Hauer s Tropes and the Enumeration of Twelve-Tone Hexachords

Transcription

1 Hauer s Tropes and the Enumeration of Twelve-Tone Hexachords Polytrope 2008 v 3 The tropes (hexachordal partitions) of Josef Matthias Hauer 1 ( ) afford a compact enumeration of the 6-chord types in a 12-pitch-class context. 2 Since 36 of the 44 tropes 3 comprise two hexachords of different types, a listing of the 44 tropes shows clearly all 80 hexachord types. Furthermore, if (as in Figures 2-8 here) the tropes are arranged in order of George Perle s enumeration of the hexachords, 4 then hexachord types of the same combinatorial character 5 appear together. This is true because, 1) being complementary, the two halves of a given trope have necessarily the same combinatorial character, and 2) combinatorial character corresponds exactly with the kinds of symmetry which guide Perle s enumeration. Each component diagram in Figures 2-8 here shows the two complementary hexachords of the Hauer trope whose number appears in the centre of the diagram. The pitch classes belonging to each half-trope are indicated by filled circles or open circles, the position of each circle indicating a corresponding pitch-class number as in Figure 1. If the Hauer number is shown with an overline, then the filled circles represent the second half-trope; otherwise the filled circles represent the first half-trope. 1 See Thanks to Christopher Butterfield for calling my attention to the affinity of my PSTP technique to Hauer s. (See my Pitch-Symmetric Tetrachordal Partitions on this website.) 2 See my Pitch-Symmetric Tetrachords, etc on this website, pp 9ff. 3 All but tropes 1, 8, 17, 19, 24, 34, 41, and Serial Composition and Atonality, University of California Press, (6th ed., rev.) 1991, pp ( Six-Note Collections ). 5 I.e. the property of comprising or not the pitch class sets of the first or last half of a tone row which is combinatorial by a particular transformation. 1

2 Below each diagram are shown the interval vector 6 common to the two half-tropes and the numbers assigned them in Perle s enumeration (P) and in that of Allen Forte 7 (F). Figures 2 and 3 also show the applicable symbols from my Pitch-Symmetric Tetrachords, etc. 8 The choice of transformation for each trope makes the pitch-class numbers agree with those in Perle s enumeration, except for the Ai / Bi diagrams in Figures 7 and 8, where my choice emphasizes the relation of pitch-inversion (mirror symmetry in the diagrams) between those and their Ap / Bp counterparts. Note that the four hexachords sharing each Perle number here also share an inverval vector. The affixed = signs call attention to this. Likewise in Figure 6. For Figures 2-5, due to the choice of transformation for each trope, either the diameter or (for P1, P3, P7, P10, P12, P13) the 0 6 diameter of each diagram is an axis of symmetry (Figures 2 and 3) or antisymmetry (Figures 4 and 5). To summarize, Figures 2-8 constitute an illustrated listing of the hexachord types by Perle number, showing also, inter alia, their Forte numbers and their position among Hauer s tropes. The appendices complete the cross-indexing of these three enumerations. Appendix A shows the Hauer tropes in their original order along with the Perle and Forte numbers for their constituent hexachords. Appendix B lists the hexachords by Forte number along with their prime forms, interval vectors, combinatorial character, Perle numbers, and position among Hauer s tropes Figure 1: Circle-of-semitones order for Figures. 6 I.e. the numbers of intervals within the hexachord comprising respectively 1, 2, 3, 4, 5, and 6 semitones. 7 The Structure of Atonal Music, Yale University Press, 1973 (prefixes 6 omitted). Note that Forte s enumeration does not distinguish between a set and its pitch-inverse; accordingly many Forte numbers apply to more than one half-trope. 8 On this website. 2

3 H 44 H 34 H 8 0, 6, 0, 6, 0, 3 : P 1; F 35; 2, 2, 2 3, 0, 3, 6, 3, 0 : P 2; F 20; 1, 3, 1 4, 2, 0, 2, 4, 3 : P 3; F 7; 1, 1, 4 H 1 H 17 H 41 5, 4, 3, 2, 1, 0 : P 4; F 1; 1, 1, 1 3, 4, 3, 2, 3, 0 : P 5; F 8; 2, 1, 1 1, 4, 3, 2, 5, 0 : P 6; F 32; 2, 2, 1 Figure 2: Tropes for Fully Combinatorial Rows 3

