – the author believes, and by experience knows, that the student of composition may be benefited by a study of this subject; that, instead of spending more or less valuable time in finding out the many minutiae of modern harmony, he may behold them here stated for his examination and possible application.

Benjamin Cutter.
Boston, June 12, 1902.

Part one.

Introductory.

Section 1. Definition.

Harmonic Analysis is the art of accounting for the various chords and foreign tones which make up the harmonic structure.

Section 2. Requirements.

To carry on the study of Harmonic Analysis successfully, the student must have learned enough harmony to be able to write exercises employing all the chords of three and four tones, and to modulate on paper, and, if possible, at the keyboard. It will indeed be found better if the whole course in harmony be finished before taking up this study. In this event the object for which this book was written will be best and most easily realized.

Section 3. On Progressions in General.

Harmonic Progressions may be summed up briefly as follows : those of the Fifth-Relationships, up or down – V, I; I, IV; II, VI, etc.; those of the Third-Relationships up or down –V, III; I, VI; II, IV, etc.; and those of the Second-Relationships, up or down – I, II; V, VI; VI, V, etc.

All harmony is composed of the elements of unrest and rest, of the progression of a chord more or less dissonant into a consonance, or into one or more successive dissonances before its final resolution – or, the reverse of this. To these two sound phenomena are attached physical and psychical impressions which, although they exist and are recognized, seem to defy a final analysis and to remain beyond satisfactory examination and explanation. These two elements, of rest and of unrest, repeated over and over, with manifold embellishment, constitute Music. In that form of the Fifth-Relationship in which the root falls, is found the most natural resolution or progression of any chord. Thus, the III goes to the VI, the VI to the II, the II to the V, the V to the I, in both major and minor, and the nearer the progression approaches the tonic harmony, the more gratifying and reposeful the effect. Hence the name, Normal Progression, sometimes applied to this peculiar succession. To modify any of these chord successions by the addition of a seventh to the first chord – as I7 IV, or I7 V – or by the chromatic alteration of the first chord, only heightens the effect; and this effect, to repeat, is the impression of rest, of satisfaction, more or less complete.

All other progressions, especially if they employ the secondary triads, have in them, in varying degrees, the element of suspense, of unrest; of motion, if we may say so. Their use in certain schools of composition is infrequent; in other schools, especially in the form of the Second-Relationships, they are very common. Still, startling and interesting as may be many of these progressions above mentioned, the student will find that the I, IV, V, and II in the Fifth and in the Second-Relationships, form the stock in trade of the composer and the material which in Analysis he will have most often to consider.

Section 4. Plan of Study.

In pursuing this course the student is expected to examine carefully the examples given, with due reference to the text; and, in working out the lessons, to indicate by the proper signs the keys as they occur, and the nature – place in the scale, and inversion – of each and every chord; and to give to each foreign tone its own distinctive mark. Also, where required, he is to reduce the tone structure to its essentials, one of the most valuable of exercises, directions for so doing being given at the proper place. To sum up: He is to account for each and every tone, whatever its duration or location. All lessons, unless otherwise specified, may be marked in the text-book itself, the signs of expression, etc., having been omitted to make sufficient room.

Section 5. Remarks to the Teacher.

This course should be taught at the keyboard, the scholars reading the figurings in turn, or, in a doubtful case, the opinion of the whole class being found before a decision is made. Any tendency toward superficiality which this method would seem to favor, may be offset by written examinations in which the examined must know – or fail! The class should enjoy the benefit of general discussion, and scholars should be led to argue for their figurings. But beware of narrow interpretations, of intolerant views; many phrases permit more than one solution.

The teacher is recommended to teach the pupil to argue backward. A doubtful passage often becomes clear if one looks on and finds the principal point toward which the doubtful passage tends.

Furthermore, to each section, beginning with Section 14, has been added a set of Preferences. These the student will do well to look up. For, unless he have studied Composition and thus gone far beyond the bounds of the Harmony Course, it cannot be possible to make him ready and expert by the use of this book alone; and so large is the field to be covered that a fair canvass of it, such as has been attempted here, fills out the bounds of an ordinary sized textbook. More material is thus necessary –as any one versed in teaching can see. And, lastly, he who studies by himself, by look-ing up these references and marking them, can give himself a liberal education in Analysis – and for him were they first designed.

