The Nature of Change and Schoenbergian Harmony

Dwight Winenger


            Change and the varied rate of change in nature are scientific facts. Chemistry, archeology, and anthropology, for example, are involved in the tracing of the nature of change and its history in their particular fields. Chemistry, in particular, studies and uses the various combinations and recombinations under various circumstances of the atomic structures of molecules to chart the changes of matter from one form to another. Physical change is evaluated in terms of relative stability and relative flux.

Composers and musicologists may tend to think of a musical form in terms of its overall configuration - as a concrete entity. The copyright laws consider a musical work as such; however, "form" is little more than a projected charting of changing sound effects over a given period of time encompassing the form. Music itself is a fleeting, ever-changing, and therefore, mysterious expression; but behind it all, music is expressed in terms of the stuff of which nature is composed: relative change.


            The writer has long felt the need to be able to objectively measure what is often intuitively structured in music. Harmonic notation in the United States is quite different from that used in Europe.1 To aid communication among composers, it would seem logical that a mathematical technique for analyzing harmonic progression and succession be devised. This paper sets forth such a system.


            Whenever a single tone is struck, the die has been cast which may develop into a work of music. A tone is considered expressive in terms of its subjective associations of pitch and timbre. Time also quickly becomes an important consideration as succeeding tones slice up lapsing time into characteristic segments of rhythm. How long will a given pitch continue? Will it be repeated? Will the pitch change slightly or greatly? Will a tone be joined by other tones to cast it into a different setting? These are questions that any piece of music suggests - then answers - from the first instant of sound.


                A musical work, from its initiation, traces symbolic change against a tapestry of human experience, measured in terms of modalities and rates of change. Romanticism capitalized on the emotional suggestions of subjectively inspired music. New-romanticism, some say, creates experience rather than expressing it. As in any subjective pursuit, however, losing touch with reality poses a real danger. It is often difficult to distinguish at close range between interpretation of reality and the never-never land of pure imagination. Chemistry for chemistry's sake can prove dangerous. Likewise, imagination for the sake of imagination can go beyond mere technical exercise to become potential psychosis. Creative imagination, however, becomes socially functional when applied to understanding and dealing with reality.


            Music alters reality just as surely as it alters the consciousness of reality. Music can dull the consciousness or sensitize it. Anyone "listening" to music allows it to affect their sensibilities. The moral question of personal integrity is raised, however, if a music bombards the senses into intoxication. As surely as gluttony is an abuse of the need for nutrition, musical intoxication is an abuse. A conscious control of the manner and rate of change expounded in music can minimize its misuse. Music can stimulate the intellect through the senses. The intellect then becomes a noble source of pleasure.


            The writer discovered in his study of a work by Arnold Schoenberg, Structural Functions of Harmony,2 striking principles which make the measurement of harmonic change in mathematical terms plausible. The writer also discovered that the computer becomes a valuable tool in quickly accomplishing complex and time-consuming procedures. Schoenberg's references to "regions" in Chapter III and to "relationships" - direct, indirect, close, remote, and distant - in Chapter VIII suggest the basis of measurable change. On page 1, Schoenberg clearly sets out the function of association in terms of orientation to the "tonality" of a tone center. He quickly qualifies his reference to tonality as relative. On page 2, he illustrates the simplest use of the root as an "interchange" of tonic and dominant, wherein tonic as root gives way to dominant as root.


            On page 4 (Chapter II), Schoenberg illustrates the relationships among the triads described by the seven degrees of the major scale as roots. Each root identifies the triad in relationship to the tone center. For example, in the key of C major, I is the C major triad with C as root. E, a scale third above C, is called "third" of the triad; and G, a scale fifth above C, is called "fifth" of the triad. B, a scale seventh above C, is called "seventh" of a quadrad or chord. Technically, however, any two or more tones sounded together constitute a chord. II is the root of the D minor chord with F as third, A as fifth, and C as seventh. III is root of the E minor  chord with G as third, B as fifth, and D as seventh. The process continues through VII.


It quickly becomes obvious that the seven chords described by Schoenberg have certain tones in common. B, for example, is seventh of I, fifth of III, third of V, and root of VII; and B, by resolving upward a half-step, can become 7 of II, 5 of IV, 3 of VI, or root of I. Based on this fact, Schoenberg then cites Anton Bruckner's law of the shortest way" and elucidates that each of the four voices should move no more than necessary. Our mathematical measurement of change leans heavily on this law.


