1. Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch www.encyclospace.org Intégration de la Set Theory américaine (AST) dans la théorie moderne européenne

2. Contents Introduction Contour Spaces Pitch Spaces Pitch Class Spaces Introduction Contour Spaces Pitch Spaces Pitch Class Spaces Robert D. Morris: Composition with Pitch-Classes Yale U Press, New Haven 1987

3. Introduction + No mumbo-jumbo theory mixture of cognitive science and psychology î $ software for AST, all on Mac: CMAP (C++) by Harris & Brinkman (& Castine) SET-SLAVE (Common LISP) Amuedo OpenMusic (Common LISP) Agon & Assayag; Andreatta. + Systematic approach (although very restricted domain): Rigorous definitions, complete tables. – Aboundance of abbreviations such as TTO, SC, ICV, pcs, ics,... – Trival „home-made“ mathematics: Elementary group theory and combinatorics No standard concepts, such as normal subgroup, etc. – No attempts to generalize results, e.g. from n = 12 to general n in Pitch classes Ÿ n.

4. Introduction Alteration Theorem of Robert M. Mason Enumeration of Synthetic Music Scales (...) J. of Music Theory 14, 1970 In cyclic pitch Ÿ 12, if E, F are two scales with card(E) = Card(F) = m ≥ 7, then either we find 2 # and 3 Ÿ 2 # and 3 Ÿ or 3 # and 2 Ÿ such that F is generated from E by some alterations in this range. „Proof“: List explicitely all possible cases of these so-called Busoni scales. In cyclic pitch Ÿ 12, if E, F are two scales with card(E) = Card(F) = m ≥ 7, then either we find 2 # and 3 Ÿ 2 # and 3 Ÿ or 3 # and 2 Ÿ such that F is generated from E by some alterations in this range. „Proof“: List explicitely all possible cases of these so-called Busoni scales.

5. Introduction Alteration Theorem (Mazzola) Gruppen und Kategorien in der Musik Heldermann, Berlin 1985 In cyclic pitch Ÿ n, if E, F are two scales with card(E) = Card(F) = m, then we find k # and t Ÿ with k+t = n-m k # and t Ÿ with k+t = n-m such that F is generated from E by some alterations in this range. F may even be generated by a sequence of elementary alterations, i.e., by shifts of one single pitch by one unit into an empty slot. Proof: By induction on n-m, 3 pages. In cyclic pitch Ÿ n, if E, F are two scales with card(E) = Card(F) = m, then we find k # and t Ÿ with k+t = n-m k # and t Ÿ with k+t = n-m such that F is generated from E by some alterations in this range. F may even be generated by a sequence of elementary alterations, i.e., by shifts of one single pitch by one unit into an empty slot. Proof: By induction on n-m, 3 pages.

6. FormsForms F = form name F = form name an identifier monomorphism in Mod @ id: Functor(F) >  Frame( √ ) an identifier monomorphism in Mod @ id: Functor(F) >  Frame( √ ) Frame( √ ) >>>> Functor(F) one of five „space“ types one of five „space“ types a name diagram √ in Mod @ a name diagram √ in Mod @ F:id.type( √ ) Introduction

7. Frame( √ )-space for type: synonyme √ = „G“  Functor(G) synonyme() =Functor(G) synonyme( √ ) = Functor(G) renamingrenaming simple √ = „“  @B simple() =@B  simple( √ ) = @B representationrepresentation limit √ = name diagram  Mod @ limit() = lim(n. diagram  Mod @ ) limit( √ ) = lim(n. diagram  Mod @ ) conjunctionconjunction colimit √ = name diagram  Mod @ colimit() = colim(n. diagram  Mod @ ) colimit( √ ) = colim(n. diagram  Mod @ ) disjunctiondisjunction power √ = „G“  Functor(G) power() =  Functor(G) power( √ ) =  Functor(G) collectioncollection Introduction

