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

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

3. Mod @ F: Mod —> Sets presheaves have all these properties introduction Sets cartesian products X x Y disjoint sums X  Y powersets X Y characteristic maps  X —>  no „algebra“ Mod direct products A ≈ B has „algebra“ no powersets no characteristic maps

4. enumeration C  Ÿ  (chords) M  — 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  GA(F) =  general affine group on F Ambient space F = Ÿ  = finite  ->Pólya & de Bruijn — 2 = infinite -> ??

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

6. enumeration x^144 + x^143 + 5x^142 + 26x^141 + 216x^140 + 2 024x^139 + 27 806x^138 + 417 209x^137 +6 345 735x^136 + 90 590 713x^135 + 1 190 322 956x^134 + 14 303 835 837x^133 +157 430 569 051x^132 + 1 592 645 620 686x^131 + 14 873 235 105 552x^130 + 128 762 751 824 308x^129 + 1 037 532 923 086 353x^128 + 7 809 413 514 931 644x^127 +55 089 365 597 956 206x^126 + 365 290 003 947 963 446x^125 +2 282 919 558 918 081 919x^124 + 13 479 601 808 118798 229x^123 +75 361 590 622 423 713 249x^122 + 399 738 890 367 674230 448x^121 +2 015 334 387 723 540 077 262x^120 + 9 673 558 570 858 327 142 094x^119 + 44 275 002 111 552 677 715 575x^118 + 193 497 799 414 541 699 555 587x^117 +808 543 433 959 017 353 438 195x^116 + 3 234 171 338 137 153 259 094292x^115 +12 397 650 890 304 440 505 241198x^114 + 45 591 347 244 850 943 472027 532x^113 + 160 994 412 344 908 368 725 437 163x^112 + 546 405 205 018 625 434 948486 100x^111 +1 783 852 127 215 514 388 216 575 524x^110 + 5 606 392 061 138 587 678 507 139 578x^109 +16 974 908 597 922 176 404 758662 419x^108 +49 548 380 452 249 950 392 015617 673x^107 + 139 517 805 378 058 810 895 892 716 876x^106 +379 202 235 047 824 659 955 968 634 895x^105 +995 405 857 334 028 240 446 249 995 969x^104 + 2 524 931 913 311 378 421 460 541 875 013x^103 +6 192 094 899 403 308 142 319 324 646 830x^102 + 14 688 225 057 065 816 000 841247 153 422x^101 +33 716 152 882 551 682 431 054950 635 828x^100 + 74 924 784 036 765 597 482 162224 697 378x^99 +161 251 165 409 134 463 248 992 354 275 261x^98 + 336 225 833 888 858 733 322 982 932 904 265x^97 +679 456 372 086 288 422 448 712 466 252 503x^96 + 1 331 179 830 182 151 403 666 404 596 530 852x^95 +2 529 241 676 111 626 447 928 668 220 456 264x^94 + 4 661 739 558 127 027 290 220 867 616 981 880x^93 +8 337 341 899 567 786 249 391 103 289 453 916x^92 + 14 472 367 067 576 451 752 984797 361 008 304x^91 +24 388 618 572 337 747 341 932969 998 362 288x^90 + 39 908 648 567 034 355 259 311114 115 744 392x^89 +63 426 245 036 529 210 051 949169 850 308 102x^88 + 97 921 220 397 909 924 969 018620 386 852 352x^87 +146 881 830 585 458 073 270 850 321 720 445 928x^86 + 214 098 939 483 879 341 610 433 150 629 060 274x^85 +303 306 830 919 747 863 651 620 555 026 700 930x^84 + 417 668 422 888 061 171 460 770 548 484 103 836x^83 +559 136 759 653 084 522 330 064 385 877 590 780x^82 + 727 765 306 194 069 123 565 702 210 626 823 392x^81 +921 077 965 629 957 077 012 552 741 715 036 692x^80 + 1 133 634 419 214 796 834 928 853 170 296 724314x^79 +1 356 926 047 220 511 677 349 073 201 120 481570x^78 + 1 579 704 950 475 555 411 914 967 237 903 930342x^77 +1 788 783 546 844 376 088 722 000 995 922 467990x^76 + 1 970 254 341 437 213 013 502 048 964 983 877090x^75 +2 110 986 794 386 177 596 749 436 553 816 924660x^74 + 2 200 183 419 494 435 885 449 671 402 432 366956x^73 +2 230 741 522 540 743 033 415 296 821 609 381912x^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 2.23.10 36.x 72 average # of stars in a galaxis = 100 000 000 000

