1. Guerino Mazzola U & ETH Zürich guerino@mazzola.ch www.encyclospace.org Topos Theory for Musical Networks Topos Theory for Musical Networks

2. Birkhäuser 2002 1365 pages, hardcover incl. CD-ROM US English ISBN 3-7643-5731-2 Networks etc. —> PNM paper Mazzola + Andreatta

3. Bach‘s „Art of Fugue“ (1924) Eine kontrapunktische Form ist eine Menge von Mengen von Mengen (von Tönen). A contrapuntal form is a set of sets of sets (of tones). Wolfgang Graeser 1906-1928

4. Structures Processes Gestures IMPLEMENT ON SOFTWARE!

6. Need recursive (possibly circular) combination of constructions such as „sequences of sets of sets of curves of sets of chords“ etc. This leads to the theory of denotators: —> concept architecture —> uses mathematical topos theory —> generalizes XML —> have implementation in Java Need recursive (possibly circular) combination of constructions such as „sequences of sets of sets of curves of sets of chords“ etc. This leads to the theory of denotators: —> concept architecture —> uses mathematical topos theory —> generalizes XML —> have implementation in Java

7. AnchorNoteAnchorNote – – OnsetOnsetLoudnessLoudnessDurationDurationPitchPitch NoteNote STRG Ÿ– – DurationDuration OnsetOnset RestRest

8. MacroNoteMacroNote MacroNoteMacroNote SatellitesSatellites Ornaments Ornaments Schenker analysis Schenker analysis AnchorNoteAnchorNote – – OnsetOnsetLoudnessLoudnessDurationDurationPitchPitch NoteNote STRG Ÿ – – DurationDurationOnsetOnset RestRest

9. MakroNoteMakroNote MakroNoteMakroNote SatellitesSatellites AnchorNoteAnchorNote – – OnsetOnsetLoudnessLoudnessDurationDurationPitchPitch NoteNote STRG Ÿ – – DurationDurationOnsetOnset ResrResr conjunction cartesian product limit selection set powerset disjunction disjoint sum colimit representation algebraic space group, module, etc. PROBLEMS!

10. x F = EHLD (onset E, pitch H, loudness L, duration D, values in — ) E H D L F = PiMod 12 (pitch classes in Ÿ 12 )  p A = 0 x: 0  — 4 A = 0 p: 0  Ÿ 12

11. space F A@F Alexander Grothendieck a point is an affine map f „address“ A

12. A = Ÿ 11, F = PiMod 12 S: Ÿ 11  Ÿ 12, S  Ÿ 11 @ Ÿ 12 ª Ÿ 12 12 Ÿ 12 S 011 Dodecaphonic Series

13. space F Prize for parametrization addresses: Parametrized objects need parametric evaluation! Prize for parametrization addresses: Parametrized objects need parametric evaluation! A@F  B@F parametric evaluation = address change = functoriality! address A address B

14. Mod @ F: Mod  Sets space functors have all these properties Mod @ F: Mod  Sets space functors have all these properties Sets cartesian products X  Y disjoint sums X   Y powersets X Y characteristic maps  X  no „algebra“ Mod direct products A ≈ B etc. has „algebra“ no powersets no characteristic maps Mod @ is a Topos

15. What is a space functor F: Mod  Sets? 1. For every address module A in Mod, we have set A@F of „A-addressed points of F“. 2. For every (affine) module homomorphism f: B —> A, have set map F(f): A@F—> B@F 3. F(Id A )= Id A@F, F(g  f)=F(f)  F(g) Example: F= @M „space of points of M“ A@F = A@M What is a space functor F: Mod  Sets? 1. For every address module A in Mod, we have set A@F of „A-addressed points of F“. 2. For every (affine) module homomorphism f: B —> A, have set map F(f): A@F—> B@F 3. F(Id A )= Id A@F, F(g  f)=F(f)  F(g) Example: F= @M „space of points of M“ A@F = A@M MA B

16. Yoneda Lemma The map @: Mod  Mod @ is fully faithful. M  @M M@F ≈ Hom(@M,F) M@F ≈ Hom(@M,F) Yoneda Lemma The map @: Mod  Mod @ is fully faithful. M  @M M@F ≈ Hom(@M,F) M@F ≈ Hom(@M,F) Mod @ Const.Const. SetsSets spaces = functors 21 st century geometry is about functors @Mod@ModModMod @

17. The Topos of Music is mainly concerned with powerset type categories and their classification: „Local and global compositions“

18. MakroNoteMakroNote MakroNoteMakroNote SatellitesSatellites AnchorNoteAnchorNote – – OnsetOnsetLoudnessLoudnessDurationDurationPitchPitch NoteNote STRG Ÿ – – DurationDurationOnsetOnset ResrResr conjunction cartesian product limit selection set powerset disjunction disjoint sum colimit representation algebraic space group, module, etc.

