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FortenotesAuthor(s): Jonathan DunsbySource: Music Analysis, Vol. 17, No. 2 (Jul., 1998), pp. 177-181Published by: WileyStable URL: http://www.jstor.org/stable/854437 .

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JONATHAN DUNSBY

FORTENOTES

I had been planning to give a presentation about why pc genera theory doesnot seem to have seized the imagination of theorists. I would have thought thistheory a steal, and yet people have not seemed to be exceptionally interested.My hunch is that this is a blip, just one of those things. Researchers may comeround to seeing pc genera theory for what it is, a significant step fotward in thetheory of the twelve-note universe. 1

I think my initial ruminations came to a halt, in favour of most of what fol-lows below, because they were more political than scientific. In order to sayanything positive, I would have had to decide whether it is the case that atonalmusic is losing the high ground, or on the other hand that the musical terrain(and culture tout court) is nowadays altogether less alpine through the erosionof the years - and thus of course easier-going for all travellers; hese processesof natural selection are hard to see clearly close up. If I imply that pc set theoryin general is a technique for the analysis of atonal music, it will be objectedthat, although this may have been its original primary purpose, its remit haswidened considerably over the years - to Skryabin, Liszt and so on - and it hasalso had its impact on contemporary composition. With the onset of pc setgenera heory the range of applicability may not have widened strictly speaking,as I suspect Forte might insist theoretically, but the species names, now includ-ing 'diatonic', are deliberately chosen to anchor pc set theory historically, in aworld where no-one is surprised at, and I hope most will welcome, moderndiscussion of, for example, chromatic completion in Mozart.2

Sticking to the positive, then, making two brief points and asking some sortof question: It seems to me that it is in a footnote - and we all know how

Forte's so-often extraordinary ootnotes are their own sort of universe - that heencapsulates what his kind of pc genera theory is about and for:

[It] organizes the universe of pitch-class sets into related 'families' on thebasis of chains of inclusion relations that begin from trichordal 'progenitors'.The result is a collection of twelve genera, each of which has distinctive in-tervallic properties reflected in the informal names assigned to each. Forexample, Genus 12 (G12) is called the dia-tonal genus. Its trichordal pro-genitors are 3-7 (e.g., C-D-F) and 3-1 1, the common triad. A major purposeof the system of genera is to permit close examination of large-scale har-monic vocabularies, particularly hose of early twentieth-century avant-gardecomposers. (Forte 1991, p. 161n)

MusicAnalysis, 17/ii (1998) 177c Blaclnwell Publishers Ltd. 1998. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK

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178 JONATHAN DUNSBY

The first question I wanted to address in connection with the present publica-tion, the event from which it derived, and the literature that underpinned it,was the simple one of why a theory of pc set genera fundamentally matters,which would be to answer, if it does matter fundamentally, how it transcends oreven in some sense supplants classic Fortean, his own, pc set theory. Forte isnot of course recanting, so we can expect to see ever more clones of that oot-note - which one is almost surprised not to meet as an automated click button,be it PC or Mac - 'cf. The Structure fAtonal Music New Haven:Yale Univer-sity Press, 1973)'. I do not know (yet?) of any way in which pc set genera theoryradically contradicts classic Fortean pc set theory, but I do feel that it may leadto a different practical, analytical focus, as mentioned below.

In his 1988 article Forte at last publicly addressed the issue of probability.Fifteenyears is a very short time in the advancement of musical understand-ing. Probability was always a niggle-and-a-half in respect of The Structure fAtonal Music. How can 50 hexachords survive the onslaught of 12 trichords?Every time a hexachord is 'used', or perhaps we may say 'found', it is just thatmuch less likely to be used or found in comparison with any particular tri-chord, which has a 1 in 12 chance of appearing among the universe of normaltrichords, a lot better than 1 in 50, which is the natural selection field of thehexachords. Or you can turn the probability issue onto its other head, and saythat any particular hexachord is more mportant as a musical instance, becauseit s statistically less

ikely to appear than is any particular trichord. This comesdown to the same point. Not only probability, but actuality, presence markedout the hexachord. After all, that '*' in a K* entry in a set-complex table istelling us three hings: 1) this Z-related set is special by virtue of being a hexa-chord; ) not only that, but it carries a special '*' by virtue of being one of twospecies f hexachord; ) and not only that, but it is actually used n this piece ofmusic. at your heart out, 3-1/9-ls everywhereNow, with the calculus 'difference quotient' Forte provides a formal expres-sion or the 'extent to which a given genus differs from the other genera' (Forte1988, p. 220) .The crucial next step is to

take account of the 'cardinalities f theconstituents' my emphasis), with the expression:Difquo= (X/Y)/4