We have drawn from the following material: Beethoven, the first Piano Sonatas; Schumann, Album for the Young, Op. 68; Bizet, L'Arlesienne, Suites I and II; Chopin, Preludes – which we would advise the student to consult constantly; also, Czerny, Op. 299; Cramer, Fifty Selected Studies (von Billow); Chopin, Nocturnes and Polonaises ; Schumann, Davidsbundler, Op. 6, and Fantasiestucke, Op. 12 ; Grieg, Poetic Tone Pictures, Op. 3, Humoresques, Op. 6, Lyric Pieces, Op. 12 ; and, lastly, and of much importance to him who has patience to examine them, Wagner's Tannhauser and The Flying Dutchman (Novello Edition).

An explanation of the figures used in making these Beferences will be found in Section 16.

Section 6. Signs Used in Marking.

A capital letter shows a major key; a small letter shows a minor key; a large Roman numeral a major triad and a small Roman numeral a minor triad; the sign + shows the augmentation, and the sign ° the diminution, of a triad.

Triads in minor.
I
II°
III+
IV
V
VI
VII°

Triads in major.
I
II
III
IV
V
VI
VII°

The inversions of triads and of seventh chords, both principal and secondary, will be indicated by the customary figurings:
6, 6, 6, 4, 4, attached to the respective Roman numerals. OR the letters a, b, c, d, meaning root-form, first, second, and third inversions, may be used with these same numerals. Thus:

I_a, I_b, I_c, II⁷_a◦, II⁷_b◦, II⁷_c◦, IV_b+, III⁷_c, etc.

The diminished seventh chord, in its various forms, will be marked:

VII^◦_7◦, VII6^◦₅, VII4^◦₃, VII4^◦₂, - or VII◦^7◦_a, VII◦^7◦_b, VII◦^7◦_c, VII◦^7◦_d.

Chromatic alterations will be discussed in their proper place.

Section 7. Principles of Analysis. (Preliminary Statement.)

No.1. Spell each chord accurately.
F# has one significance, F another. An analysis may be made incorrect by carelessly calling F#, F. Too great stress cannot be laid on this point.
No.2. Build up the chords in thirds.
Seize that interval most apparent, third or fifth, and build from it in thirds until the whole chord be found.

Section 8. Triads and Seventh Chords. (No Modulations.)

"Write the numerals and the signs of inversion. (It has been found impossible to divide this book into lessons, as is done in most text-books, and the plan has been followed of numbering the exercises and examples straight through. The amount of work to be assigned to a pupil is thus left to the decision of the teacher.)

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Section 9. The Broken Chord.

All music is derived from the scale and the chord, the latter element predominating. The chord may be plain, all its tones sounding simultaneously, or broken in the many forms of the arpeggio; that is, its members may be sounded one after the other in a great variety of order. For instance, the chord structure given below may be broken in the following and in other ways without making a change of chord.

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Section 10. Reduction.

It will be observed that Examples e and f show a compass greater than that of the primary chord form; which leads to the General Statement, that All broken chords may be reduced, with the voices which accompany them, if there be such, to a simple four-part structure. In making such a Reduction, so-called, the extreme notes of the florid phrase must be brought into proper vocal compass, after which the inner parts may be added.

Mark each chord with its key and numeral, and in the case of an extended broken chord indicate by small notes the Reduction to the primary chord form.

Listen

Section 11. The Broken Chord, the Appoggiatura, the Passing Tone, and the Embellishment.

The tones of a broken chord may be preceded, all or some of them, by tones foreign to the chord. The most common of these foreign tones are those mentioned in the above heading.

The Appoggiatura is a foreign tone which enters by a skip. A skip is any interval greater than a major second. The Appoggiatura may enter from above or below, and must move a second, major or minor, up or down. In its simplest form – that here given – it moves into a chord tone, and may have any time value, long or short, and may be on or off the accent. Other varieties will be explained later.