8. Frame( √ ) >> Functor(F) Form F DenotatorsDenotators K  @ K  @Functor(F) „A-valued point“ D = denotator name A address A K D:A@F(K)D:A@F(K) Introduction

9. PointSpace 0  PointSpace n „Linear“: Ÿ „Cyclic“: Ÿ n D:0@ PointSpace n (d) D: Ÿ k-1 @ PointSpace n (d 0, d 1,... d k-1 ) single „points“ serial „motif“ PointSpace n :id.Syn(PiMod n ) PiMod n :id.Simple( Ÿ n ) Introduction

10. X:0@SetSpace n (d,e,f,g,...) S: Ÿ k-1 @SetSpace n (s 0, s 1,... s k-1 ) Sets of zero-addressed „points“ Sets of Ÿ k-1 -addressed serial motives SetSpace n :id.Power(PointSpace n ) PointSpace n :id.Syn(PiMod n ) Introduction

11. C-Spaces A c-space is a space which describes contour parameters, typically of pitch-related quality X. It has no precise relation to the physical/symbolic parameters, it is just a kind of integer quantification of abstract ordering relations. c-space(X):id.Syn(PiMod 0 ) c-setspace(X):id.Power(c-space(X)) I n :0@ c-setspace(X) (0,1,2,... n-1) D:0@ c-space(X)( d), d  I n cp = contour pitch of order n 0 n-1

12. C-Spaces S:0@c-setspace(X)(s,t,u,...) S  I n cpset = contour pitch set of order n contour for c-space(X) length(C) = k CONT n,k (X) contour for c-space(X) length(C) = k CONT n,k (X) C: Ÿ k-1 @c-space(s 0, s 1,... s k-1 ), s i  I n 0 k-1 miniature global composition

13. C-Spaces K 4  CONT n,k (X)  CONT n,k (X) C: Ÿ k-1 @c-space(s 0, s 1,... s k-1 ) K 4 ={Id, R, I, IR} I R IR Id Contour Symmetries

14. C-Spaces K 4 ={Id, R, I, IR} COM: CONT n,k (X)  Ã k  k ( Ÿ ) COM(C) i,j = sign(C j-1 -C i-1 ) COM: CONT n,k (X)  Ã k  k ( Ÿ ) COM(C) i,j = sign(C j-1 -C i-1 ) COM(C) COM(C) = Comparison Matrix K 4  Ã k  k ( Ÿ )  Ã k  k ( Ÿ ) I.M = M t = transposed matrix R.M = M r = 180 degree rotated matrix IR.M = M k = codiagonal transposed matrix K 4  Ã k  k ( Ÿ )  Ã k  k ( Ÿ ) I.M = M t = transposed matrix R.M = M r = 180 degree rotated matrix IR.M = M k = codiagonal transposed matrix CONT n,k (X) Ã k  k ( Ÿ ) COM K 4 \CONT n,k (X) K 4 \ Ã k  k ( Ÿ ) K4\K4\K4\K4\ K4\K4\K4\K4\ Contour classes Similarity

15. C-Spaces M,N  Ã k  k ( Ÿ ) skew-symmetric matrices SIM(M,N) = card{(i,j)|i<j,M i,j = N i,j }.2/k(k-1) d COM (M,N) = (2/k)  { ∑ i<j (M i,j - N i,j ) 2 } Pseudo-metric on set CONT n,k (X) of contours of length k The order n is not relevant here, forget it, take union CONT k (X) of all CONT n,k (X). M,N  Ã k  k ( Ÿ ) skew-symmetric matrices SIM(M,N) = card{(i,j)|i<j,M i,j = N i,j }.2/k(k-1) d COM (M,N) = (2/k)  { ∑ i<j (M i,j - N i,j ) 2 } Pseudo-metric on set CONT n,k (X) of contours of length k The order n is not relevant here, forget it, take union CONT k (X) of all CONT n,k (X). Distances and Similarity of Contours Apply this to define a topology on the set MOT all motives!

