129 138 228 147 237 156 246 336 165 255 345 174 264 354 444 183 273 192
135 Bacon does precisely the same thing, and comes up with a very similar system.136 The only significant difference between Bacon’s taxonomy and that employed in this paper is that Bacon does not normally cover the full octave. Of the 214 sets comprising between 2 and 6 elements, Bacon lists the final interval-class on only 34 occasions, in brackets: he does not deem it essential, as I do.
If one: a) considers ‘pitch-class sets’ in terms of what they actually specify – not sets of pitch-classes but sets of interval-classes between adjacent pitch-classes –, and b) represents these interval-classes as numbers, one inevitably ends up with something close to Bacon’s classification, as I have done.
136 Bacon, Ernst Lecher: op.cit. (1917), pp.580, 592-603. 137 In the Fortean or Schoenbergian sense, that is.
138 This is only possible if one includes the final interval-class, to make up the octave. It is the primary reason for my having done so. That Bacon does not accommodate this possibility in 1917, prior to Schoenberg’s development of serialism, is hardly surprising. Bacon does consider Rameauan inversion.139
The 43 tetrachords may be classified thus: 1119 1128 1218 1137 1227 1317 1146 1236 1326 1416 1155 1245 1335 1425 1515 1164 1254 1344 1434 1524 1173 1263 1353 1443 1533 1623 1182 1272 1362 1452 1542 1632 1722 2226 2235 2325 2244 2334 2424 2253 2343 2433 3333
The remaining sets need not be listed here, since the principle is now clear enough.
One feature of Forte’s system may serve our purposes more usefully, on rare occasions: the interval-class vector.140 This vector lists the total number of interval-classes between the various
pitch-classes of a given set, rather than simply the interval-classes between adjacent pitch-classes. For example: [2, 1, 2, 3, 2, 0] indicates two semitones, one tone, two minor 3rds, three major 3rds, two
5ths (strictly speaking, 4ths),141 and no tritones.
We may thereby quantify two useful characteristics of a given interval-class set. First, the number of sensory dissonances:142143144145
• Semitones (and therefore major 7ths and minor 9ths) are strong sensory dissonances.142
140 Forte, Allen: The Structure of Atonal Music, New Haven and London: Yale University Press (1973), pp.15-18 and 179-181. Forte’s term is ‘interval vector’, but these are interval-classes and not intervals.
141 That is: a 4th will often be heard as an inverted 5th. For example, within a dyad consisting of C and F, F forms a more natural root than C in most spacings and contexts. Therefore, although the 4th is the smaller interval, it is more useful to think of this interval-class harmonically as a 5th, rather than a 4th. With the other five interval- classes, however, it makes better musical sense to think of the smallest representative interval – the semitone, tone, minor 3rd, major 3rd and tritone respectively.
142 In this, I am following Plomp and Levelt (op.cit., 1965). See this paper, p.20. My terminology is partly borrowed from Ernst Krenek (1940), who classified semitones as ‘sharp dissonances’143 and tones as ‘mild dissonances’.144 However, Krenek considered the tritone a ‘neutral interval’.145 From the later research of Plomp and Levelt (1965), we can establish that on this last point, Krenek was mistaken.
143 Krenek, Ernst: Studies in Counterpoint Based on the Twelve-Tone Technique, New York: Schirmer (1940), p.7. 144 Ibid., p.7.
• Tones (also, therefore, minor 7ths and major 9ths) and tritones are mild sensory
dissonances.
Second, the number of interval-classes conducive to Klangverwandtschaft: • 5ths (strictly speaking, 4ths) are strongly conducive.
• Major 3rds, tones and minor 3rds are fairly conducive.
5. Techniques of Chordal Spacing
Within equal temperament, each of the 352 interval-class sets, and therefore each of the 4,096
sets of pitch-classes, contains latent affinities (Klangverwandtschaften) between subsets of pitch-
classes, and/or among the entire set. If a given chord is spaced with sufficient transparency to allow the ear to pick out some of these latent affinities, this will establish one or more audible roots, albeit to varying degrees of strength. That is: I advance that the clearest way to render these inherent relationships readily audible within a single vertical sonority is often to space it as a polychord. Evidently, the larger the set of pitch-classes, the truer this becomes. Of the 352 interval-class sets, well over 300 are intrinsically polychordal.
For all sets larger than trichords, there are numerous spacing solutions, allowing the composer to control: a) voice-leading; b) in most cases, which of several possible pitch-classes are heard as roots; and c) the perceptible strength of, and balance between, such roots. I know of no better route to coherence in ostensibly ‘atonal’ harmony than to consistently illuminate sonorities in this manner. One may also clarify and control the affinities of dyads and trichords through spacing – the obvious differences being: a) almost invariably, one generates only a single root; and b) the range of solutions is narrower.
If, besides clarifying Klangverwandtschaften, one also exploits spacing and register to attenuate the effect of sensory dissonances to whatever degree is expressively necessary, one can also achieve euphony with any of the 4,096.
We will begin by briefly considering the relatively straightforward case of the 19 trichordal interval-class sets. Thereafter, as we examine progressively larger sets over Chapters 5 and 6, our focus will turn increasingly towards techniques of polychordal spacing. By Chapter 6, all sonorities will be polychords.
It is of course one thing to dissect the roots and affinities of sonorities as I hear them, and another to consider other listeners’ perceptions. In the final part of this chapter, we shall reflect on the latter aspect, albeit only briefly and speculatively, drawing on a single study carried out by William Forde Thompson and Shulamit Mor (1992)146 in a related area. But for now, the focus remains on my
chordal spacing techniques themselves, for which I am guided by my own ears.