Concepts locaux et globaux Premire partie Thorie objective

Concepts locaux et globaux. Première partie: Théorie ‚objective‘ Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola. ch www. encyclospace. org

contents • Introduction • Enumeration • Théorie d‘adresse zéro locale • Théorie d‘adresse zéro globale • Construction d‘une sonate • Adresses générales • Classification adressée globale

introduction Sets cartesian products X x Y disjoint sums X È Y powersets XY characteristic maps c: X —> 2 no „algebra“ Mod@ F: Mod —> Sets presheaves have all these properties Mod direct products A≈B has „algebra“ no powersets no characteristic maps

enumeration C Í Ÿ 12 (chords) MÍ — 2 (motives) Enumeration = calculation of the number of orbits of a set C of such objects under the canonical left action H¥C ® C of a subgroup H Í GA(F)� general = affine group on F Ambient space F = Ÿ 12 = finite -> Pólya & de Bruijn — 2 = infinite -> ? ?

enumeration 1973 A. Forte (1980 J. Rahn) • List of 352 orbits of chords under the translation group T 12 = eŸ 12 and the group TI 12 = eŸ 12. ± 1 of translations and inversions on Ÿ 12 1978 G. Halsey/E. Hewitt • Recursive formula for enumeration of translation orbits of chords in finite abelian groups F • Enumeration of orbit numbers for chords in cyclic groups Ÿn, n c 24 1980 G. Mazzola • List of the 158 affine orbits of chords in Ÿ 12 • List of the 26 affine orbits of 3 -elt. motives in (Ÿ 12)2 and 45 in Ÿ 5 ¥ Ÿ 12 1989 H. Straub /E. Köhler List of the 216 affine orbits of 4 -element motives in (Ÿ 12)2 1991. . . H. Fripertinger • Enumeration formulas for Tn, TIn, and affine chord orbits in Ÿn, n-phonic k-series, all-interval series, and motives in Ÿn ¥ Ÿm 2 • Lists of affine motive orbits in (Ÿ 12) up to 6 elements, explicit formula. . .