19. Henry Klumpenhouwer: Deep Structure in K-net Analysis with Special Reference to Webern‘s Opus 16,4 J 1, J 2, J 3, J 4, J 5, J 6, J 7, J 8 J k  0@PiMod 12 J 1  J 2  J 3  J 8 class 19 J 4  J 7 class 22 J 5 class 28 J 6 class 31 I1I1I1I1 I2I2I2I2 I3I3I3I3 I4I4I4I4 I5I5I5I5 I6I6I6I6 I7I7I7I7 I8I8I8I8 1-dimensional nerve of global composition Atlas for global composition made of 8 local compositions I 1, I 2, I 3, I 4, I 5, I 6, I 7, I 8

20. Db} J1J1J1J1 J2J2J2J2 J3J3J3J3 J4J4J4J4 Klumpenhouwer (hyper)networks

21. Guerino Mazzola: Circle chords (Lectures on Group-theoretic Methods in Music Theory, U Zurich 1981, Gruppen und Kategorien in der Musik, Heldermann, Berlin 1985) Ch =  f  (x) = {x, f(x), f 2 (x),...} „orbit“ of x under monoid generated by affine map f Guerino Mazzola: Circle chords (Lectures on Group-theoretic Methods in Music Theory, U Zurich 1981, Gruppen und Kategorien in der Musik, Heldermann, Berlin 1985) Ch =  f  (x) = {x, f(x), f 2 (x),...} „orbit“ of x under monoid generated by affine map f

22. Db} J1J1J1J1 J2J2J2J2 J3J3J3J3 J4J4J4J4 Klumpenhouwer (hyper)networks

23. Ÿ 12 T4T4T4T4 T2T2T2T2 T 5.-1 T 11.-1 D3724     Ÿ 12 T4T4T4T4 T2T2T2T2 T 5.-1 T 11.-1 (3, 7, 2, 4)  lim( D )

24. Z i = Ÿ 12 f ij t  Z i @ Z j Fact: lim( D )  U U = (empty or) subgroup of ( Ÿ 12 ) n If f ** * = isomorphisms card (U) (= 0 or) divides 12 If f ** * = isomorphisms card (U) (= 0 or) divides 12Fact: lim( D )  U U = (empty or) subgroup of ( Ÿ 12 ) n If f ** * = isomorphisms card (U) (= 0 or) divides 12 If f ** * = isomorphisms card (U) (= 0 or) divides 12 lim( D ) ZiZiZiZi ZjZjZjZj ZlZlZlZl ZmZmZmZm f ij t D f il q f jm s f li p f jl k f ll r

25. Z i = module f ij t  Z i @ Z j diaffine morphism lim( D )  (empty or) module ZiZiZiZi ZjZjZjZj ZlZlZlZl ZmZmZmZm f ij t D f il q f jm s lim( D ) f li p f jl k f ll r

26. David Lewin: Analysis of Stockhausen‘s Klavierstück III (Musical Form and Transformation: 4 Analytic Essays, Yale U Press, 1993) B1B1B1B1 B2B2B2B2 B3B3B3B3 B4B4B4B4 B5B5B5B5 A1A1A1A1 A2A2A2A2 A3A3A3A3 A4A4A4A4 A5A5A5A5

27. Z i = P(M i ), M i = module f ij t = powerset map P( g ij t ) induced by diaffine morphisms g ij t M i  P(M i ):x ~> {x} special case of singletons! ZiZiZiZi ZjZjZjZj ZlZlZlZl ZmZmZmZm f ij t D f il q f jm s lim( D ) f li p f jl k f ll r (B 1, B 2, B 3, B 4, B 5 )  B1B1B1B1 B2B2B2B2

28. Z i = space functors over modules f ij * = natural transformation lim( D )  space functor ZiZiZiZi ZjZjZjZj ZlZlZlZl ZmZmZmZm f ij t D f il q f jm s lim( D ) f li p f jl k f ll r Is this necessary?