By simply moving one or two tones of a chord, progression to another chord is effected. For example, I becomes IV simply by moving 3 and 5 upward a scale degree to form an "ascending" progression: E, 3rd, a half-step to F, root; and G, 5th, a whole-step to A, 3rd. C, root of I, remains stationary as its role changes to 5th of IV. Although the note remains the same pitch, its "sound" in relation to its brothers changes. Our problem is to measure such delicate changes in terms of mathematical symbols which will make them palpable. The full, authentic cadence from V7 to I is accomplished by moving B, 3rd, upward a half-step to C, root; D, fifth, downward a whole-step to C, root (doubling root), and F, 7th, downward a half-step to E, 3rd. G remains stationary, changing role from root of V7 to 5th of I.


The dominant seventh, just illustrated, is an important link between keys for two reasons. There is only one major triad with a minor seventh in any traditional key, and it resolves strongly to the tonic triad with a stationary tone for stability and with two tones a tritone apart resolving by complementary half-steps to a major third interval, the basis of the tonic triad.


The above examples illustrate that as tones move in relationship to one another, the role of each may change in terms of the "family." Key relationships are often recognized in terms of the patterns of contradictory

functions among tonal centers of a particular tonic.3 Subgroupings, therefore, exist within the tonal family, and they are normally recognized in terms of modal scales rooted on "secondary" tonics called "roots."4 Each degree of the scale can become a modal root of the key. For example, Ill of C major may orient a considerable segment of music to the Phrygian modality of the major key of C if E becomes, temporarily, a sort of "secondary" tonic.


            The term "key," therefore, does not necessarily apply only to a major or a minor key. The term "key" suggests a limited palette of tones in relation to a tone center. If E were to become sufficiently identified as root to be accepted as tone center while still in its Phrygian relationship to the other tones, the result is a Phrygian key, just as surely as major is Ionian

and minor is essentially Aeolian.5 A seven-tone modal scale can be constructed on any degree of the scale, and the triad rooted on the same degree is a common harmonic abstraction of each modal scale.


            The foregoing does not suggest that a triad must be present to suggest harmony. A single tone following another may suggest harmonic relationships and thus a tonal center. The common designation of harmonic progression by Roman numerals standing for chords built on degrees of a scale suggests that modal roots, as described above, are functional as they identify triads in relation to key center; however, any tone of a triad may be deleted or altered with only relative effect on the orientation of the remaining tones.6 Tones may be added to the triad without changing its general orientation, especially if such tones are chosen from among those of the modal scale based on the root of the particular triad. Chords of addition, for example 7ths, 9ths, 11ths, and 13ths (within the key), change the intensity of the harmonic entity, but not its identity. The root is the most important tone in any grouping of tones because it is related directly to tonic through its own modal scale. Roots will usually be identified in relation to a third above (or a sixth below) or a fourth below. The third of the triad is the next most important tone of identification. Triads are either major of minor as identified by the third. We must recognize the fifth of the triad as less important than the third because the fifth may be deleted in most cases with little effect on the orientation of the triad. In most cases, the ear will supply a missing fifth, especially a perfect fifth. The seventh may serve to identify the orientation of a triad to tonic. Major sevenths suggest I or IV. Minor sevenths suggest II, Ill, VI, or even V, but, as mentioned above, tones of addition are not as much of importance in identification as in intensity. Increased intensity adds to the value of a progression, but we are here discussing orientation as a basis for determining value. A diminished triad, using only tones of the key will normally be oriented through VII (VI10, Locrian).


            Roles may be assigned mathematical equivalents with the root the highest in value. To demonstrate the practicability of the mathematical evaluation of progression, the writer has made available a computer program (PRO-EVA) using only the four most common roles: root, third, fifth, and seventh ,(other roles can be substituted, however). The program is in Microsoft Basic, designed for TRS 80 Color Computer; and it is available from this author in cassette form. Print-outs of the program to be translated to other languages may be bought from the writer (more about that later). In PRO-EVA, we have assigned the value of role change to root as 4,7 role change to third as 3, role change to fifth as 2, and role change to seventh as 1. The distance - or extent - of change we append to this basic figure as a prefixed decimal. A change to root from third is not as great as a change to root from seventh, so the change from third to root is indicated as 2.4. Root to root, a reiteration - with value - is 1.4. The change from seventh to root is 4.4. (See Figure 1 for a Table of Values.)