enumeration x^144 + x^143 + 5 x^142 + 26 x^141 + 216 x^140 + 2 024 x^139 + 27 806 x^138 + 417 209 x^137 +6 345 735 x^136 + 90 590 713 x^135 + 1 190 322 956 x^134 + 14 303 835 837 x^133 +157 430 569 051 x^132 + 1 592 645 620 686 x^131 + 14 873 235 105 552 x^130 + 128 762 751 824 308 x^129 + 1 037 532 923 086 353 x^128 + 7 809 413 514 931 644 x^127 +55 089 365 597 956 206 x^126 + 365 290 003 947 963 446 x^125 +2 282 919 558 918 081 919 x^124 + 13 479 601 808 118798 229 x^123 +75 361 590 622 423 713 249 x^122 + 399 738 890 367 674230 448 x^121 +2 015 334 387 723 540 077 262 x^120 + 9 673 558 570 858 327 142 094 x^119 + 44 275 002 111 552 677 715 575 x^118 + 193 497 799 414 541 699 555 587 x^117 +808 543 433 959 017 353 438 195 x^116 + 3 234 171 338 137 153 259 094292 x^115 +12 397 650 890 304 440 505 241198 x^114 + 45 591 347 244 850 943 472027 532 x^113 + 160 994 412 344 908 368 725 437 163 x^112 + 546 405 205 018 625 434 948486 100 x^111 +1 783 852 127 215 514 388 216 575 524 x^110 + 5 606 392 061 138 587 678 507 139 578 x^109 +16 974 908 597 922 176 404 758662 419 x^108 +49 548 380 452 249 950 392 015617 673 x^107 + 139 517 805 378 058 810 895 892 716 876 x^106 +379 202 235 047 824 659 955 968 634 895 x^105 +995 405 857 334 028 240 446 249 995 969 x^104 + 2 524 931 913 311 378 421 460 541 875 013 x^103 +6 192 094 899 403 308 142 319 324 646 830 x^102 + 14 688 225 057 065 816 000 841247 153 422 x^101 +33 716 152 882 551 682 431 054950 635 828 x^100 + 74 924 784 036 765 597 482 162224 697 378 x^99 +161 251 165 409 134 463 248 992 354 275 261 x^98 + 336 225 833 888 858 733 322 982 932 904 265 x^97 +679 456 372 086 288 422 448 712 466 252 503 x^96 + 1 331 179 830 182 151 403 666 404 596 530 852 x^95 +2 529 241 676 111 626 447 928 668 220 456 264 x^94 + 4 661 739 558 127 027 290 220 867 616 981 880 x^93 +8 337 341 899 567 786 249 391 103 289 453 916 x^92 + 14 472 367 067 576 451 752 984797 361 008 304 x^91 +24 388 618 572 337 747 341 932969 998 362 288 x^90 + 39 908 648 567 034 355 259 311114 115 744 392 x^89 +63 426 245 036 529 210 051 949169 850 308 102 x^88 + 97 921 220 397 909 924 969 018620 386 852 352 x^87 +146 881 830 585 458 073 270 850 321 720 445 928 x^86 + 214 098 939 483 879 341 610 433 150 629 060 274 x^85 +303 306 830 919 747 863 651 620 555 026 700 930 x^84 + 417 668 422 888 061 171 460 770 548 484 103 836 x^83 +559 136 759 653 084 522 330 064 385 877 590 780 x^82 + 727 765 306 194 069 123 565 702 210 626 823 392 x^81 +921 077 965 629 957 077 012 552 741 715 036 692 x^80 + 1 133 634 419 214 796 834 928 853 170 296 724314 x^79 +1 356 926 047 220 511 677 349 073 201 120 481570 x^78 + 1 579 704 950 475 555 411 914 967 237 903 930342 x^77 +1 788 783 546 844 376 088 722 000 995 922 467990 x^76 + 1 970 254 341 437 213 013 502 048 964 983 877090 x^75 +2 110 986 794 386 177 596 749 436 553 816 924660 x^74 + 2 200 183 419 494 435 885 449 671 402 432 366956 x^73 +2 230 741 522 540 743 033 415 296 821 609 381912 x^72 + …. … . . . + 2024. x 5 + 216. x 4 + 26. x 3 + 5. x 2 + x + 1 = cycle index polynomial 2 230 741 522 540 743 033 415 296 821 609 381 912. x 72 ª 2. 23. 1036. x 72 average # of stars in a galaxis = 100 000 000

enumeration From generalizations of the main theorem by N. G. de Bruijn, we have (for example) the following enumerations: k= 0 1 2 3 4 5 6 7 8 9 10 11 12 T 12 1 1 6 19 43 66 80 66 43 19 6 1 1 TI 12 1 1 6 12 29 38 50 38 29 12 6 1 1 GA(Ÿ 12) 1 1 5 9 21 25 34 25 21 9 5 1 1 k 2 3 4 5 6 # of orbits of (k, 12)-series 6 30 275 2 000 14 060 k 7 8 9 10 11 12 # of orbits of (k, 12)-series 83 280 416 880 1 663 680 4 993 440 9 980 160 9 985 920 (dodecaphonic)

affine category • Fix commutative ring R with 1. For any two (left) R-modules A, B, let A@B = e. B. Lin(A, B) be the R-module of R-affine morphisms F(a) = eb. F 0(a) = b + F 0(a) F 0 = linear part, eb = translation part. • Example: R = —, A = — 3, B = — 2 A@B = e. Lin(— 3, — 2) ª — 2 x M 2 x 3(—) — 2 eh. G 0. eb. F 0 = eh + G 0(b). F 0. G 0

local compositions The category Locom. R of local compositions over R: objects = couples (K, A) of subsets K of R-modules A, morphisms = f: (K, A) ® (L, B) = set maps f: K ® L which are induced by an affine morphism F in A@B. A B f K L

exampoles retrograde including duration reflection transvection

counterpoint a + e. b K = Ÿ 12 +e. {0, 3, 4, 7, 8, 9} = consonances Ÿ 12[e] = Ÿ 12[X]/(X 2) dual numbers in algebraic geometry e e. 2. 5 D = Ÿ 12 +e. {1, 2, 5, 6, 10, 11} = dissonances