4 H 4 H 7 H 11 4, 3, 2, 3, 2, 1 : P 7A; F Z37; 1, 1 : P 7B; F Z4; 1, 1, 2 4, 2, 1, 2, 4, 2 : P 8A; F Z38; 1, 4, 1 : P 8B; F Z6; 1, 1, 3 3, 2, 4, 2, 2, 2 : P 9A; F Z13; 1, 2, 1 : P 9B; F Z42; 3, 1, 1 H 36 H 33 2, 3, 2, 3, 4, 1 : P 10A; F Z48; 1, 3 : P 10B; F Z26; 1, 2, 2 2, 2, 4, 2, 3, 2 : P 11A; F Z50; 2, 3, 1 : P 11B; F Z29; 1, 2, 3 H 28 H 31 2, 3, 4, 2, 2, 2 : P 12A; F Z45; 1, 2 : P 12B; F Z23; 2, 1, 2 2, 2, 4, 3, 2, 2 : P 13A; F Z28; 2, 1 : P 13B; F Z49; 1, 3, 2 Figure 3: Tropes for Rows Semicombinatorial by Retrograde Inversion. 4

5 H 38 H 2 H 9 H 6 2, 2, 4, 2, 2, 3 : P 14p; F 30 : P 14i; F 30 4, 4, 3, 2, 1, 1 : P 15p; F 2 : P 15i; F 2 3, 4, 2, 2, 3, 1 : P 16p; F 9 : P 16i; F 9 4, 2, 2, 2, 3, 2 : P 17p; F 5 : P 17i; F 5 H 18 H 39 H 40 3, 2, 3, 4, 2, 1 : P 18p; F 15 : P 18i; F 15 2, 4, 1, 4, 2, 2 : P 19p; F 22 : P 19i; F 22 2, 4, 2, 4, 1, 2 : P 20p; F 21 : P 20i; F 21 H 21 H 23 3, 2, 2, 4, 3, 1 : P 21p; F 16 : P 21i; F 16 3, 2, 2, 2, 4, 2 : P 22p; F 18 : P 22i; F 18 Figure 4: Tropes for Rows Semicombinatorial by Inversion (1 of 2). 5

6 H 30 H 43 H 32 H 42 2, 2, 5, 2, 2, 2 : P 23p; F 27 : P 23i; F 27 1, 4, 3, 2, 4, 1 : P 24p; F 33 : P 24i; F 33 2, 2, 3, 4, 3, 1 : P 25p; F 31 : P 25i; F 31 1, 4, 2, 4, 2, 2 : P 26p; F 34 : P 26i; F 34 Figure 5: Tropes for Rows Semicombinatorial by Inversion (2 of 2). H 24 H 19 3, 2, 3, 4, 3, 0 = : P 27p; F 14 = 3, 2, 3, 4, 3, 0 : P 27i; F 14 Figure 6: Tropes for Rows Semicombinatorial by Transposition. 6

7 H 5 H 3 H 14 H 15 4, 3, 3, 2, 2, 1 = : P 28Ap; F Z36 : P 28Bp; F Z3 = 4, 3, 3, 2, 2, 1 : P 28Ai; F Z36 : P 28Bi; F Z3 3, 3, 3, 2, 3, 1 = : P 29Ap; F Z40 : P 29Bp; F Z11 = 3, 3, 3, 2, 3, 1 : P 29Ai; F Z40 : P 29Bi; F Z11 H 12 H 16 H 13 H 10 3, 3, 3, 3, 2, 1 = : P 30Ap; F Z39 : P 30Bp; F Z10 = 3, 3, 3, 3, 2, 1 : P 30Ai; F Z39 : P 30Bi; F Z10 3, 3, 2, 2, 3, 2 = : P 31Ap; F Z41 : P 31Bp; F Z12 = 3, 3, 2, 2, 3, 2 : P 31Ai; F Z41 : P 31Bi; F Z12 Figure 7: Tropes for Noncombinatorial Rows (1 of 2). 7

8 H 29 H 27 H 22 H 20 2, 3, 3, 3, 3, 1 = : P 32Ap; F Z46 : P 32Bp; F Z24 = 2, 3, 3, 3, 3, 1 : P 32Ai; F Z46 : P 32Bi; F Z24 3, 2, 2, 3, 3, 2 = : P 33Ap; F Z17 : P 33Bp; F Z43 = 3, 2, 2, 3, 3, 2 : P 33Ai; F Z17 : P 33Bi; F Z43 H 37 H 35 H 26 H 25 2, 3, 3, 2, 4, 1 = : P 34Ap; F Z47 : P 34Bp; F Z25 = 2, 3, 3, 2, 4, 1 : P 34Ai; F Z47 : P 34Bi; F Z25 3, 1, 3, 4, 3, 1 = : P 35Ap; F Z44 : P 35Bp; F Z19 = 3, 1, 3, 4, 3, 1 : P 35Ai; F Z44 : P 35Bi; F Z19 Figure 8: Tropes for Noncombinatorial Rows (2 of 2). 8

