Recognizing the motive as the basis for all composition, baroque theorists devised a system, known as loci topici, in which the various affetti (sadness, joy, passion, etc.) were associated with specific musical motives. Bukofzer describes this systemization as beginning with the writings of: "Nucius, Crüger, Schönsleder, and Herbst," continuing with "the more explicit treatise of Bernhard, and finally crystallized in definitive form with Vogt, Mattheson, and Scheibe." The term that these writers used to describe the motive was inventio. Implied by the inventiois the idea that such motives had the potential to be transformed into new ideas (development) and therefore permeate the entire composition (saturation). This process of variation was known by the baroque theorists as elaboratio. Motivic elaboration became so important, especially to the Germans, that later generations invented a word for it: Fortspinnung, meaning the "spinning out" of a motive, or ideas generated from it, throughout the duration of a musical work. Partly because the baroque term inventio in association with our word "motive" is suggestive of Bach's own inventionen & sinfonien, I have chosen to begin this study of motivic development and saturation with a peek at one of his Two-Part Inventions.

Two-Part Invention in C Major (BWV 772)

The techniques represented in the foregoing paragraph represent the limit of motivic variation techniques employed in the C Major Invention. But in this lesson we are after something else...how does the composer DEVELOP the motive, and its variations, in a tonal context? We wish to consider, further, some of the formal operations Bach employs to SATURATE the composition with not only motivic, but also tonal and contrapuntal variants? Bach uses four techniques:

1. Scale Degree Development:
   If Bach had restricted himself to repetition of the motive (do re mi fa re mi do) or its variants on the same scale degree, his invention could by no means have been considered "inventive." To be inventive, it is necessary for him to develop the motive by stating it on other scale degrees, in other keys or modes, and employing textural contrasts in addition to the variations inherent in the motive itself. The goal of an invention is to saturate itself with the motive, but never the same way twice. So, after repeating the motive in the second voice, the composer proceeds to develop it, tonally, by stating it on a different scale degree. It may seem trivial, but this statement is clearly recognizable as "the motive" albeit the scale degree functions are quite distinct: (sol la ti do la ti sol).

2. Development of Related Tonal Centers:
   Later, Bach will repeat the original two-measure chunk, but in the key of G, a fourth lower. Notice that the motive in the first half of this new key is a repetition of the same pitches (an octave lower) as the second half of the old key. Although this fact often goes unnoticed, it is significant that, though the pitch classes of these two segments are identical, they do not sound the same. The difference is not attributable to a change of register. The reason they sound different is because these pitches function, in the new key, as "do re mi fa re mi do," whereas in the old key they had functioned as "sol la ti do la ti sol." Our ear hears this change of scale degree function because, in the counterpoint accompanying the motive, Bach has changed all the F's to F-sharps, causing our ear to interpret the same pitches in the light of a new tonal center. The business of compelling the ear to move through a series of related tonal centers--known as modulation--is the most important tonal developmental technique in the western tradition.

3. Modal Development:
   Notice that each time the motive has appeared so far it has been in the major mode. Another technique of tonal development involves stating the motive in the opposing mode--in this case minor. This Bach does, beginning in measure 11, where an inverted variant of the motive, repeated in sequence, commences in d, and propels forward to a cadence in the key of a minor. This technique for tonal variation is known as "change of mode," or "mutation," and is distinct from modulation because it does not necessarily involve the imposition of a new tonal center.

4. Textural Development:
   By "textural development" we describe the sort of thing a computer artist does when she creates a soldier then copies it thousands of times to make him look like an army. She might make this army appear less staged by cloning the soldier first in mirror image, so that half the host holds its spears in the left hand, and the other half in the right. If this two-measure excerpt from the invention is "right-handed" then this fragment is a "south-paw" for it retains the motive in the same voices, but transforms it by means of contrary motion. The opposite type of textural development involves retaining an earlier form of the motive (all the soldiers are right-handed), but causing the high and low voices to exchange music (phalanx alpha-bravo switches places with company D). This type of textural development, known as "contrapuntal inversion" (or "double counterpoint") is ubiquitous in Bach developments. Finally, the artist could have ordered phalanx alpha-bravo and company D (all holding spears in their left hands) to swap positions while at the same time grabbing spears with their right hands. In musical terms such a development would be represented by two voices having exchanged registers (the high voice having taken what was in the low and visa versa) AND both voices having turned all of their intervals in the opposite direction (i.e. a switch to the contrary motion form of the motive in BOTH voices). Bach called this species of textural development the evolutio. If, for example, this were the original, then this would be the evolutio.