S‘ – 3 log(5) p b b a e b f c g d b a b f Í f# d e b P S* Í – 2 log(3) F S Í just theory Major and chromatic scales S in just tuning: — = pitch axis S Í —— ? p = c + o. log(2)+ q. log(3) + t. log(5) = F(o, q, t) o, q, t Î – —–

just theory tonal inversion F = eq. -1 -1 0 1 a f e b c g d

just theory just major and minor 1800 -rotation = Uq turbidity = Uq. F a f e b c g ab eb d bb *

just theory 12 -tempered C-chromatic log(5) dbc one octave ebd abg bba b log(2) log(3) f# f e There is exactly one automorphism of the octave

just theory Just (Vogel) C-chromatic There is exactly one automorphism of the octave e a g f# b d c eb f bb log(5) ab db log(3) log(2)

concatenation Concatenation Theorem Mus. Gen = {T, Dm (m Î Ù ), K, S, Ps (s = 2, 3, . . . , n)} Set of endomorphisms of Ÿn as follows: • • T = et, t = (0, 1, 0, . . . , 0) translation in 2 nd axis. Dm = m-fold dilatation in direction of first axis K = D-1 = reflection in first axis S = transvection or shearing of the second coordinate in direction of the first axis • Ps = parameter exchange of first and sth coordinates Then every affine endomorphism on Ÿn is a concatenation of some elements of Mus. Gen. Affine automorphims are a concatenation of elements of Mus. Gen except the types Dm (m Î Ù ).

local classification Theorem (local geometric classification for a semi-simple ring) Let R be semi-simple and n any natural number. Then there is an R-algebraic scheme Cln such that the set Ob. Lo. Classn, R of isomorphism classes of local compositions of cardinality n in any R-module is in bijection with the set Cln(R) of R-valued points of Cln Ob. Lo. Classn, R ª Cln(R)

classification algorithm Application to orbit algorithms for rings • • • R of finite length R local self-injective E. g. R = Ÿsn , s = prime soc(Rn) Í V subspace V Í subgroup G Í Rn Sn+1 soc(V) π soc(Rn) V = soc(V) π soc(Rn) V π soc(V) V/soc (Rn) Í (R/soc(R))n VÍ (R/Rad(R))n I(V) Í Rn (direct factor) I(V) ª Rm m < n G : = Iso(I(V)) V Í Rm

motive classes 1 2 3 4 5 6 7 8 9 10 11 12 generic 13 14 15 16 17 18 19 20 21 22 23 24 25 26 0: 05 -0: 33 Classes of 3 -element motives M Í (Ÿ 12)2

globalization K K t Í Ct Ci ◊ K i ◊ ◊ Kit local iso Kti

scales 11 0 Space Ÿ 12 of pitch classes in 12 -tempered tuning 1 2 10 C 9 3 8 Scale = part of Ÿ 12 4 7 6 5 Twelve diatonic scales: C, F, Bb , Eb , Ab , Db , Gb , B, E, A, D, G

triadic interpretation I II IV V VI VII

nerves The class nerve cn(K) of global composition is not classifying 2 5 II 2 10 5 10 2 10 6 2 V VII 10 2 VI IV 2 6 2 2 5 15 2 2 10 I 2 6 5 2 2 2 10 III

meters n/16 0 a b c d e 2 3 4 6 8 10 12

nerves c 3 0 a 4 6 e 12 2 10 d nerve of the covering {a, b, c, d, e} x dominates y iff simplex(y) Í simplex(x) b

composition Sonate für Klavier „Aut. G(Messiaen III)DIA(3) “ (1981) Gruppen und Kategorien in der Musik Heldermann, Berlin 1985 Construction on 58 pages 99 bars, 12/8 metrum, C-major

scheme Overall Scheme Op. 106 Op. 3 AutŸ(C# -7 ) = {+1} x e 3Ÿ 12 AutŸ(C# +) = {+1} x e 4Ÿ 12 minor third ~ 2 nd Messiaen scale „limited transposition“ major third ~ 3 nd Messiaen scale „limited transposition“

modulators Modulators in op. 3 Exposition Development Recapitulation Coda C® B b ® G b ® Ab ® E E® A® F F® C Uc# e-4 Ua e-4 *

motivic principle Motivic Zig-Zag in op. 106 ( 1 0 -2 1 ) Bars 75 -78

motivic model Motivic Zig-Zag Scheme minor third ~ 2 nd Messiaen scale „limited transposition“ major third ~ 3 nd Messiaen scale „limited transposition“