Listen

The Passing Tone is a foreign tone which stands between two chord tones. These two tones may belong to one and the Dame, or to two different chords, and the Passing Tone may be diatonic or chromatic, accented or not. The interval to be filled out may be a second, a third, or even a fourth; in the last case more than one diatonic Passing Tone will be required. Other conditions will be discussed later.

The examples here given will show the Passing Tone used in connection with one chord only. SIGNS: Accented Passing Tone, o ; Unaccented Passing Tone, +.

Listen

The Embellishment is the ripper or lower neighbor of a chord tone, and proceeds from its principal, or harmonic, tone, and returns to this harmonic tone. The Embellishment may move a major or a minor second.

Listen

Mark first the chords in an example, then the foreign tones.

Listen

Section 12. Modulation.

Modulation is a change of key. It is made ordinarily by a dominant harmony with its resolution, and, according to the older and some recent text-books, the presence of this progression, whenever it occurs and whatever its effect con-stitutes a Modulation. In regard to this matter, however, views have become modified very materially, and the theorists, as in many other things, would seem to be, on the whole, behind the practicists, the composers. Indeed there is ground for belief that from the time of Haydn, perhaps from that of Bach, composers have thought it possible to raise and to lower certain scale steps, and to obtain thus non-modulating chromatic harmonies, which sound like modu-lations but do not leave the key.

In a piece in С major, for instance, we may find the dominant or diminished seventh of D minor, with its resolution, and this followed by С major chords. Some theorists contend that such a progression is a modulation; that any chord which has the intervals of a dominant seventh, and is properly resolved, is an undoubted dominant. Other theorists contend that a real modulation is made only when the modulatory process is confirmed by a stay in its evident key; that these seeming modulations are only intensifications of triads of the key other than the primary tonic, generally the subordinate triads, accomplished by the use of their seventh chords; and that these chords or progressions in question are only altered chords, or progressions in the primary key, and are to be so marked. They quote that most startling and familiar example – the end of the Lohengrin Prelude, by Wagner – which, while seemingly moving through, or, as they sometimes say, "touching upon" such and such keys, really gives the ear the impression of A major, wonderfully enriched, but A major, all the time; many of them conceding, however, that this is an extreme case. A third party of theorists, acknowledging the difficulties of the matter, holds that while this passage in question may be in A major, to mark the many chromatic harmonies, as chromatic alterations in this key, is to strain the key unwarrantably; they would call each apparent change of key a real change, with a mental reservation as to the correctness of the analysis; would, perhaps, write two figurings, each one tenable, and depending on the point of view.

In opposition to all this, the old-school men say that the modern ear has heard so many modulations that it has become blunted, dazed; that if the ear were fresher and keener, it would call each progression in question an undoubted change of tonality; and they refer their opponents back to the impressions of youth, when each chord change ravished the auditory nerve, and each seeming shift in the seat of key, however fleeting, was felt as a genuine thing. They say, further, that the short and fleeting modulation, the Digression, so-called, is as much a part of the composer's stock in trade as that deliberate modulatory procedure in which the forces of the key are drawn up in array.

In this matter it is difficult to lay down a hard and fast rule. In many instances the analyst must use his own judgment, and the ear, which is plainly the last court of appeal, must be called upon to decide. And as ears do not always hear alike, the validity of more than one interpretation, based on the individual point of view, is evident. We would, however, add, that while the theory of altered chords may often be the theory most plausible, cases will occur in which the only satisfactory explanation is that of a sudden and passing but unmistakable change of key.

Section 13. Principles of Analysis.

