Guerino Mazzola

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Concepts locaux et globaux. Première partie
Théorie objective

• Guerino Mazzola
• U ETH Zürich
• Internet Institute for Music Science
• guerino_at_mazzola.ch
• www.encyclospace.org

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• Introduction
• Enumeration
• Théorie dadresse zéro locale
• Théorie dadresse zéro globale
• Construction dune sonate
• Adresses générales
• Classification adressée globale

contents
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Sets cartesian products X x Y disjoint sums X È
Y powersets XY characteristic maps c X gt 2 no
algebra
introduction
Mod direct products AB has algebra no
powersets no characteristic maps
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enumeration
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 Ÿ12 finite
-gt Pólya de Bruijn 2
infinite -gt ??
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• 1973 A. Forte (1980 J.Rahn)
• List of 352 orbits of chords under the
  translation group T12 eŸ12 and the group
  TI12 eŸ12. 1 of translations and inversions on
  Ÿ12
• 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 Ÿ12
• List of the 26 affine orbits of 3-elt. motives
  in (Ÿ12)2 and 45 in Ÿ5 Ÿ12
• 1989 H. Straub /E.Köhler List of the 216
  affine orbits of 4-element motives in (Ÿ12)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
• Lists of affine motive orbits in (Ÿ12)2 up to 6
  elements, explicit formula...

enumeration
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x144 x143 5x142 26x141 216x140 2
024x139 27 806x138 417 209x137 6 345
735x136 90 590 713x135 1 190 322 956x134
14 303 835 837x133 157 430 569 051x132 1
592 645 620 686x131 14 873 235 105 552x130
128 762 751 824 308x129 1 037 532 923 086
353x128 7 809 413 514 931 644x127 55 089 365
597 956 206x126 365 290 003 947 963 446x125
2 282 919 558 918 081 919x124 13 479 601 808
118798 229x123 75 361 590 622 423 713 249x122
399 738 890 367 674230 448x121 2 015 334 387
723 540 077 262x120 9 673 558 570 858 327 142
094x119 44 275 002 111 552 677 715 575x118
193 497 799 414 541 699 555 587x117 808 543 433
959 017 353 438 195x116 3 234 171 338 137 153
259 094292x115 12 397 650 890 304 440 505
241198x114 45 591 347 244 850 943 472027
532x113 160 994 412 344 908 368 725 437
163x112 546 405 205 018 625 434 948486
100x111 1 783 852 127 215 514 388 216 575
524x110 5 606 392 061 138 587 678 507 139
578x109 16 974 908 597 922 176 404 758662
419x108 49 548 380 452 249 950 392 015617
673x107 139 517 805 378 058 810 895 892 716
876x106 379 202 235 047 824 659 955 968 634
895x105 995 405 857 334 028 240 446 249 995
969x104 2 524 931 913 311 378 421 460 541 875
013x103 6 192 094 899 403 308 142 319 324 646
830x102 14 688 225 057 065 816 000 841247 153
422x101 33 716 152 882 551 682 431 054950 635
828x100 74 924 784 036 765 597 482 162224 697
378x99 161 251 165 409 134 463 248 992 354 275
261x98 336 225 833 888 858 733 322 982 932 904
265x97 679 456 372 086 288 422 448 712 466 252
503x96 1 331 179 830 182 151 403 666 404 596
530 852x95 2 529 241 676 111 626 447 928 668
220 456 264x94 4 661 739 558 127 027 290 220
867 616 981 880x93 8 337 341 899 567 786 249
391 103 289 453 916x92 14 472 367 067 576 451
752 984797 361 008 304x91 24 388 618 572 337
747 341 932969 998 362 288x90 39 908 648 567
034 355 259 311114 115 744 392x89 63 426 245
036 529 210 051 949169 850 308 102x88 97 921
220 397 909 924 969 018620 386 852 352x87 146
881 830 585 458 073 270 850 321 720 445 928x86
214 098 939 483 879 341 610 433 150 629 060
274x85 303 306 830 919 747 863 651 620 555 026
700 930x84 417 668 422 888 061 171 460 770 548
484 103 836x83 559 136 759 653 084 522 330 064
385 877 590 780x82 727 765 306 194 069 123 565
702 210 626 823 392x81 921 077 965 629 957 077
012 552 741 715 036 692x80 1 133 634 419 214
796 834 928 853 170 296 724314x79 1 356 926 047
220 511 677 349 073 201 120 481570x78 1 579
704 950 475 555 411 914 967 237 903 930342x77 1
788 783 546 844 376 088 722 000 995 922
467990x76 1 970 254 341 437 213 013 502 048
964 983 877090x75 2 110 986 794 386 177 596 749
436 553 816 924660x74 2 200 183 419 494 435
885 449 671 402 432 366956x73 2 230 741 522 540
743 033 415 296 821 609 381912x72 . ...
2024.x5 216.x4 26.x3 5.x2 x 1 cycle
index polynomial
enumeration
average of stars in a galaxis 100 000 000 000
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From generalizations of the main theorem by N.G.
de Bruijn, we have (for example) the following
enumerations
enumeration
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• Fix commutative ring R with 1. For any two
  (left) R-modules A,B, let
• A_at_B eB.Lin(A,B)
• be the R-module of R-affine morphisms
• F(a) eb.F0(a) b F0(a)
• F0 linear part, eb translation part.
• Example R , A 3, B 2
• A_at_B e2.Lin(3, 2) ª 2 x M2 x 3()
• eh.G0. eb.F0 eh G0(b).F0 .G0