möbius Motivic strip of Zig-Zag (15) 4 5 6 (19) 7 3 (2) 1 (11) 8 (20) 2 9 (10) (15) (16)

main theme C Main Theme IC Bars 3 -5 0: 10 -0: 20

kernel Kernel of Development 4 2 1 7 3 8 9 A‘ 7 8 2 1 3 5 9 7 8 6 4 A‘ 2 1 5 8 4 2 1 6 3 7 8 6 4 2 3 5 9 7 B 5 9 B‘ 1 3 9 7 A 8 4 6 4 2 1 3 C‘ 9 7 C 5 9 7 8 6 4 2 1 7 8 2 D 8 7 4 8 6 E‘ 8 5 7 F 3 1 2 5 4 6 3 9 E 9 7 6 4 2 1 5 3 9 D‘ 7 6 4 2 1 5 1 6 3 9 3 5 8 2 A 6 U 2 3 1 9 5 5 6 8 4 2 3 5 9 7 6 8 4 6 F‘

kernel ABCDEF 648791 564879 356487 235648 Kernel Matrix Dr db a f db A‘ B‘ C‘ D‘ E‘ F‘ Dl

kernel Dr 4: 18 -4: 43 D = Dr È Dl Dl

kernel moduation Kernel Modulation Ua : Gb ® Ab 4: 44 -5: 10 Ua U a (D l ) Dr Dl

addresses KÍ B B set module B @ 0Ÿ@B K Í A 0Ÿ@B • A = Ÿn: sequences (b 0, b 1, …, bn) • A = B: self-addressed tones Need general addresses A

motivic intervals B MÍ A B @B A@B = e. B. Lin(A, B) A=R R@B = e. B. Lin(R, B) ª B 2

series Ÿ 12 S A@B = e. B. Lin(A, B) R = Ÿ, A = Ÿ 11, B= Ÿ 12 Series: S Î Ÿ 11 @ Ÿ 12 = e Ÿ 12. Lin(Ÿ 11, Ÿ 12) ª Ÿ 12 12

0 Ÿ 12 @ 0 @ Ÿ 12 X={ Ÿ 12 @ Ÿ 12 ◊ ◊ self-addressed tones Ÿ 12 @ Ÿ 3 x Ÿ 4 } Int(X)

time spans David Lewin‘s time spans: (a, x) Î — x —+ a = onset, x = (multiplicative) duration increase factor Interval law: int((a, x), (b, y)) = ((b-a)/x, y/x) =(i, p) (b, y) = (a, x). (i, p) = (a+x. i, x. p) eb. y = ea. x. ei. p = ea+x. i. x. p is multiplication of affine morphisms ea. x, ei. p: — —> — Think of ea. x, ei. p Î — @ —, i. e. self-addressed onsets

global copmpositions The category Ob. Locom. A of local objective A-addressed compositions has as objects the couples (K, A@C) of sets K of affine morphisms in A@C and as morphisms f: (K, A@C) ® (L, A@D) set maps f: K ® L which are naturally induced by affine morphism F in C@D The category Ob. Glocom. A of global objective A-addressed compositions has as objects KI coverings of sets K by atlases I of local objective A-addressed compositions with manifold gluing conditions and manifold morphisms ff: KI ® LJ, including and compatible with atlas morphisms f: I ® J ! y f i s s a l C

resolutions Have universal construction of a „resolution of KI“ res: ADn* ® KI It is determined only by the KI address A and the nerve n* of the covering atlas I. res ADn* KI

non-interpretable 3 6 1 4 5 0 Dn* res 2 4 6 2 2 1 5 1 4 d c a b 4 5 KI 6 3 1 3 6 3 2 5

classification Theorem (global addressed geometric classification) Let A = locally free of finite rank over commutative ring R Consider the objective global compositions KI at A with (*): • the chart modules R. Ki are locally free of finite rank • the function modules G(Ki) are projective (i) Then KI can be reconstructed from the coefficient system of retracted functions res*n. G(KI) Í n. G(ADn*) (ii) There is a subscheme Jn* of a projective R-scheme of finite type whose points w: Spec(S) ® Jn* parametrize the isomorphism classes of objective global compositions at address SƒRA with (*).

fin théorie objective