9 A Hauer s Tropes in Original Order The diagrams in this appendix (Figures 9-12) are generally as above, but arranged so as to represent directly TAFEL I on In particular, assuming pitch-class numbers as in Figure 1, 0 here corresponds to the pitch class D =E, and closed circles indicate first half-tropes, open circles second half-tropes. H 1 H 2 H 3 H 4 : P 4; F 1 : P 15p; F 2 : P 15i; F 2 : P 28Bi; F Z3 : P 28Ai; F Z36 : P 7B; F Z4 : P 7A; F Z37 H 5 H 6 H 7 H 8 : P 28Bp; F Z3 : P 28Ap; F Z36 : P 17p; F 5 : P 17i; F 5 : P 8B; F Z6 : P 8A; F Z38 : P 3; F 7 Figure 9: Hauer Tropes

10 H 9 H 10 H 11 H 12 : P 16p; F 9 : P 16i; F 9 : P 31Bi; F Z12 : P 31Ai; F Z41 : P 9A; F Z13 : P 9B; F Z42 : P 30Bp; F Z10 : P 30Ap; F Z39 H 13 H 14 H 15 H 16 : P 31Bp; F Z12 : P 31Ap; F Z41 : P 29Bp; F Z11 : P 29Ap; F Z40 : P 29Bi; F Z11 : P 29Ai; F Z40 : P 30Bi; F Z10 : P 30Ai; F Z39 H 17 H 18 H 19 H 20 : P 5; F 8 : P 18p; F 15 : P 18i; F 15 : P 27i; F 14 : P 33Bi; F Z43 : P 33Ai; F Z17 Figure 10: Hauer Tropes

11 H 21 H 22 H 23 H 24 : P 21p; F 16 : P 21i; F 16 : P 33Ap; F Z17 : P 33Bp; F Z43 : P 22p; F 18 : P 22i; F 18 : P 27p; F 14 H 25 H 26 H 27 H 28 : P 35Bi; F Z19 : P 35Ai; F Z44 : P 35Bp; F Z19 : P 35Ap; F Z44 : P 32Bi; F Z24 : P 32Ai; F Z46 : P 12B; F Z23 : P 12A; F Z45 H 29 H 30 H 31 H 32 : P 32Bp; F Z24 : P 32Ap; F Z46 : P 23p; F 27 : P 23i; F 27 : P 13B; F Z49 : P 13A; F Z28 : P 25i; F 31 : P 25p; F 31 Figure 11: Hauer Tropes

12 H 33 H 34 H 35 H 36 : P 11B; F Z29 : P 11A; F Z50 : P 2; F 20 : P 34Bi; F Z25 : P 34Ai; F Z47 : P 10B; F Z26 : P 10A; F Z48 H 37 H 38 H 39 H 40 : P 34Bp; F Z25 : P 34Ap; F Z47 : P 14p; F 30 : P 14i; F 30 : P 19p; F 22 : P 19i; F 22 : P 20i; F 21 : P 20p; F 21 H 41 H 42 H 43 H 44 : P 6; F 32 : P 26p; F 34 : P 26i; F 34 : P 24i; F 33 : P 24p; F 33 : P 1; F 35 Figure 12: Hauer Tropes

13 B Hexachord Types by Forte Number combin. Forte prime form interval vector T I RI Perle Hauer 1 {0, 1, 2, 3, 4, 5} 5, 4, 3, 2, 1, I, II 2 {0, 1, 2, 3, 4, 6} 4, 4, 3, 2, 1, I, II Z3 {0, 1, 2, 3, 5, 6} 4, 3, 3, 2, 2, 1 28B 3.I, 5.I Z4 {0, 1, 2, 4, 5, 6} 4, 3, 2, 3, 2, 1 7B 4.I 5 {0, 1, 2, 3, 6, 7} 4, 2, 2, 2, 3, I, II Z6 {0, 1, 2, 5, 6, 7} 4, 2, 1, 2, 4, 2 8B 7.I 7 {0, 1, 2, 6, 7, 8} 4, 2, 0, 2, 4, I, II 8 {0, 2, 3, 4, 5, 7} 3, 4, 3, 2, 3, I, II 9 {0, 1, 2, 3, 5, 7} 3, 4, 2, 2, 3, I, II Z10 {0, 1, 3, 4, 5, 7} 3, 3, 3, 3, 2, 1 30B 12.I, 16.I Z11 {0, 1, 2, 4, 5, 7} 3, 3, 3, 2, 3, 1 29B 14.I, 15.I Z12 {0, 1, 2, 4, 6, 7} 3, 3, 2, 2, 3, 2 31B 10.I, 13.I Z13 {0, 1, 3, 4, 6, 7} 3, 2, 4, 2, 2, 2 9A 11.I 14 {0, 1, 3, 4, 5, 8} 3, 2, 3, 4, 3, I, II; 24.I, II 15 {0, 1, 2, 4, 5, 8} 3, 2, 3, 4, 2, I, II 16 {0, 1, 4, 5, 6, 8} 3, 2, 2, 4, 3, I, II Z17 {0, 1, 2, 4, 7, 8} 3, 2, 2, 3, 3, 2 33A 20.II, 22.I 18 {0, 1, 2, 5, 7, 8} 3, 2, 2, 2, 4, I, II prime form: see Forte, op. cit. interval vector: see footnote 6 above. combin.: combinatoriality by transposition, inversion, retrograde inversion. Table 1: Hexachord Types by Forte Number (1 of 3). 13