affine category
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• The category LocomR 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_at_B.

local compositions
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retrograde including duration
exampoles
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counterpoint
a e.b
K Ÿ12 e.0,3,4,7,8,9 consonances
Ÿ12e Ÿ12X/(X2) dual numbers in algebraic
geometry
D Ÿ12 e.1,2,5,6,10,11 dissonances
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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
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tonal inversion
just theory
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just major and minor
just theory
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12-tempered C-chromatic
just theory
There is exactly one automorphism of the octave
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Just (Vogel) C-chromatic
There is exactly one automorphism of the octave
just theory
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• Concatenation Theorem
• MusGen 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 2nd
  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 MusGen.
• Affine automorphims are a concatenation of
  elements of MusGen except the types Dm (m Î Ù
  ).

concatenation
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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
ObLoClassn,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
local classification
ObLoClassn,R ª Cln(R)
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• Application to orbit algorithms for rings
• R of finite length
• R local
• self-injective
• E.g. R Ÿsn , s prime

classification algorithm
subspace V Í Rn subgroup G Í Sn1
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motive classes
005-033
Classes of 3-element motives M Í (Ÿ12)2
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globalization
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scales
Space Ÿ12 of pitch classes in 12-tempered tuning
Twelve diatonic scales C, F, Bb , Eb , Ab , Db ,
Gb , B, E, A, D, G
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triadic interpretation
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The class nerve cn(K) of global composition
is not classifying
nerves
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meters
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nerves
3
6
2
nerve of the covering a,b,c,d,e
x dominates y iff simplex(y) Í simplex(x)
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Sonate für Klavier AutG(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
composition
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Overall Scheme
scheme
AutŸ(C -7 ) 1 x e 3Ÿ12
AutŸ(C ) 1 x e 4Ÿ12
minor third 2nd Messiaen scale limited
transposition
major third 3nd Messiaen scale limited
transposition
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Modulators in op. 3
modulators
C Bb Gb Gb Ab E E
A F F C
Uc e-4 Ua e-4
e-4

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Motivic Zig-Zag in op.106
motivic principle
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Motivic Zig-Zag Scheme
motivic model
minor third 2nd Messiaen scale limited
transposition
major third 3nd Messiaen scale limited
transposition
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Motivic strip of Zig-Zag
möbius
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Main Theme
main theme
010-020
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Kernel of Development
kernel
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Kernel Matrix
kernel
Dr
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kernel
418-443
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Kernel Modulation Ua Gb Ab
kernel moduation
444-510
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addresses
B _at_ 0Ÿ_at_B
K Í 0Ÿ_at_B
A

• A Ÿn sequences (b0,b1,,bn)
• A B self-addressed tones
• Need general addresses A

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motivic intervals
A_at_B eB.Lin(A,B) A R R_at_B eB.Lin(R,B) ª B2
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series
A_at_B eB.Lin(A,B) R Ÿ, A Ÿ11, B
Ÿ12 Series S Î Ÿ11 _at_ Ÿ12 e Ÿ12.Lin(Ÿ11,
Ÿ12) ª Ÿ12 12
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Ÿ12 _at_ Ÿ3 x Ÿ4
self-addressed tones
Ÿ12 _at_ 0 _at_ Ÿ12
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David Lewins 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)
(ax.i,x.p) eb.y ea.x. ei.p eax.i.x.p is
multiplication of affine morphisms ea.x, ei.p
gt Think of ea.x, ei.p Î _at_ , i.e.
self-addressed onsets
time spans
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The category ObLocomA of local objective
A-addressed compositions has as objects the
couples (K, A_at_C) of sets K of affine morphisms
in A_at_C and as morphisms f (K, A_at_C) (L,
A_at_D) set maps f K L which are naturally
induced by affine morphism F in C_at_D The
category ObGlocomA 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
global copmpositions
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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.
resolutions
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non-interpretable
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• 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
• resnG(KI) Í nG(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 ().

classification
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fin théorie objective