16. M M H E L C (M) M EH E H C-Spaces

17. COM-shape type. General shape-type theory: topologies on motive spaces, topologies on motive spaces, group actions, group actions, orbit distances, etc. orbit distances, etc. developed by Chantal Buteau & G.M. See ToM or Musicae Scientiae Vol. IV, No.2, 2000 COM-shape type. General shape-type theory: topologies on motive spaces, topologies on motive spaces, group actions, group actions, orbit distances, etc. orbit distances, etc. developed by Chantal Buteau & G.M. See ToM or Musicae Scientiae Vol. IV, No.2, 2000 com k : MOT k  CONT k (X)  Ã k  k ( Ÿ ) skew com k (M) = COM(C(M)) k = 2,3,... com k (M) = COM(C(M)) k = 2,3,... com k : MOT k  CONT k (X)  Ã k  k ( Ÿ ) skew com k (M) = COM(C(M)) k = 2,3,... com k (M) = COM(C(M)) k = 2,3,... Abstract motive space

18. C-Spaces Open Balls around Motives M Let  > 0 and M  MOT. U  (M) = {N  MOT | ex. M*  N, card(M*) = card(M), d COM (M, M*) <  }

19. Ÿ s  Ÿ t s ≤ t, define morphism f: Ÿ s  Ÿ t e 0 ~> e i(0) e 1 ~> e i(1)................. e s ~> e i(s) S1S1 SkSk Ÿ 12 e0e0 e1e1 eses ŸsŸsŸsŸs S 1.f S k.f Ÿ 12 f@K^ Functorial Locs Address change!

20. C-Spaces Theorem: The system of open balls {U  (M) | M  MOT,  > 0 } is the base of a topology T com on MOT, i.e., for any two such balls U  (M), U  ‘ (M‘), we have U  (M)  U  ‘ (M‘) =  U  (i) (M i ) Theorem: The system of open balls {U  (M) | M  MOT,  > 0 } is the base of a topology T com on MOT, i.e., for any two such balls U  (M), U  ‘ (M‘), we have U  (M)  U  ‘ (M‘) =  U  (i) (M i ) This is due to the inheritance property of com shape type: „If two motives are similar, then so are their submotives.“ Such a property is not true for Morris‘ INT shape type of successive intervals.

21. C-Spaces Theory Implemented in RUBATO‘s MeloRubette Theory of motivic weights, sheaves of weight functions on motivic topologies. Theory of motivic weights, sheaves of weight functions on motivic topologies.

22. P-Spaces x:0@[M 1/n ](x) PiMod 0 :id.Simple( Ÿ ) [M 1/n ]:id.Syn(PiMod 0 ) pitch(x) = F 0.M x/n M = 2, n = 12 p-spacep-space pitchpitch S: Ÿ k-1 @[M 1/n ](S 0, S 1,... S k-1 ) pseg(ment)pseg(ment)

23. P-Spaces PiMod 0 :id.Simple( Ÿ ) [M 1/n ]:id.Syn(PiMod 0 ) [M 1/n ]sets:id.Power([M 1/n ]) X:0@[M 1/n ]sets (a,b,c,d,...) Unordered pitch set „pset“ X: Ÿ k-1 @ [M 1/n ]sets (a,b,c,d,...) X: Ÿ k-1 @ [M 1/n ]sets (a,b,c,d,...) Set of psegs SEG k [M 1/n ]: Ÿ k-1 @ [M 1/n ]sets (all segments) abelian group

24. P-Spaces [M 1/n ]:id.Syn(PiMod 0 ) Ordered p-space interval ip  a,b  = b-a Unordered p-space interval ip{a,b} =  ip  a,b   For unordered pset P: 0@ [M 1/n ]sets (u,v,x,y,...) Interval content: int(P) = (ip{a,b}) (a,b)  P  P Ordered p-space interval ip  a,b  = b-a Unordered p-space interval ip{a,b} =  ip  a,b   For unordered pset P: 0@ [M 1/n ]sets (u,v,x,y,...) Interval content: int(P) = (ip{a,b}) (a,b)  P  P x:0@[M 1/n ](x) Pitch Intervals