Schoenberg: Klavierstücke Opus 11, No. 1

In the two hundred years separating Bach and Schoenberg, the techniques for writing organically structured music had evolved to the point that Schoenberg would be able to articulate relationships unimaginable in the eighteenth century. Whereas Bach's music was organic at the motivic level, it relied upon tonal centricity for development. The first part of this lesson listed some of these techniques as found in the C-Major Invention: repetition of the motive on other scale degrees and in other keys or modes. Thus, from Bach to Schoenberg, tonality was the sine qua non of development; it was assumed that no musical idea could be developed without it.

By the turn of the 20th century, however, tonality as the primary means of motivic development had broken down. Richard Wagner's brinkmanship with the tonal system had left composers with nowhere to go and they were looking for new models. The impressionists, the "Russian Five," Charles Ives and others had experimented with non-western and nationalistic idioms, but it was Arnold Schoenberg who grasped the fundamental significance of the motive itself, apart from its tonal centeredness, as cornerstone of the great organic tradition. Picking up where Brahms and Wagner had diverged (Brahms too having understood the importance of the motive) Schoenberg showed that it was possible to develop the motive without recourse to a tonal center.

Schoenberg's twelve-tone system of composition was nothing less than a systematic attempt to insure that a composition was fully saturated with variations of the motive. To be sure, his tone rows were calculated to avoid the hint of tonality (dominants, triadic formulations, leading tones etc.) it is unfortunate, in contemporary descriptions of this "new system" as Schoenberg himself called it, that the emphasis is often placed upon the absence of tonality rather than the presence of motive. Schoenberg is often called, for example, the "grandfather of the atonal system" as if his main contribution to western art was to have rid music of tonal centricity. In truth this riddance had happened long before Schoenberg's invention of the twelve-tone row. Schoenberg himself might have preferred to have been called the "grandson" of the motivic tradition (Bach having been the father, and Brahms the son). It is this sophisticated concept of the motive that Schoenberg called Grundgestalt elaborated in the first paragraph of this section.

If motivic development and saturation were to exist "beyond Bach and Brahms" it was necessary to find ways of expressing the motive as a static--as opposed to temporal--entity. In the baroque, the tonal center provided an element of stasis against which it was possible to hear the motive--essentially a melodic/rhythmic construct--as having relationship to it. This relationship we have expressed in terms like: "the motive is now reiterated on the fifth scale degree," or "the motive is now stated in the relative minor key." Statements like these are possible only in the presence of something bigger than motive, something the motive is subservient to, something that operates independently of motive. That something, in the music of Bach and Brahms, had been tonality. But Schoenberg's "new system" admitted no such something, for, without tonal centeredness, there are no scale degrees, no keys and no modes. In terms of development, this left him only the techniques of transposition (not to a new key, but to a new tessitura) and textural development. However, the baroque resources of variation inherent in the motive itself--contrary motion and augmentation--remained intact.

Schoenberg's solution was to make the motive itself sine qua non, subservient to nothing, and compared to which all else was subsidiary. To the excision of tonal stasis, Schoenberg responded by extending the baroque techniques of motivic variation to the chord. If it was possible, he reasoned, to express a melody in contrary motion, then it should be possible to express a chord (a static and atemporal element) in the same manner. Now this may seem, on the face of it, to be a contradiction in terms...for there to be motion there must also be time. But Schoenberg's insight was his perception of the fundamental motivic relationship as consisting, at its most primal level, not only in the ORDER and DIRECTION in which intervals are expressed, but also in the QUALITY of the INTERVAL itself. This quality, Schoenberg reasoned, could provide the element of stasis vacated by the tonal system.