Less rucial in the abstract perhaps, but vital in practice, is Forte's extension ofprobability xclusion (my phrase) by means of an index termed a status quo-tient,which 'is a further step toward refining and interpreting the matrix struc-ture or the analysis of actual compositions' (Forte 1988, p. 232). For therecord, he squo formula iS:3((XIY)IZ)- 10

Whatwe have here is an explanation of the semitonal, or half-step universethat eally is a theory, in that it not only frees us from the philosophically dis-

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FORTENOTES 179

tracting world of compositional practice (the K* syndrome), but also frees it-self also from the statistical spin of the hexachord in theory, of which evenmore anon.

In the introversive world of pc matters I obviously have a second question:Never mind probability (theory), what about segmentation (analysis?)? I haveput that parenthetical question mark there because it is not at all obvious thatanalysis and theory can be segregated in the convenient way that some Ameri-can colleagues assume or have been led to assume. The Structure fAtonal Mu-sicmay be a bit long in the tooth, yet there is no publication or downloadable orcommercially available software that I know of which essentially faults it: ifother theories asking similar questions were better, there are an awful lot ofpeople to whom they never got through; and yet it made little ingress into theworld of segmentation.

My second question, then, is how the theory of pc set genera may affectsegmentation. I cannot be alone in having taught, StrAMly, urreptitiously,'hunt the hexachord'. That's the way you made a set-complex work, asking astudent to interrogate whether that embarrassing challenger-set really mat-tered so much and could not perhaps be excluded as a feature of the music, orwhether there were not many more lurking hexachords that s/he had heard/seen (I am almost tempted to add '/played', but presumably in this forum I canwrite shielded from the Perception Police); and research showed again and

again that SecondViennese composers were, in different ways, composing withhexachords. The seduction of an overlap between the poietics and the esthesicswas of course compelling.4 It seems to me that in pc set genera heory, althoughhexachords have by definition lost none of their special properties, hey have nospecial place n the results of an analysis, and thus there is no danger of themhaving a special place in the segmentation on which the analysis of the results isbased. My study of Forte's analyses of Schoenberg, Ravel, Musorgsky, Chopin,Messiaen, Webern, Stockhausen, Carter, Stravinsky and Debussy (the rela-tively extensive treatment of La terrasse es audiences u clair de lune), all in

Forte (1988, pp. 238-63), and Chris Kennett's detailed work on music byFrank Bridge (Kennett 1995) that it was my privilege to supervise, suggest anapproach that is palpably closer to some kind of'neutral' level than what Iearlier called 'classic' pc set approaches.

Finally, it makes sense to look at large sets (seeWalker 1989), and I do notneed to cite any literature that puts its focus on small sets.5The grand questionof why there should be twelve trichordal progenitor sets is addressed in thesepages by Richard Parks and was defended vigorously by Forte in the discussionat CUMAC 97. Hovering somewhere in that discussion is the question of what

it is about the trichord that makes it so fundamental, asked here for the nth ime(I don't really even ask it, so much as refer to it) - from where oes hat magicthree ome?6

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180 JONATHAN DUNSBY

REFERENCES

Baker, James, 1990: 'Chromaticism n Mozart's 'Jupiter' Symphony', in Mozart-ffarAbuch 991, ed. RudolphAngermuller et al. (Kassel: Barenreiter), pp.1050-55.