25. P-Spaces Give a pseg S: Ÿ k-1 @[M 1/n ](S 0, S 1,... S k-1 ) INT(S): Ÿ k-2 @[M 1/n ](S 1 -S 0, S 2 -S 1,... S k-1 -S k-2 ) Interval succession of pseg S C(S): Ÿ k @[M 1/n ](S 0, S 1,... S k-1, S 0 ) „Cyclic extension“ of pseg S INT: SEG k [M 1/n ]  SEG k-1 [M 1/n ] C: SEG k [M 1/n ]  SEG k+1 [M 1/n ] CINT=INT  C: SEG k [M 1/n ]  SEG k [M 1/n ] Cyclic intervall succession

26. P-Spaces  : SEG k [M 1/n ]  SEG k [M 1/n ] PermutationsPermutations  (S): Ÿ k-1 @[M 1/n ](S  (0),... S  (i),...S  (k-1) )  : SEG k [M 1/n ]  SEG k [M 1/n ] Canonical operators  (S): Ÿ k-1 @[M 1/n ](  (S 0 ),...  (S i ),...  (S k-1 )) Rotation r t Retrogression R Rotation r t Retrogression R Ÿ k-1  Ÿ k-1  Ÿ  Ÿ S SEG k [M 1/n ] Sym k Ÿ@ŸŸ@ŸŸ@ŸŸ@Ÿ Address change! Functorial!   Sym k  =T n M m = e n.m  Ÿ @ Ÿ

27. P-Spaces Proposition: On SEG k [M 1/n ], the actions of canonical operators and permutations commute. On SEG k [M 1/n ], the actions of canonical operators and permutations commute. INT is invariant under translations: INT  T n = INT INT is invariant under translations: INT  T n = INT INT commutes with multiplications:INT  M n = M n  INT INT commutes with multiplications:INT  M n = M n  INT INT anticommutes with retrogression: INT  R = —R  INT INT anticommutes with retrogression: INT  R = —R  INT Remark: such results, also valid for modular pitch, are used to classify n-phonic series and all-interval series by Harald Fripertinger, e.g. n = 12: G 12 =  retrogression, all affine automorphisms of Ÿ 12  Card (G 12 \ALLSER 12 ) = 267

28. s: Ÿ k-1 @ pc-space n (s 0, s 1,... s k-1 ) PC-Spaces PiMod n :id.Simple( Ÿ n ) pc-space n :id.Syn(PiMod n ) x:0@ pc-space n (x) Pitch class segment pcseg Pitch class pc pc-spaces

29. U n :0@ pc-setspace n (all pcs...) PC-Spaces pc-space n :id.Syn(PiMod n ) X:0@ pc-setspace n (x,y,z,...) (Unordered) pitch class set pcset The aggregate pc-setspace n :id.Power(pc-space n )

30. PC-Spaces PiMod n :id.Simple( Ÿ n ) pc-space n :id.Syn(PiMod n ) A@pc-space n Ÿn@ŸnŸn@ŸnŸn@ŸnŸn@Ÿn For any address A have canonical actions A  A  Ÿn  ŸnA  A  Ÿn  ŸnA  A  Ÿn  ŸnA  A  Ÿn  ŸnS A@AA@AA@AA@A TTO  =T n M m = e n.m m=1,5,7,11; 48 elements Twelve Tone Operators TTO  =T n M m = e n.m m=1,5,7,11; 48 elements Twelve Tone Operators n = 12 1  e Ÿ 12   Ÿ 12   1 1  e Ÿ 12  TTO  Ÿ 12   1 „Split exact sequence“ Monoid and Group Actions

31. PC-Spaces For a pcset X:0@ pc-setspace 12 (x,y,z,...) have orbit.X = (G-)set class SC have orbit G.X = (G-)set class SC For a pcset X:0@ pc-setspace 12 (x,y,z,...) have orbit.X = (G-)set class SC have orbit G.X = (G-)set class SC A@pc-space n G A@pc-setspace n G induces canonical action G = subgroup of TTO, e.g. G = {tranpositions T n }, {transpositions and inversions T n,T n.I} Classification for more general situations (incl. general addresses): see ToM...!