7. enumeration From generalizations of the main theorem by N.G. de Bruijn, we have (for example) the following enumerations: k = 0123456789101112 T 12 11619436680664319611 TI 12 11612293850382912611 GA( Ÿ 12 )115921253425219511 k# of orbits of (k,12)-series 26 330 4275 52 000 614 060 k# of orbits of (k,12)-series 783 280 8416 880 91 663 680 104 993 440 119 980 160 129 985 920 (dodecaphonic)

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

9. 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 K B L f

10. exampoles retrograde including duration reflection transvection

11.  = Ÿ 12 +  = consonances D = Ÿ 12 +  {1,2,5,6,10,11} = dissonances e .2.5 Ÿ 12 [  Ÿ 12 [X]/(X 2  dual numbers in algebraic geometry  b a +  b counterpoint

12. ebebebeb abababab bbbbbbbb f#f#f#f# dbdbdbdb just theory Major and chromatic scales S in just tuning: — = pitch axis S  — — ? p = c + o.log(2)+ q.log(3) + t.log(5) = F(o,q,t) o,q,t  – f cg daeb log(5) log(3) —–—–—–—–S –3–3–3–3S‘ F f –2–2–2–2 S*  Pp

13. just theory f cg d a eb F = e q. -1 -1 0 1 0 1 tonal inversion

14. just theory f cg daeb abababab ebebebeb bb*bb*bb*bb* just major and minor 180 0 -rotation = U q turbidity = U q.F

15. one octave log(5)log(3) log(2) just theory c dbdbdbdb b d ebebebeb e f f#f#f#f# g abababab a bbbbbbbb 12-tempered C-chromatic There is exactly one automorphism of the octave There is exactly one automorphism of the octave

16. log(5)log(3) log(2) just theory Just (Vogel) C-chromatic There is exactly one automorphism of the octave There is exactly one automorphism of the octave c dbdbdbdb b d ebebebeb e f f#f#f#f# g abababab a bbbbbbbb

17. concatenation Concatenation Theorem Concatenation Theorem MusGen = {T, D m (m  Ù ), K, S, P s (s = 2,3,...,n)} MusGen = {T, D m (m  Ù ), K, S, P s (s = 2,3,...,n)} Set of endomorphisms of Ÿ n as follows: T = e t, t = (0,1,0,...,0) translation in 2nd axis. T = e t, t = (0,1,0,...,0) translation in 2nd axis. D m = m-fold dilatation in direction of first axis D m = m-fold dilatation in direction of first axis K = D -1 = reflection in first axis K = D -1 = reflection in first axis S = transvection or shearing of the second coordinate in direction of the first axis S = transvection or shearing of the second coordinate in direction of the first axis P s = parameter exchange of first and s th coordinates P s = parameter exchange of first and s th coordinates Then every affine endomorphism on Ÿ n is a concatenation of some elements of MusGen. Affine automorphims are a concatenation of elements of MusGen except the types D m (m  Ù ). Concatenation Theorem Concatenation Theorem MusGen = {T, D m (m  Ù ), K, S, P s (s = 2,3,...,n)} MusGen = {T, D m (m  Ù ), K, S, P s (s = 2,3,...,n)} Set of endomorphisms of Ÿ n as follows: T = e t, t = (0,1,0,...,0) translation in 2nd axis. T = e t, t = (0,1,0,...,0) translation in 2nd axis. D m = m-fold dilatation in direction of first axis D m = m-fold dilatation in direction of first axis K = D -1 = reflection in first axis K = D -1 = reflection in first axis S = transvection or shearing of the second coordinate in direction of the first axis S = transvection or shearing of the second coordinate in direction of the first axis P s = parameter exchange of first and s th coordinates P s = parameter exchange of first and s th coordinates Then every affine endomorphism on Ÿ n is a concatenation of some elements of MusGen. Affine automorphims are a concatenation of elements of MusGen except the types D m (m  Ù ).

18. 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 Cl n such that the set ObLoClass n,R of isomorphism classes of local compositions of cardinality n in any R-module is in bijection with the set Cl n (R) of R-valued points of Cl n 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 Cl n such that the set ObLoClass n,R of isomorphism classes of local compositions of cardinality n in any R-module is in bijection with the set Cl n (R) of R-valued points of Cl n ObLoClass n,R ª Cl n (R)

19. classification algorithm Application to orbit algorithms for rings Application to orbit algorithms for rings R of finite length R of finite length R local R local self-injective self-injective E.g. R = Ÿ s n, s = prime subspace V  R n subgroup G  S n+1 subspace V  R n subgroup G  S n+1 soc(R n )  V V/soc (R n )  R/soc(R)) n soc(V) π  soc(R n ) V = soc(V) V   R/Rad(R)) n soc(V) π  soc(R n ) V π soc(V) I(V)  R n (direct factor) I(V) ª R m m < n G := Iso(I(V)) V  R m I(V)  R n (direct factor) I(V) ª R m m < n G := Iso(I(V)) V  R m