To understand the possibilities afforded by interval stasis as surrogate for tonal centricity, it shall be helpful to begin with a concept familiar to all musicians--that of chord quality. A major triad, for example is comprised of three pitch classes in the following relationship. The triad's third is a Major third, while the fifth is a Perfect fifth, above the root. In the 18th century, Bach's contemporary, Rameau, was the first to recognize that regardless of how such a chord was voiced, its "fundamental bass" was always the root of the chord. This eventuated in what we now call the "invertibility" of the triad, meaning that whether the triad sounded with its root, third, or fifth in the bass, and regardless of which triad factors the upper voices took, the chord had a qualitative "sameness" about it. This sameness expressed itself in the perception that a major triad sounds "different" from a minor triad, which sounds different from diminished, etc.

Schoenberg's departure from tonality hinged upon a recognition that, while major and minor triads do sound different, they, too, have a fundamental "sameness" based upon the intervals of which they are made. Whereas a major triad is made of a M3 and P5 ABOVE the bass, a minor triad is made of a M3 and P5 BELOW the fifth. Thus, the interval resources of the major and minor triad are identical. What is different, Schoenberg reasoned, is that they are mirror images of each other. While theorists also call this mirroring "inversion," it is a species of inversion exceedingly different from that of Rameau and the common practice. Eighteenth-century inversion involved re voicing of the same sonority by the placement of a different pitch in the bass (e.g. C major triad with E in the bass rather than C). In other words, 18th-century inversion resulted when the same PITCHES were put into a different order. By contrast, Schoenbergian inversion resulted when the same INTERVALS were put into a different order (mirror image) to produce a new set of pitch classes (e.g. C major becomes f minor). In Schoenberg's mind, this fundamental equivalence of INTERVAL content was one of motivic sameness or Grundgestalt, and could function as the static element unifying a composition as the tonal system had functioned before it.

Returning now to the skip/step idea of our lesson on motivic variation, here is how Schoenberg expresses, and develops, the same idea in his Klavierstücke Opus 11, Number 1. Let the step be a minor second, and let the skip be a Major third. Now let us express this skip/step motive as a chord of three pitches. Let us represent the first pitch--it could be any pitch--by the number zero [0]. The minor second is ONE half step away and therefore could be represented by the number 1--the two pitches of the set now being represented by [0,1]. The Major third is FOUR half steps away and therefore represented by the number 4--the set now being [0,1,4]. Notice that, when the m2 is subtracted from the M3 [4-1], the motive also allows the possibility of a m3 (interval class 3). The [0,1,4] set is what we call the "P-form" of the motive (P for Prime) because it expresses the set with the smallest interval toward the left. Inversion involves reconfiguration of the P-form in mirror image, so that the smallest interval appears at the other end: [0,3,4]. The inversion is called the "I-form" of the set. P- and I-forms of a motive may sound as different as major and minor, but, because they are unified by interval content, the ear hears them, in the context of non-tertian sonorities, as motivic variations of each other.

The following represents the first five measures of No. 1 from Schoenberg's Klavierstücke Op. 11. Follow the animation as it illustrates imbrications of the [0,1,4] motive (yellow) and its inversion [0,3,4] (green). Notice that every pitch, whether it be in the melody or a chord, is generated from the motive. Although the animation is not synchronized with the CD, you can play the movement by clicking on the score. At the beginning of this "beyond" module you were asked to listen to this work. Did you recognize, at that time, its motivic tightness? While Bach's Invention was motivically dense, it was, by comparison, only partially saturated. It is in the music of "beyond" that composers found the words to complete a chapter they had begun to write three hundred years earlier--in the baroque.

Schoenberg Klavierstücke Op. 11, No. 1
motive P-form [0,1,4] in yellow; motive I-form [0,3,4] in green
Animation and analysis ©1997 Tim Smith with kind permission of Universal Edition
A.G., Wien and Belmont Music Publishers (USA distributors)
(To restart the animation, reload the page from your web browser.)