29. ZiZiZiZi ZjZjZjZj ZlZlZlZl ZmZmZmZm f ij t f il q f jm s lim( D ) f li p f jl k f ll r A@ Z l A@ Z i A@ Z j A@ Z m A@ f il q A@ f ll r A@ f li p A@ f ij t A@ f jm s A@ f jl k A@ lim( D ) 0@ Z l 0@ Z i 0@ Z j 0@ Z m 0@ f il q 0@ f ll r 0@ f li p 0@ f ij t 0@ f jm s 0@ f jl k 0@ lim( D ) 0@N = {(di)affine maps f: 0  N}  N N N N Replace module N by its space functor @N: A ~> A@N Replace powerset P(N) by the space functor P(N): A ~> A@P(N)= P(A@N) advantages? Avoid reinventing the whole theory for each address A Avoid reinventing the whole theory for each address A Have transition mechanisms of address change B  A Have transition mechanisms of address change B  A

30. Ÿ 11 @ Ÿ 12 T4T4T4T4 T2T2T2T2 T 5.-1 T 11.-1 D RUS V     r: Ÿ 11  Ÿ 11 Ÿ 11 @ Ÿ 12 S.r  Networks of Dodecaphonic Series => network of retrograde series

31. D = diagram of space functors F i P( F ) = one of these power spaces:  F 2 F Fin( F )... P( D ) = canonical power space diagram with above power spaces P( F i ) FiFiFiFi FjFjFjFj FlFlFlFl FmFmFmFm f ij t D f il q f jm s f li p f jl k f ll r F D :.Limit(P( D )) „Network space of D “

32. Ÿ 12 T 5.-1 T 9.-1 T0T0T0T0 T4T4T4T4 Ÿ 12 T 11.-1 T 3.-1 T0T0T0T0 T4T4T4T4 Ÿ 12 T4T4T4T4 T4T4T4T4 T0T0T0T0 T0T0T0T0 T2T2T2T2 T2T2T2T2 T4T4T4T4 T4T4T4T4

33. smsmsmsm sjsjsjsj slslslsl sisisisi FiFiFiFi FjFjFjFj FlFlFlFl FmFmFmFm f ij t D f il q f jm s f li p f jl k f ll r GiGiGiGi GjGjGjGj GlGlGlGl GmGmGmGm g ij t D*D*D*D* g il q g jm s g li p g jl k g ll r s FDFDFDFD FD*FD*FD*FD* FsFsFsFs

34. Summarizing recursive procedure: 1.Given are -> Category C of spaces -> Diagram scheme D. 2.Consider category Dia(D, C ) of diagrams D over D with values in C, plus natural transformations S: D  D *. 3.Define category C (1) =D  P C of limits F D of power diagrams P( D ) of one of the power types P =  D, 2 D, Fin( D ),..., together with morphisms F S : F D  F D *. 4.Restart the procedure from 1. with the new category C (1) and a new diagram scheme D (1), defining the category C (2) =D (1)  P (1) C (1), and so on... Problem: ¿Is there an operation  P on diagram schemes with ( D (1)  P D)  P C =D (1)  P (D  P C )

35. global compositions chartsatlases

36. B1B1B1B1 B2B2B2B2 B3B3B3B3 B4B4B4B4 B5B5B5B5 A1A1A1A1 A2A2A2A2 A3A3A3A3 A4A4A4A4 A5A5A5A5 A1A1A1A1 A2A2A2A2 A3A3A3A3 A4A4A4A4 A5A5A5A5 2-dimensional nerve OnPi( Ÿ 2 ) PiMod 12 Morphism f of global compositions 3-dimensional nerve B3B3B3B3 B5B5B5B5 B1B1B1B1 B2B2B2B2 B4B4B4B4 nerve( f )