20. motive classes Classes of 3-element motives M  ( Ÿ 12 ) 2 1 2 3456 789101112 13 14 15161718 19 20 21222324 25 26 generic 0:05-0:33

21. K globalization local iso  C i  K i  K t  C t K it  K ti 

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

23. I IVVIIIIIVIVII triadic interpretation

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

25. meters n/16 a b c d e 0234681012

26. 0 6 4 2 nerves b nerve of the covering {a,b,c,d,e} x dominates y iff simplex(y)  simplex(x) e c a d 12 10 3

27. 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

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

29. C  B b  G b G b  A b  E E  F F  C U c # e -4 U a e -4 * e -4 * Modulators in op. 3 DevelopmentExpositionRecapitulation Coda Coda modulators

30. motivic principle Motivic Zig-Zag in op.106    Bars 75-78

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

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

33. main theme Main Theme CC C Bars 3-5 0:10-0:20

34. kernel Kernel of Development 6 7 4 1 9 8 2 5 3        ‘‘‘‘  U2U2U2U2 A B C D E F A‘ B‘ C‘ D‘ E‘ F‘                                                                          

35. dbdbdbdb Kernel Matrix 6 4 8 7 9 1 5 6 4 8 7 9 3 5 6 4 8 7 2 3 5 6 4 8 A B C D E F dbdbdbdb f a DrDrDrDr DlDlDlDl A‘ B‘ C‘ D‘ E‘ F‘ kernel

36. DrDrDrDr DlDlDlDl D  =  D r  D l kernel 4:18-4:43

37. kernel moduation DrDrDrDr DlDlDlDl Kernel Modulation U a : G b  A b UaUaUaUa Ua(Dl)Ua(Dl)Ua(Dl)Ua(Dl) 4:44-5:10

38. addresses K  B B setmodule B  Ÿ @B A = Ÿ n : sequences (b 0,b 1,…,b n ) A = B: self-addressed tones Need general addresses A  Ÿ K  Ÿ @B 

39. motivic intervals B A@B = e B.Lin(A,B) A = R R@B = e B.Lin(R,B) ª B2 ª B2 ª B2 ª B2 M  B M  A@B

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

41. Ÿ 12  0 @ Ÿ 12 0 Ÿ 12 @ Ÿ 12 X = { }  self-addressed tones Ÿ 12   Ÿ 3 x Ÿ 4 Int(X) 

42. time spans David Lewin‘s time spans: (a,x)  — x — + David Lewin‘s time spans: (a,x)  — 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) e b.y = e a.x. e i.p = e a+x.i.x.p is multiplication of affine morphisms e a.x, e i.p: — —> — Think of e a.x, e i.p  — @ —, i.e. self-addressed onsets Think of e a.x, e i.p  — @ —, i.e. self-addressed onsets

43. Classify! The category ObLocom 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 ObGlocom A of global objective A-addressed compositions has as objects K I coverings of sets K by atlases I of local objective A-addressed compositions with manifold gluing conditions and manifold morphisms f  : K I  L J, including and compatible with atlas morphisms  : I  J global copmpositions

44. resolutions Have universal construction of a „resolution of K I “ res:  A  n*  K I It is determined only by the K I address A and the nerve n* of the covering atlas I. A  n* KIKIKIKI res

45. non-interpretable a d b c 1234 1265 5 6 34 1 2 3 4 6 5 KIKIKIKI 6 5 234 1 0  n* res

46. classification Theorem (global addressed geometric classification) Let A = locally free of finite rank over commutative ring R Consider the objective global compositions K I at A with (*): the chart modules R.K i are locally free of finite rank the function modules  (K i ) are projective the function modules  (K i ) are projective (i) Then K I can be reconstructed from the coefficient system of retracted functions res*n  (K I )  n  ( A  n* ) (ii) There is a subscheme J n* of a projective R-scheme of finite type whose points  : Spec(S)  J n* parametrize the isomorphism classes of objective global compositions at address S ƒ R A with (*). Theorem (global addressed geometric classification) Let A = locally free of finite rank over commutative ring R Consider the objective global compositions K I at A with (*): the chart modules R.K i are locally free of finite rank the function modules  (K i ) are projective the function modules  (K i ) are projective (i) Then K I can be reconstructed from the coefficient system of retracted functions res*n  (K I )  n  ( A  n* ) (ii) There is a subscheme J n* of a projective R-scheme of finite type whose points  : Spec(S)  J n* parametrize the isomorphism classes of objective global compositions at address S ƒ R A with (*).

47. fin théorie objective fin théorie objective