            For role changes to:

Root:                            Thirds 8                      Fifth:                           Seventh:

from 7th :                    4-1=4.4           4-2=4.3  4-3=4.2                 4-4-4.1

from 5th :                    3-1=3.4           3-2=3.3  3-3=3.2                 3-4=3.1

from 3rd :                    2-1=2.4           2-2=2.3  2-3-2.2                  2-4=2.1

from root:                    1-1=1.4           1-2=1.3  1-3=12                  1-4-1.1


Figure 1: Table of Values

            By using this system, we can objectively evaluate the strength of modal progressions in terms of role change as suggested by Schoenberg's principles. He further suggests that voice-leading influences the strength of progression by activity.9 For example, if a role change takes place in a voice without a change in pitch, the effect is limited in comparison with that in which a pitch change occurs in conjunction with the role change.10 And leading-tones by half-steps appear to have a stronger effect than other intervals.11 Because of paging and program limitations, PRO-EVA utilizes only stationary tones, half-steps, and whole-steps; although a program for triads only, utilizing minor thirds, is available.

When other intervals are infrequently encountered as in A flat+ - Bo or in B flat - D flato, other intervals may be hand-figured, as the whole-step, at twice the value given for the particular role change (see Figure 1). Remember, also, that we are measuring activity and not necessarily propriety. In certain instances, the stability of a common tone would be preferred to high

activity, as in what Schoenberg terms "Superstrong Progressions."12


            To demonstrate the effectiveness of PRO-EVA, with or without the computer, we may analyze the so-called full, authentic cadence, G7 - C. We assign the numerical values to the twelve tones of the keyboard (which some may consider simplistic, but pragmatic): C = 1, D flat or C# = 2, D = 3, etc. The penultimate chord in our progression (G7) is spelled:

Role:            Tone:               Numerical Value:

root             (1) = G                (8)

3rd              (2) = B                (12)

5th               (3) = D                (3)

7th               (4) = F                (6)

Figure 2

The ultimate chord (C major) is spelled:

Role:              Tone:                Numerical Value:

root            (1) = C            (1)

3rd             (2) = E            (5)

5th             (3) = G            (8)

7th             (4) = 0             (none present)

Figure 3

If we then set the chords side by side:


                                              Voicing Values:


8 -1                         (72)             (6.8)

12-5                       (12.9)

3-8                          (1,2)




                                       28.1 = PRO-EVA

            Figure 4

            We see that 12 is only a half-step away from 1 (B - C); therefore, the third (12) ascends to root (a value of 2.4, see Figure 1) by a half-step (times 3 = 7.2). 8 remains stationary at 8 but with a role descendancy (or degradation) from root to fifth (value: 1.2). 6 moves to 5 by half-step (7th to 3rd, an ascendancy: 4.3 times 3 = 12.9). Since G7 has four voices and C major only three, the root will be doubled in the ultimate. An ascension from fifth to root (3.4) by whole-step (times 2) equals 6.8. The total of the four voicing values results in PRO-EVA (pro gression eve luation) for the progression G7 - C. Summary: 7.2 + 6.8 + 12.9 + 1.2 = 28.1.


            The computer program prints out all possible voicings, allowing specialized voicings for special effects. The program also includes a subroutine for eliminating redundant voicings under most circumstances. The best way to test preferred voicings is at the keyboard. A little practice soon develops facility in equating letter names and numerical value at the keyboard. The computer does it for you otherwise.


            Intuition might indicate that the value of a return to a chord would be the same as the progression from the chord in the case of a succession. A little reason, however, would indicate that the half-cadence C-G7 would not be similar in effect to the full cadence, G7-C. Reversing a progression seldom results in the same value because of changes in the values of role ascendancy. In C-G7, for example, root becomes fifth by, whole-step rather than by stationary tone. This lowers the activity value of a potentially high-value ascendancy. The seventh becomes a factor for decendency as third becomes seventh instead of vice versa. Thus, the half-cadence PRO­EVA's at 18, whereas the full cadence is 28.1.


