Letters on P-Relations, 1992–1997

Abstract

Jack Douthett wrote a number of letters to John Clough and Richard Cohn concerning Cohn's “P-Relations,” single-semitone voice-leading relationships. The ideas in these letters led to graph-theoretic and geometric models. The following selection has been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank.

Keywords:

• P-relations
• Voice leading
• Group actions
• Neo-Riemannian theory
• Maximal evenness

2010 Mathematics Subject Classification:

• 00A65

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 David Clampitt was a PhD student at SUNY Buffalo in 1992.

2 The first six are transpositionally equivalent, as are the last six. The two groups represent the two inversions of the prime form.

3 Here, sets are numbered cardinally, not based on their included elements. Following his earlier rules, the second orbit should be O${}_{x_6}$.

4 Presumably, O${}_{x_k}$ and O${}_{x_{k+1}}$ refer to O${}_{x_k}\left({\sigma }_1\right)$ and O${}_{x_{k+1}}\left({\sigma }_1\right)$, respectively.

5 “Tonal” meaning that we can think of the inversion-related orbits as “major” and “minor” by analogy with consonant triads.

6 Douthett wrote E^♭s in both cycles, but undoubtedly intended E natural.

7 Cohn (1996) refers to the invariant hexachords as the hexatonic scale, and the paired triads as hexatonic poles. The invariant hexachord is equivalent to the augmented scale in jazz theory.

8 There should only be six “major” and six “minor” orbits.

9 Clough & Douthett (1991). M$(c,d)$ refers to the set of maximally even sets of size d in a c-element universe. The J-set is an ordering of these sets following the Clough–Myerson algorithm (Clough & Myerson 1985), adding one to each element's numerator of each subsequent set. (c, d) is the greatest common factor of c and d.

10 In other words, which set classes are interesting under P-relations that are not maximally even?

11 SC($\{0,2,5,8,10,12\}$) is the interesting mod-15 set class that Douthett was seeking.

12 This refers to the M5 and M7 operations described in Morris (1987).

13 Cubes are central to Douthett's eventual model, but this one behaves inconsistently and is not replicated in subsequent work. Note that Cdim is P-related to neither Caug nor Cm, as asserted. The problem here is that changes in coordinate do not involve consistent changes in direction. In this variation, a coordinate shift simply means that the note has moved, and the 1-position isn't necessarily +1 from the 0-position (often, it's $-1$ away).

14 In the trichords, Douthett uses “A” and “B” in place of 10 and 11, respectively. Thus A designates pitch class B^♭, and B (by coincidence) designates the eponymous pitch class.

15 While differently oriented, these cubes are identical to the ones eventually used in Cube Dance, mapping a dimension on the graph to the toggling of one element of the trichord. However, like the cube-graph in the first letter, this graph is inconsistent in mapping the binary digits to direction (i.e. 1 is not always the higher pitch).

16 The two ?-labelled chords are French 6ths.

17 The T${}_n/$I set class with dominant 7ths also includes their inversion, half-diminished 7ths.

18 In “Power Towers,” the diminished 7ths alone provide transportation. The French 6ths occupy the same radius as the minor 7ths.

19 Technical-Vocational Institute, now Central New Mexico Community College, in Albuquerque, New Mexico.

20 The accompanying illustrations that Douthett describes have been lost, but they are similar to ones eventually published in Douthett and Steinbach (1998)

21 See footnote 16.

22 This letter is not precisely dated. It was written sometime between February 1993 and early 1994.

23 The theorem referred to, “If a set class is PP [has the PP property], then so is its complement,” was proposed by Cohn in a letter to Paul Isihara on 8/15/1992. Isihara wrote back with a proof on 10/14/1992.

24 Agmon (1991).

25 Clough, Cuciurean, and Douthett (1997)

26 ${}^{\circ }7$ denotes a half-diminished 7th chord.

27 Compare Douthett and Steinbach (1998), p. 258.

28 See Agmon (1991)

29 Gollin (1998)

30 This is a version of the figure published in Douthett and Steinbach (1998), p. 250.

31 See Douthett and Steinbach (1998), p. 248.

32 Childs (1998).

33 See Clough and Douthett (1991).

34 Lewin (1996).

35 Lewin (1996).

36 Clampitt (1997).