Burnett, Henry and O'Donnell, Shaugn, 1996: 'Linear Ordering of the Chro-matic Aggregate in Classical Symphonic Music', Music Theory Spectrum,18/i, pp. 22-50.Dawkins, Richard, 1995: River out of Eden:A Darwinian View of Life (London:Weidenfeld & Nicolson).Dunsby, Jonathan, 1997: 'Acts of Recall', The Musical Times, 138 (No. 1847), pp.12-17.Forte, Allen,1973: The Structure fAtonal Music New

Haven:Yale niversity Press).1988: 'Pitch-Class Set Genera and the Origin of Modern Harmonic Species',3'ournal fMusic Theory, 2/ii, pp.187-271.1991: 'Debussy and the Octatonic', MusicAnalysis, 10/i-ii, pp.125-69.

Jalowetz, Heinrich, 1944: 'On the Spontaneity of Schoenberg's Music', TheMusical Quarterly, 0/iv, pp. 385-408.Kennett, Chris, 1995: 'The Harmonic Species of Frank Bridge: An Assessment ofthe Applicability of Pitch-Class Generic Theory to Analysis of a Corpus ofWorks by a Transitional Composer' (PhD diss., University of Reading) .Mithen, Steven, 1996: The Prehistory f the Mind:A Search or the

Origins of Art,Science and Religion London: Thames and Hudson).Nattiez, Jean-Jacques, 1990: Music and Discourse, rans. Carolyn Abbate (Prince-ton, NJ: Princeton University Press).Pople, Anthony, 1991: SetBrowser analysis software for Apple' Macintosh com-puters] (Lancaster: CTI Centre for Music).Walker, Rosemary, 1989: 'Modes and Pitch-Class Sets in Messiaen: A Brief Dis-cussion of "Premiere ommunion e la Vierge"', Music Analysis, 8/i-ii, pp. 159-68.

NOTES

1. I was delighted to hear pc genera theory discussed so expertly at the CUMAC 97round-table session under the authoritative guidance of Craig Ayrey. As Chair ofthe CUMAC 97 Programme Committee, I express my thanks to him for organis-ing such a fascinating session. In this written contribution I have tried to retainsome flavour of the original oral 'article'.2. See Baker (1990), and Burnett and O'Donnell (1996). I say 'modern' but thistopic goes back at least to the 1940s: see Jalowetz (1944), p. 387, concerning the'twelve-tone line' at the beginning of the development section of the Finale ofMozart's G minor Symphony, K. 550.3. Forte

explains X,Y and Z with his customary precision (Forte 1988, p. 232). Ihave lingered long and hard over 'what Forte is really saying' type statements,concluding that what Forte is really saying is what he says.

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FORTENOTES 181

4. At the risk of annoying some readers, but since thankfully music analysis is notjust one new world, I add this reference: to the index of Nattiez's Music and Dis-course Nattiez 1990, pp. 266 and 263 respectively).

5. If software is a good test of opinion, I note that the seemingly widely disseminatedUK software in Pople (1991) (version 1.2 appeared in 1994), figures sets fromcardinal 2 up to and including 10, without, needless to say, revealing any sub- orsuperset of cardinal 2 or cardinal 10 sets respectively.

6. My deep distrust of twos rather than threes is evidenced in my 'Acts of Recall'(Dunsby 1997). Recent interpretation of archaeological evidence suggests that thecurrent fashion for dualistic thinking, though it must by definition have its place inevolution, may be misguided. In The Prehistory f the Mind (Mithen 1996), StevenMithen argues that the early development of human cognition involved at least

three areas of awareness - roughly speaking survival, communication and aware-ness of others (we might say, again in my words: I know I have to eat that animal; Iknow I have to pretend to make the sound of this animal to stop that animal killingme; I know I like this animal - maybe s/he can help me solve my problems?). Ofcourse, you can say that consciousness is a duality (I/the world). But most of us inour everyday lives do not find matters so simple. And composers in any case talk ofmodern music as being, as Schoenberg often said, 'on a higher plane', when per-haps the human mind has evolved beyond the 'eat it/not eat it?' mentality. If Forteinsists that nowadays, millions of years into the evolving descendants of our fewand all of them perfect ancestors (see Dawkins 1995,pp. 1-2), we cannot figureout notes but in threes or more, he may after all be right. It would be intriguing tocompare pc set genera theory with current ethnomusicological theories of thetritonic scale in prehistoric civilizations And the relevance? I can let Allen Forteanswer that himself: 'They don't seem to realize', he once said to me about pc setgenera theory, and never mind where and when, and who they were, 'this is ahuman thing'.

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