            A few reversals are equal in value to the original progression. Most superstrong progressions (Em-F, Dm-Em, etc.) are equal in both directions because all tones are eliminated in each direction. A little experimentation with PRO-EVA will yield surprising, but reasonable, results. PRO-EVA, for example, proves that some of the conventions of progression are not absolutely reliable. On page 8 of Structural Functions of Harmony, Schoenberg asserts that "ascending progressions can be used without restriction, but the danger of monotony, as, for example, in the circle of consecutive fifths, must be kept under control." The following calculations demonstrate that the "circle of consecutive fifths" as described by Schoenberg is not as monotonous as one might expect.


I - (15) - IV - (17.8) - viia - (12.6) -     - (15.9) - vi m - (15.9) - ii m - (12.6) - V - (15) - I


Figure 5. Circle of Consecutive Fifths


            Schoenberg may have assumed that the root consistently becoming the fifth in a progression to a root a fourth higher is the sole determining factor in measuring activity. In doing so, he overlooked principles he had formerly set out regarding the effects of leading-tones and other combinations of motion and distance which occur when a progression moves from major to minor chords or from major to diminished. The following "Circle of Consecutive Perfect Fifths" utilizing secondary dominance illustrates true monotony, because it moves by similar distances and similar directions.


C7 - (27.2) - F7 - (27.2) - B flat 7 - (27.2) - E flat 7 - (27.2) - A flat - (27.2) - D flat 7 - (27.2) - G  flat 7 - (27.2) - B7 - (27.2) - E7 - (27.2) -


A7 - (27.2) - D7 - (27.2) - G7 - (28.1) - C


Figure 6:Circle of Consecutive Perfect Fifths


            PRO-EVA lends itself to calculating the value of any progression without relationship to key orientation because tones are entered directly; however, there are similarities, as illustrated in Fig. 6, among values of similar progressions within keys and between keys. For example, any progression with root descending a fifth and with a penultimate dominant seventh and ultimate major triad (such as V7-I) is valued at 28.1, whether G7-C, F7-Bb, or Eb7-Ab. With the aid of PRO-EVA, the writer has developed a chart of ninety progressive patterns with forty-two values ranging from 8.1, for root movement of triads upward a minor third (major to major or diminished to minor) and upward a major third (major to diminished), to 29.2, for the relative cadence, b1119+ - I. Oddly enough this relative cadence is more active than the full, authentic cadence. PRO-EVA results in easily handled figures over a wide range for the control of progression and succession within any key and between key-related chords and other keys.


            Schoenberg reconciles all chords of indefinite or altered orientation to the scale of seven degrees as "transformations."13 He admits that many transformations have "multiple meanings;" however, he prefers to call them  "vagrant harmonies"14 rather than allow for ambiguity as a lyric process. Acceptability, however, does not necessarily depend on being reconciled to the current tone center through its traditional modalities. Relatively ambiguous orientation has been deliberately exploited for centuries in diminished seventh chords, augmented triads, and whole-tone scales. Early examples of ambiguous orientation may have been considered as the sacrifice of harmonic tonality in favor of melodic voice-leading as in Bach, but Debussy, with the whole-tone scale, made it clear that indefinite tonality is a valid melodic and harmonic effect in relation to an absolute tonality.


            Certain "transformations" are abstracted from 8-tone modal scales related to the diminished scale, which is composed of alternating major and minor seconds throughout the octave (C, D, E flat, F, G flat, A flat, A, B, C). The abstractions of the five "chromatic root" scales are bIll+, bVl+, b110, bVll and bV0. Their unfamiliarity may make them seem of indefinite orientation, but major and minor scales are not the be-and-end-all of scale organization. The so-called "chromatic" tones of the major key may be treated as functional roots. The "Greater Perfect System" of the Dark Ages incorporated  b  Vll, and vestiges of it still exist in European notation, with "h" denoting Bb. The five "chromatic-root" modes are melodically and harmonically active, possessing leading-tones ascending and descending to root.15

            Once we have acclimated ourselves to the idea of twelve modal roots in a "chromatic key," the function of key change takes on a different aspect. The twelve modal scales are related by similarity, not of tones, but of scale degree relationships. Mixolydian (V) is distinguished from Ionian (1) only by a minor seventh degree. All other degrees are common to both modes. Dorian (ii m) is distinguished from Mixolydian (V) by its minor third. Aeolian (vi m) is distinguished from Dorian (ii m) by b13 (minor sixth), and so forth.