37. I IVVIIIIIVIVII

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

39. n/16 a b c d e 0234681012

40. nerve of the covering {a, b, c, d, e} b e d a c n/16 a b c d e 0234681012

41. K local iso  C i  K i  K t  C t K it  K ti  KIKIKIKI

42. 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 (weighted) nerve n* of the covering atlas I. A  n* KIKIKIKI res

43. 1234 1265 5 6 34 1 2 3 4 6 5 KIKIKIKI 6 5 234 1 0  n* res a d b c

44. A@R 1 2 34 6 5 KiKiKiKi 6 534 1 iiii 2  (K i )  res   (  i ) A@R (Ki)(Ki)(Ki)(Ki)  (K i ) = module of affine functions on K i

45. 6 5 3 4 1 A@R 2  (K i )  res   (  i ) R A@R iiii  (K i  K j )  res   (  i   i )

46. n  ≈ n  (K I ) = nerve of K I ,  n  ,  n  ,  n  ,  n    A  n* (  ) —»  A  n* (  ) homomorphism of R-modules  A  n* (  ) »  A  n* (  )  K I (  )  res »  K I (  )  res  K I (  )  res »  K I (  )  res  res*n  (K I )  n  ( A  n* )

47. D (n  ) = diagram of function spaces F  P(F  ) = SubMod(F  ) P( D (n  )) = canonical power space diagram with the above power spaces P(F  ) FFFF FFFF FFFF F f(/)f(/)f(/)f(/) D (n  ) f(/)f(/)f(/)f(/) f (  / ) f( /)f( /)f( /)f( /) F D (n  ) :.Limit(P( D (n  ) )) „Network space of D (n  ) “ K I -Functions:A@ F D (n  ) (res*n  (K I ))

48.  network isomorphisms vivivivi vjvjvjvj vlvlvlvl vmvmvmvm vivivivi vjvjvjvj vlvlvlvl vmvmvmvm vivivivi vjvjvjvj vlvlvlvl vivivivi vjvjvjvj vlvlvlvl cartesian chart XiXiXiXi XjXjXjXj XlXlXlXl XiXiXiXi XjXjXjXj XlXlXlXl YiYiYiYi YjYjYjYj YlYlYlYl YiYiYiYi YjYjYjYj YlYlYlYl YmYmYmYm

49. Theorem: There is functor |?|: GlobLim red A —> GloCom A 1 2 3 4 6 5K 1 2 3 4 6 5|K|~> Corollary: There are proper global networks (limits)

50. Structures Processes Gestures ! ? ?

51. Remark on automorphism groups on Ÿ 12 Aut( Ÿ 12 ) = T Ÿ 12  | Ÿ 12 *  Ÿ 12  | Ÿ 12 * T/I group = subgroup of Aut( Ÿ 12 ) T/I group = T Ÿ 12  | (±1)  Ÿ 12  | Ÿ 2 Int: Aut( Ÿ 12 )  Aut(T/I group)  Ÿ 12  | Ÿ 12 * Ker(Int) =  Ÿ 2 (T u ) = T nu, n  Ÿ 12 * ={1, 5, 7, 11} (-1) = T j (-1) Int(T s (n)) = Im(Int)  2 Ÿ 12  | Ÿ 12 * (T u ) = T nu, n  Ÿ 12 * ={1, 5, 7, 11} (-1) = T j (-1) Int(T s (n)) = Im(Int)  2 Ÿ 12  | Ÿ 12 *

52. Remark on isographies on Ÿ 12 An isography between two diagrams D, D * : D  T/I-group over the same diagram scheme D with values in category T/I-group with the single object Ÿ 12 the single object Ÿ 12 morphisms = T/I-group isomorphisms morphisms = T/I-group isomorphisms is a group automorphism f (= a functorial automorphism) f: T/I-group  T/I-group such that D * = f  D DT/I-group T/I-group D*D*D*D*Df ~

53. Unfortunately, an isography f between two diagrams D, D * : D  T/I-group does not tell anything about the underlying limits! Example: <2q+1,1> x = 2q+1 -x x = 2(2q+1) -x lim( D ) = Ø lim( D *)  Ø However, an isography with, stemming from conjugation, preserves limits, i.e., it is a natural transformation! Ÿ 12 T 2q+1 (-1) T 2q+1 (-1)D Ÿ 12 T 2(2q+1) (-1) T 2(2q+1) (-1) D*D*D*D*