14 combin. Forte prime form interval vector T I RI Perle Hauer Z19 {0, 1, 3, 4, 7, 8} 3, 1, 3, 4, 3, 1 35B 25.I, 26.I 20 {0, 1, 4, 5, 8, 9} 3, 0, 3, 6, 3, I, II 21 {0, 2, 3, 4, 6, 8} 2, 4, 2, 4, 1, I, II 22 {0, 1, 2, 4, 6, 8} 2, 4, 1, 4, 2, I, II Z23 {0, 2, 3, 5, 6, 8} 2, 3, 4, 2, 2, 2 12B 28.I Z24 {0, 1, 3, 4, 6, 8} 2, 3, 3, 3, 3, 1 32B 27.I, 29.I Z25 {0, 1, 3, 5, 6, 8} 2, 3, 3, 2, 4, 1 34B 35.I, 37.I Z26 {0, 1, 3, 5, 7, 8} 2, 3, 2, 3, 4, 1 10B 36.I 27 {0, 1, 3, 4, 6, 9} 2, 2, 5, 2, 2, I, II Z28 {0, 1, 3, 5, 6, 9} 2, 2, 4, 3, 2, 2 13A 31.II Z29 {0, 2, 3, 6, 7, 9} 2, 2, 4, 2, 3, 2 11B 33.I 30 {0, 1, 3, 6, 7, 9} 2, 2, 4, 2, 2, I, II 31 {0, 1, 4, 5, 7, 9} 2, 2, 3, 4, 3, I, II 32 {0, 2, 4, 5, 7, 9} 1, 4, 3, 2, 5, I, II 33 {0, 2, 3, 5, 7, 9} 1, 4, 3, 2, 4, I, II 34 {0, 1, 3, 5, 7, 9} 1, 4, 2, 4, 2, I, II 35 {0, 2, 4, 6, 8, 10} 0, 6, 0, 6, 0, I, II Z36 {0, 1, 2, 3, 4, 7} 4, 3, 3, 2, 2, 1 28A 3.II, 5.II Z37 {0, 1, 2, 3, 4, 8} 4, 3, 2, 3, 2, 1 7A 4.II Z38 {0, 1, 2, 3, 7, 8} 4, 2, 1, 2, 4, 2 8A 7.II Z39 {0, 2, 3, 4, 5, 8} 3, 3, 3, 3, 2, 1 30A 12.II, 16.II Z40 {0, 1, 2, 3, 5, 8} 3, 3, 3, 2, 3, 1 29A 14.II, 15.II Z41 {0, 1, 2, 3, 6, 8} 3, 3, 2, 2, 3, 2 31A 10.II, 13.II Z42 {0, 1, 2, 3, 6, 9} 3, 2, 4, 2, 2, 2 9B 11.II prime form: see Forte, op. cit. interval vector: see footnote 6 above. combin.: combinatoriality by transposition, inversion, retrograde inversion. Table 2: Hexachord Types by Forte Number (2 of 3). 14

15 combin. Forte prime form interval vector T I RI Perle Hauer Z43 {0, 1, 2, 5, 6, 8} 3, 2, 2, 3, 3, 2 33B 20.I, 22.II Z44 {0, 1, 2, 5, 6, 9} 3, 1, 3, 4, 3, 1 35A 25.II, 26.II Z45 {0, 2, 3, 4, 6, 9} 2, 3, 4, 2, 2, 2 12A 28.II Z46 {0, 1, 2, 4, 6, 9} 2, 3, 3, 3, 3, 1 32A 27.II, 29.II Z47 {0, 1, 2, 4, 7, 9} 2, 3, 3, 2, 4, 1 34A 35.II, 37.II Z48 {0, 1, 2, 5, 7, 9} 2, 3, 2, 3, 4, 1 10A 36.II Z49 {0, 1, 3, 4, 7, 9} 2, 2, 4, 3, 2, 2 13B 31.I Z50 {0, 1, 4, 6, 7, 9} 2, 2, 4, 2, 3, 2 11A 33.II prime form: see Forte, op. cit. interval vector: see footnote 6 above. combin.: combinatoriality by transposition, inversion, retrograde inversion. Table 3: Hexachord Types by Forte Number (3 of 3). 15