1. Spell each chord accurately.
F# has one harmonic significance, F another. And an analysis may be made incorrect by carelessly calling F#, F.
2. Build up chords in thirds.
Seize that interval most apparent, third or fifth, and build from it in thirds until the whole chord be found.
3. The identity of a chord depends on its resolution.
In other words: judge a chord by what it does! Always look ahead. Any chord may be taken as a harmony in one key and quitted as a harmony in another key; or it may belong to one key alone; or it may be an altered chord; it is well to remember these three possibilities.
4. Reduce harmonies to principal chords, if possible.
The III and the VI usually occur in sequences. The I, V, IV, and II, with their various derivatives, will generally be found sufficient for both classical and modern music. Hence, in analyzing, first get out the chord structure, making it as simple as possible, and then go back and mark the foreign tones. 5. Place as many chords as possible in one key.
6. A major triad used as an opening chord should be regarded as a tonic harmony.
Instances to the contrary are rare, although they do occur.
7. The normal chord change falls on the accent.
The use of this principle will simplify many places otherwise complicated and difficult. If, for instance, the V7 be before the bar-line or before the third beat of a measure in four-four time, and only one member of the chord of resolution be on the following accent, this one tone, especially if it be the bass tone, defines the chord and causes the other tones to be unessential, – suspensions, appoggiaturas, etc., – provided that the other members of the rightful chord of resolution enter later in the measure. 8. A six-four on the accent may be regarded, in nearly every case, as a tonic harmony.
9. Any scale step may be changed chromatically without causing a desertion of the key, if the chromatic chord be followed by a principal chord of the reigning key.
The "identity of a chord," then, "depends on its resolution."
10. Modulation.
Made (1) by the V, in its various forms; (2) by a ⁶₄ on the accent, the root becoming a tonic and drawing after it a cadence, — a progression which may also follow an unaccented ⁶₄, although infrequently; (3) by the II in its various forms, this chord, often called a " chord of approach," moving into a cadence through its tendency toward the dominant; (4) by any of the triads of a key, even the weakest, the III in which especial case the establishment of the key may require several characteristic chords, most often the succession, III, VI, II, V, I, altered or diatonic, with or without sevenths, inverted or not; (5) by taking a tonic, and afterwards establishing it, the so-called Assumption of a Key; (6) by enharmonic means; and (7) by a change of mode, the change being usually made from a tonic harmony.
11. An apparent dominant seventh must be tested as to its resolution and its surroundings.
A chromatic chord, apparently a dominant seventh – having all its intervals – will often appear and disturb the analysis. Unless it fixes the key strongly on the mind through legitimate resolution and sufficient duration, it may best be regarded as an altered chord, in most cases as a supertonic seventh, chromatically changed. Principle Number 3 must be borne in mind. If an apparent V7 of G major stands between two strong С major chords, it is an altered chord in С major, and nothing else. 12. Modulations should be marked as belonging to the next related keys; i. е., from С one goes to G, e, a, F, d. If, for example, in С major the D major tonic appears after its dominant, the dominant must be marked as D minor and the major tonic as D major, involving a change of mode.
13. "When a seventh chord does not contain the elements of a V7 – major third, perfect fifth, minor seventh nor of a VII^◦_7◦ – minor third, diminished fifth, diminished seventh – this chord may be regarded as a supertonic seventh. In the great majority of cases this interpretation will be found to be correct, for the seventh chords on the other steps of the scale are rarely used save in sequence progressions.
14. "When in a rapid tempo a chord is repeated with more than one bass note, that is, with change of inversion, the first appearance of the chord is usually the one to be figured.
15. Any member of a chord may be omitted without causing the chord to lose its identity; an incomplete principal harmony must not be regarded, however, as a secondary chord.
16. Chromatic alterations may be indicated, if necessary, by placing in brackets below the Roman chord-numerals, the figures for the chord accompanied by the proper accidentals.
Thus, II⁶₅ [^#I_#3] signifies that the supertonic seventh is in its first inversion with raised root and third.
17. The nature of a major triad – whether dominant or tonic – when standing at the end of a phrase, may often be best decided by ear.
The effect of a Tonic is that of rest, of finality; the effect of a Dominant is that of suspense, of something to come.
18. The chromatic chord which precedes a modulating dominant seventh or tonic six-four chord – usually a diminished seventh or an augmented chord – is to be placed in the key of this modulating seventh or tonic six-four chord.
This is the natural outcome of Principles Numbers 3 and 5.

Section 14. Real Modulations.

Shown by an extended and deliberate progression in the new key, or by rapidly shifting key clusters. Mark as before. No further directions would seem to be needful as to this point.

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