That Opus 11 predates Schoenberg's system of composition by twelve-tones, shows that his conception of motivic preeminence was fully formed before he conceived of his "new system" for composition. For Schoenberg, the composing out of the "fundamental Gestalt" required more than the concatenation of pitches developed from the motive. Measures 1-10 illustrate how the [0,1,4] motive manifests itself at a higher level. These measures contain four phrases in which Schoenberg articulates four chords. These chords employ the bass notes Gb-Bb-Bb-G, the P-form [0,1,4] of the motive. Similarly, the highest melodic pitches of each phrase are B, G, G, G#, not only the P-form of the motive, but also the first three pitches of the piece! Thus, the "fundamental shape" replicates itself not only in contiguous melodic and harmonic entities, but also in non-contiguous units analogous to each other by means of texture, function, or register.

Assignment

Listen to the Two-Voice Invention in C-Major while following the score. After you have become thoroughly familiar with the invention, listen again while following the motivic analysis. You will have noticed that my explanation of textural development did not specify WHERE, in the Bach invention, each of these two-measure chunks were located. This omission was deliberate inasmuch as your assignment, now, is to locate them yourself. You will find this assignment difficult to do without a thorough understanding of the textural variations and repositioning of voices illustrated by the animated analysis. Begin by reviewing the discussion of textural development, training your ear to distinguish material in the high voice from material in the low. Write down what you hear. Next listen for the motive variant that sounds in each voice--is it the original, or does the motive appear in contrary motion? Now do the same for the analogous section, then compare notes.

1. Identify the measure in which Bach first develops the motive's initial figure (the rising scale idea) in note values that are twice as long?

2. With respect to the question above, in which measure does Bach first develop this augmented figure in contrary motion?

3. In mm. 1-2 the motive is stated in both voices beginning on "do" then repeated in both voices beginning on "sol" All four statements are in the key of C. Identify two measure that represent a repetition of this chunk, somewhat modified, in a different key. What is the key? Whereas the new key provides TONAL variety. what does Bach do to provide TEXTURAL variety in this new chunk?

4. Identify two measures in which the motive appears in the same voices as an earlier two-measure chunk, but in which the motive has been varied by means of contrary motion. Identify the two measures from which this chunk was copied.

5. Measure 3-4 represent a falling sequence made of the contrary version of the motive in the high voice, and figure 1 of the motive in the low voice. Identify two subsequent measures where these same ideas return, but in the opposite voices. (i.e. contrapuntally inverted). Identify the interval by which the high-voice idea in mm. 3-4 has been lowered. Identify the interval by which the low-voice idea in mm-3 has been raised. For extra credit, read about how to identify the type of contrapuntal inversion, then identify the type represented in the excerpt under consideration.

6. Identify two measures in which the motive appears melodically inverted (that is, in contrary motion) AND in the opposite voices (that is, contrapuntally inverted). Identify the two measures from which this chunk was copied.

Questions pertaining to the Schoenberg Klavierstücke Op. 11, No. 1:

1. On a separate sheet of paper sketch out nine clock faces indicating the hours as zero to eleven rather than one to twelve. (Or, if you would prefer, print out this page then use the "back" button to return to here.) Label clock faces as "a, b, c, d" etc. Study the segmentations of the first phrase of the Schoenberg. With C=0, C#=1, D=2, D#=3, E=4 etc., the pitches of the first segment of the Schoenberg (segment "a") would be [7,8,11]...circle those numbers on your first clock diagram. On the remaining clock diagrams circle the numbers representing the pitches of remaining Klavierstücke segments (b) through (i). The numbers are an abstraction representing "pitch-classes" with strict enharmonic equivalence: A# = Bb = 10. The collections of numbers representing each segmentation are called "pitch-class sets." In 20th-century music, pitch-class sets undergo variation, and tend to saturate compositions, much like motives of the baroque period. Notice that we could have represented each set as having starting at three different points on the clock diagram and moving clockwise. Set (a), for example, could have been written as [7,8,11] or [8,11,7] or [11,7,8]. While all of the foregoing are in clockwise order, [7,8,11], representing the shortest circumference on the clock face, is the set's "normal order." What would be the normal order of Klavierstücke sets (b) through (i)?

