Transcript No Slide Title

Concepts locaux et globaux.
Première partie:
Théorie ‚objective‘
Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science
[email protected]
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 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 Ÿ12
(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 = Ÿ12= finite ->Pólya & de Bruijn
—2 = infinite -> ??
enumeration
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
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^12 2 +
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^11 0 +
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 673 x^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 9 95 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.x5 + 216.x4 + 26.x3 + 5.x2 + x + 1 = cycle index polynomial
2 230 741 522 540 743 033 415 296 821 609 381 912.x72 ª 2.23.1036 .x72
average # of stars in a galaxis = 100 000 000 000
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
T12
1
1
6
19
43
66
80
66
43
19
6
1
1
TI12
1
1
6
12
29
38
50
38
29
12
6
1
1
GA(Ÿ12)
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 = 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@B = e .Lin(—3, —2) ª —2 x M2 x 3(—)
—2
eh.G0. eb.F0 = eh + G0(b).F0 .G0
local compositions
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@B.
A
B
f
K
L
exampoles
retrograde including duration
reflection
transvection
counterpoint
a + e.b
K = Ÿ12 +e.{0,3,4,7,8,9} = consonances
Ÿ12[e]=Ÿ12[X]/(X2)
dual numbers in
algebraic geometry
e e.2.5
D = Ÿ12 +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
f#
d
e
b
P
S*  –2
log(3)
F
S

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 –
—–
just theory
tonal inversion
F = eq. -1 -1
0
a
f
e
b
c
g
1
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
MusGen = {T, Dm (m  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  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
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
ObLoClassn,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) V
subspace V Rn
subgroup G 
Sn+1
soc(V) π soc(Rn)
V = soc(V)
soc(V) π soc(Rn)
V π soc(V)
V/soc (Rn) R/soc(R))n
V 
R/Rad(R))n
I(V) Rn (direct
factor)
I(V) ª Rm m < n
G := Iso(I(V))
V 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  (Ÿ12)2
globalization
K
Ci  Ki
Kt Ct


Kit
Kti
local iso
scales
11
0
Space Ÿ12 of pitch classes in
12-tempered tuning
1
2
10
C
9
3
8
Scale = part of Ÿ12
4
7
6
5
Twelve diatonic scales: C, F, Bb , Eb , Ab , Db , Gb , B, E, A,
D, G
triadic interpretation
I
II
III
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)  simplex(x)
b
composition
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
scheme
Overall Scheme
Op. 106
Op. 3
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“
modulators
Modulators in op. 3
Exposition
Development
Recapitulation
Coda
CBb  Gb
Gb Ab E
E F
F C
Uc# e-4
Ua e-4
*
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 ~
2nd Messiaen
scale
„limited transposition“
major third ~
3nd Messiaen
scale
„limited transposition“
möbius
Motivic strip of Zig-Zag
(15)
5
6
4
(19)
(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
‘
7
2
1
8
3
A
5
9
7
6
8
1
7
6
8
7
4
6
5
7
6
8
4
3
7
C‘9
C
5
9
6
8
3
9
4
2
1
4
2
B‘1
5
9
B
5
8
3
3
9
4
A‘ 2
2
1

6
U2
6
4
4
2
1
6
8
7
D‘ 7
D
6
8
7
4
6
8
7
E‘ 8
7
6
F
3
1
6
4
5
E
5
9
3
9
4
2
2
1
5
1
5
3
9
3
9
2
1
5
8
5
3
9
3
7
2
1
4
2
1
3
2
9
5
5
6
1
8
4
2
3
5
9
7
6
8
4
F‘
kernel
ABCD EF
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 Dl
Dl
kernel moduation
Kernel Modulation Ua : Gb Ab
4:44-5:10
Ua
Ua(Dl)
Dr
Dl
addresses
K B
B
set
module
B @0Ÿ@B
 Ÿ@B
K 0
• A = Ÿn: sequences
(b0,b1,…,bn)
• A = B: self-addressed tones
Need general addresses A
motivic intervals
B
M A@B
B
A@B = eB.Lin(A,B)
A=R
R@B = eB.Lin(R,B)
ª B2
series
Ÿ12
S
A@B = eB.Lin(A,B)
R = Ÿ, A = Ÿ11, B= Ÿ12
Series: S  Ÿ11 @ Ÿ12 = e Ÿ12.Lin(Ÿ11, Ÿ12)
ª Ÿ12 12
0
Ÿ12 @ 0 @ Ÿ12
X={
Ÿ12 @ Ÿ12


self-addressed tones
Ÿ12 @ Ÿ3 x Ÿ4
}
Int(X)
time spans
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)
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  — @ —, i.e. self-addressed
onsets
global copmpositions
The category ObLocomA 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 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
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
0Dn*
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*nG(KI) 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 (*).
fin théorie objective