A BIT OF THEORY
As in the case of Schoenberg, whose creative methods were later explained by analysts using concepts (like the rudimentary “set theory” we explored in chapter 6) that the composer never knew or needed, so Bartók’s symmetrical structures were rationalized long after his death according to “paradigms” or models with which he was probably unfamiliar as such. The fact that analytical methods are sometimes anachronistic does not necessarily lessen their appropriateness or their explanatory potential (or else we would have long since stopped using Roman numerals to label chord functions in Bach or Mozart). As explained with reference to Schoenberg, we often need them in order to infer and then demonstrate to our own satisfaction the premises on which the composer was relying a priori.
A case in point is a handy method devised by the composer George Perle (b. 1915) to represent symmetrical arrays like the one Bartók employed in the second Bagatelle, and more importantly, to compare them. It may already have occurred to the reader that, with its “pluses and minuses” arranged around a stationary center, axial symmetry à la Bartók or Strauss or Webern is a “zero-sum game.” That is, whatever is added on one side is taken away on the other, so that the “sum”—in this case A (or E≭) —does not change. If that sum could be represented numerically, there would be a means of classifying axes, of measuring progressions from one to another, and (most important, analytically) of assigning any interval or chord one may chance to encounter to its potential place within a symmetrical array.
The trick is done by numbering the pitch classes from an arbitrary starting point on C (like the “fixed doh” of French-style solfège, or sight-singing technique), thus,
and then calculating the sums of the two chromatic scales whose intersections define the symmetrical array. That is what is done in Ex. 7-14b. The array in Ex. 7-14a begins with one of the points of intersection: 2 As. A being nine semitones above C, the sum of two As is 9 + 9, or 18. But since we are dealing with idealized pitch classes rather than actual pitches, we need to conceptualize everything within a single ideal octave; hence 12 is subtracted from all sums 12 or above, since 11 defines the limit of an octave, after which “zero” comes again. Thus the index number or “sum” of our two As is 6. And so is the index number for the reciprocal point of intersection, the two E≭s (3 + 3), around which the same series of dyads take their place but in the opposite order.C = 0
C♯D♯ = 1
D = 2
D♯E♯ = 3
E = 4
F = 5
F♯G♯ = 6
G = 7
G♯A♯ = 8
A = 9
A♯B♯ = 10
B = 11
As Perle’s method represents them, all axes (points of intersection between chromatic scales in contrary motion) have the same index numbers as their reciprocals. In other words, whereas with individual pitches we assume “octave equivalency,” in the case of axes we assume “tritone equivalency,” thus:
Table 7.1
AXIS PAIR |
INDEX NUMBER |
SUMS |
---|---|---|
C−F♭G≭ |
0 |
0 + 0 or (6 + 6 − 12) |
C♯D≭ − G |
2 |
1 + 1 or (7 + 7 − 12) |
D − G♯A≭ |
4 |
2 + 2 or (8 + 8 − 12) |
D♯E≭ − A |
6 |
3 + 3 or (9 + 9 − 12) |
E − A♯B≭ |
8 |
4 + 4 or (10 + 10 − 12) |
F − B |
10 |
5 + 5 or (11 + 11 − 12) |
The odd index numbers belong to arrays in which the chromatic scales in contrary motion cross without actually intersecting on unisons or octaves. Whereas the intervals in an intersecting array like the one in Ex. 7-14 are limited to the intervals that contain an even number of semitones, namely M2/m7, M3/m6, and aug4/dim5 in addition to the octave/unison (o), the intervals in a nonintersecting array, like the one in Ex. 7-15, will have the complementary set—m2/M7, m3/M6, P4/P5—in which all the intervals contain an odd number of semitones. As noted above, a symmetrical array can have either a single or a dual axis, depending on whether it contains an odd or even number of members. Thus the array of dual axes and their tritone reciprocals (taking the minor second as the axis since it is closest to the unison) will look like this:
TABLE 7.2
AXIS PAIR |
INDEX NUMBER |
SUMS |
---|---|---|
C/C♯D≭—F♯G≭/G |
1 |
0 + 1 or (6 + 7 − 12) |
C♯D≭/D—G/G♯A≭ |
3 |
1 + 2 or (7 + 8 − 12) |
D/D♯E≭—G♯A≭/A |
5 |
2 + 3 or (8 + 9 − 12) |
D♯E≭/E—A/A♯B≭ |
7 |
3 + 4 or (9 + 10 − 12) |
E/F—A♯B≭/B |
9 |
4 + 5 or (10 + 11 − 12) |
F/F♯G≭—B/C |
11 |
5 + 6 or (11 + 0) |
- Citation (MLA):
- Richard Taruskin. "Chapter 7 Social Validation." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 30 Mar. 2020. <https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-007004.xml>.
- Citation (APA):
- Taruskin, R. (n.d.). Chapter 7 Social Validation. In Oxford University Press, Music in the Early Twentieth Century. New York, USA. Retrieved 30 Mar. 2020, from https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-007004.xml
- Citation (Chicago):
- Richard Taruskin. "Chapter 7 Social Validation." In Music in the Early Twentieth Century, Oxford University Press. (New York, USA, n.d.). Retrieved 30 Mar. 2020, from https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-007004.xml