            Schoenberg, and others, in interpreting key relationships as "regions," follows the traditional view that a similar key has similar inflections, roughly related to key signature. But he treats as polished entities such altered keys as melodic and harmonic minor. This rationale ignores the fact that each minor key in its pure form has the same key signature as its relative major and that alterations in the key, whether reflected in the signature or not, are temporary. Schoenberg uses these variant keys in his "Chart of the Regions,"16 but he does not include other temporary inflections, which he terms "substitutions."


            The only reasonable difference between C major and pure A minor is the distribution of leading-tones in relation to tonic. C major's extraordinary utility is due to its strong harmonic resolution to tonic. Its two leading-tones, respectively, ascend to tonic and descend to mediant, the basic identifiers of the triad. The leading tones of Aeolian A minor ascend to mediant and descend to dominant for an "imperfect cadence" effect.


            If we view all the modal "keys," as in the case of Aeolian, as relatively "imperfect" versions of their relative major, with the possibility of twelve roots in each key, it should not be necessary to apologize for any modality by consistently "doctoring" it to imitate its strong Big Brother, the relative major. In actuality, in his "Chart of the Regions," Schoenberg suggests both modal key and the chromatic root modes by including Dorian and Neopolitan ( b II) as keys. It seems reasonable to expand on the already existent practice.


            Any key can be altered as required for specific melodic or harmonic purposes. Melodic bending is a traditionally accepted practice. Such temporary inflections, however, need not be incorporated in a chart of "key" relationships. In short, if minor is considered a distinct key, why not accept Aeolian as pure minor and also admit Dorian, Phrygian, and the other modal phases of the "chromatic key?"


          Schoenberg's principles suggest further logical applications, but for the present, PRO-EVA is a step taken toward the objective analysis of harmonic structure for the composer who wishes to forecast accurately the effect of a particular sequence of tones in relation to a tonal center. PRO­EVA, however, is not limited to tonal music. Atonal or non-tonal music, since it uses the same pitches, can be interpreted in terms of relative ambiguity - assuming the deletion or substitution of otherwise important tones. PRO-EVA may be used by the composer who wishes to work toward specific effects, and it may be altered to handle structures more complex than the four-note chord. We invite our readers to experiment with the concept.17 T



        1 An occasional European jazz musician with international experience may be familiar with such chord designations as Bb for Bb major, C°7 for C diminished seventh, Dm7 for D minor seventh; but the traditional figured bass and even the use of "h" for "B flat" are more common in Europe.


        2Arnold Schoenberg, Structural Functions of Harmony, revised edition, ed. by Leonard Stein, W. W. Norton & Co., NY, 1969.


        3 ibid., p. 2.


        4ibid.,p. 4.


       51bid., pp. 15, 16.


        61bid., p. 36 (Transformations).


        7 ibid, p. 6. Since "structural functions are exerted by root progressions (italics ours). Schoenberg's term "to promote the advancement of inferior tones" suggests the basis for measuring relative advancement. To achieve the results Schoenberg anticipates, we must alter some of his assumptions. For example, V - I = 15.2 (ascending) and I - V = 10.3 (descending) only if we weight "advancement" of role changes rather than "degradation" of role changes.


        8 ibid., p. 6, fin. Schoenberg appears to feel that the fifth is more important than the third. The text above explains the logic of the

arrangement used here. Also, the conventional doubling of minor thirds, not fifths, supports our thesis.


          9 ibid, pp. 4-5.


        10 ibid., p. 7, fn. 1. Schoenberg suggests that the "conquering power" of a role change accompanied by pitch change is not present when a role change is in a stationary voice.


        11 George Thadeus Jones, Music Theory, Barnes & Noble NY, 1974, p. 33.


        12 Schoenberg, op. cit, pp. 7-9,

13 ibid., p. 35.


14 ibid., p. 44.


15Dwight Winenger, Music for Dummies, Minuscule University Press, Inc., 1981, p. 53. This citation charts the modal scales and describes their use in the surrounding text.


16 Schoenberg, ibid., p. 20.

        17 The program is in full color and with sound. A print-out of the program described in this article is available (for $9.95) from Dwight Winenger, Minuscule University Press, Inc., 66358 Buena Vista Ave., Desert Hot Springs, CA. 92240-3914.