Guerino Mazzola U & ETH Zürich Internet Institute for Music Science Concepts locaux et globaux. Première partie: Théorie ‚objective‘ Concepts locaux et globaux. Première partie: Théorie ‚objective‘

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

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

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 -> ??

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 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   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...

enumeration x^144 + x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^72 + …. … x x x x 2 + x + 1 = cycle index polynomial x 72 ª x 72 average # of stars in a galaxis =

enumeration From generalizations of the main theorem by N.G. de Bruijn, we have (for example) the following enumerations: k = T TI GA( Ÿ 12 ) k# of orbits of (k,12)-series k# of orbits of (k,12)-series (dodecaphonic)

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 = e B.Lin(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 = 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

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 K B L f

exampoles retrograde including duration reflection transvection

 = 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

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

just theory f cg d a eb F = e q tonal inversion

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

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

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

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 ).

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)

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

motive classes Classes of 3-element motives M  ( Ÿ 12 ) generic 0:05-0:33

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

scales Space Ÿ 12 of pitch classes in 12-tempered tuning 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

I IVVIIIIIVIVII triadic interpretation

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

meters n/16 a b c d e

nerves b nerve of the covering {a,b,c,d,e} x dominates y iff simplex(y)  simplex(x) e c a d

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

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

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

motivic principle Motivic Zig-Zag in op.106    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) (15) (10) (11) (19) (19) (20) (2) (16)

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

kernel Kernel of Development        ‘‘‘‘  U2U2U2U2 A B C D E F A‘ B‘ C‘ D‘ E‘ F‘                                                                          

dbdbdbdb Kernel Matrix A B C D E F dbdbdbdb f a DrDrDrDr DlDlDlDl A‘ B‘ C‘ D‘ E‘ F‘ kernel

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

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

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

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

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

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

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

Classify! The category ObLocom A of local objective A-addressed compositions has as objects the couples (K, of sets K of affine morphisms in and as morphisms f: (K,  L, set maps f: K  L which are naturally induced by affine morphism F in 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

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

non-interpretable a d b c KIKIKIKI  n* res

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 (*).

fin théorie objective fin théorie objective