2. Pitch-class set analytical technique enables us to determine whether a set is a TRANSPOSITION or an INVERTED TRANSPOSITION of another set. Here are the basic mathematical operations and what they imply.
   • If you can SUBTRACT one normal-ordered set from another, and the DIFFERENCE is the same value for each number in the set, then the two sets are transpositions of each other and the difference between them is the T-FACTOR. If you subtract a larger number from a smaller, convert the negative difference to a positive value by adding 12 to it. It is customary to subtract the original from the derived set, as reversing the order would indicate how many half-steps DOWNWARD the derived set is from from its progenitor. But subtracting the original set from its variant shows how many half-steps UPWARD (clockwise on the clock diagram) the variation is from its original. For example, if we wanted to determine how set (b) is generated from set (a), we would subtract set (a) from set (b) and NOT (b) from (a):

     [5, 6, 9 ] set b
     [7, 8, 11] set a
     [-2,-2,-2] difference

     Convert the negative to a positive value by adding 12 to it, and the T-factor is 10. Set (b) is generated from set (a) by a T-factor of 10...which is another way of saying that set (b) is a transposition of set (a) up 10 half steps. This relation could be expressed as follows: set (b) is generated from set (a) at T10.

   • If you can ADD opposite members of normal-ordered sets to each other, and the SUM is the same, then the two sets are inversions of each other and the sum is the T-FACTOR. By "opposite members" we mean, add the FIRST number of the first set to the LAST number of the second set, the 2nd number of the first set to the 2nd-to-last, middle to middle, etc. For example, if we wanted to determine if the following sets were related by inversion, we would first put them into normal order, then add opposite members as follows:

     

     The fact that the sum is the same value all the way across indicates that set (y) is generated from set (x) by a process of inversion with a T-factor of 4...another way of saying that set (y) is a mirror image (counterclockwise reading on the clock diagram) of the intervals of set (x), transposed up 4 half steps (T4I). For every set there will exist one inversion that is untransposed, in which case we would say that the T-factor is zero: i.e. the relation of each set to the other is T0I.

     Which sets in the Schoenberg example represent TRANSPOSED INVERSIONS of set (a)? Which sets represent TRANSPOSITION of (a) without inversion?

3. Complete the following statements. In each case "process" will either be "transposition" or "inversion (with or without transposition)." Use Jay Tomlin's Set Theory Machine, if you wish, to complete this assignment.
   • Set (b) is generated from set (a) by a process of TRANSPOSITION with a T-factor of TEN.
   • Set (c) is generated from set (b) by a process of _____________ with a T-factor of ____.
   • Set (d) is generated from set (c) by a process of _____________ with a T-factor of ____.
   • Set (e) is generated from set (d) by a process of _____________ with a T-factor of ____.
   • Set (f) is generated from set (e) by a process of _____________ with a T-factor of ____.
   • Set (g) is generated from set (f) by a process of _____________ with a T-factor of ____.
   • Set (h) is generated from set (g) by a process of _____________ with a T-factor of ____.
   • Set (i) is generated from set (h) by a process of _____________ with a T-factor of ____.

4. While Schoenberg is best known as a composer, he was also an accomplished music theorist. His most important theoretical writing is found in Harmonielehre ("Theory of Harmony"). This work was begun in the 1920's and underwent significant revision at about the same time that Schoenberg wrote the Klavierstücke studied in this unit. While the author was adamant that Harmonielehre was not intended to be a systematic theory, the work was seminal in that it prepared the way for Schoenberg's eventual abandonment of tonality and turn toward what he called his "method of composing with twelve tones which are related only with one another." In 1949 Schoenberg commented upon his abandonment of tonal centricity by saying: "I myself and my pupils, Anton Webern and Alban Berg, and even Alois Hába believed that now music could renounce motivic features [as well] and remain coherent and comprehensible nevertheless (from "My Evolution," Musical Quarterly XXXVIII, No. 4, pp. 524-5). Please comment on what you see as the feasibility of Schoenberg's envisioned abandonment of motive. What do you think such music might have sounded like? If no tonal center and no motive, what might Schoenberg have used as a tool to make the music "coherent and comprehensible?" How might this music have been related (or not) to the music of Bach and the Baroque? Can you name any composers whose music could be said to have done this?

Links

1. Schoenberg Institute and Archives

2. Ducan Vinson's "Schoenberg the Romantic"

Notes

1. Manfred Bukofzer, Music in the Baroque Era (W. W. Norton, 1947), p. 388.

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