32. 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... PC-Spaces

33. Pitch Class Intervals Ordered pc interval i  a,b  = b-a Unordered pc interval i{a,b} = min(i  a,b , i  b,a  ) ic  k  = {(x,y)  U 12  i  x,y  = k} „i  x,y  is ic  k  “ ic  k  = {(x,y)  U 12  i  x,y  = k} „i  x,y  is ic  k  “ ic{k} = {(x,y)  U 12  i{x,y} = k} „i{x,y} is ic{k}“ ic{k} = {(x,y)  U 12  i{x,y} = k} „i{x,y} is ic{k}“ Ordered pc interval i  a,b  = b-a Unordered pc interval i{a,b} = min(i  a,b , i  b,a  ) ic  k  = {(x,y)  U 12  i  x,y  = k} „i  x,y  is ic  k  “ ic  k  = {(x,y)  U 12  i  x,y  = k} „i  x,y  is ic  k  “ ic{k} = {(x,y)  U 12  i{x,y} = k} „i{x,y} is ic{k}“ ic{k} = {(x,y)  U 12  i{x,y} = k} „i{x,y} is ic{k}“ x:0@ pc-space n (x) pc-space n :id.Syn(PiMod n )

34. PC-Spaces Ordered pc interval i  a,b  = b-a Unordered pc interval i{a,b} = min(i  a,b , i  b,a  ) ic  k  = {(x,y)  U 12  i  x,y  = k} ic  k  = {(x,y)  U 12  i  x,y  = k} ic{k} = {(x,y)  U 12  i{x,y} = k} ic{k} = {(x,y)  U 12  i{x,y} = k} Ordered pc interval i  a,b  = b-a Unordered pc interval i{a,b} = min(i  a,b , i  b,a  ) ic  k  = {(x,y)  U 12  i  x,y  = k} ic  k  = {(x,y)  U 12  i  x,y  = k} ic{k} = {(x,y)  U 12  i{x,y} = k} ic{k} = {(x,y)  U 12  i{x,y} = k} Let A,B:0@ pc-setspace 12 (...) be two pcsets. (Ordered) multiplicity Mult  (A,B,k) = card(A  B  ic  k  ) (Unordered) multiplicity Mult {} (A,B,k) = card(A  B  ic{k}) Clear: Mult {} (A,B,k) = 0 for k > 6 Interval vector IV(A,B) = (Mult  (A,B,k)) k= 0,1,2,...11 Interval class content vector IVC(A,B) = (Mult {} (A,B,k)) k= 0,1,2,...6 Let A,B:0@ pc-setspace 12 (...) be two pcsets. (Ordered) multiplicity Mult  (A,B,k) = card(A  B  ic  k  ) (Unordered) multiplicity Mult {} (A,B,k) = card(A  B  ic{k}) Clear: Mult {} (A,B,k) = 0 for k > 6 Interval vector IV(A,B) = (Mult  (A,B,k)) k= 0,1,2,...11 Interval class content vector IVC(A,B) = (Mult {} (A,B,k)) k= 0,1,2,...6 Interval Vectors

35. PC-Spaces Complement Theorem (for n=12): Let A,B:0@ pc-setspace n (...) be two pcsets and A‘, B‘ their complements in the aggregate U n. Then we have Mult  (A‘, B‘,k) = Mult  (A,B,k) + n - (card(A)+card(B)) (Hexachord Theorem) In particular, if n=2m, and if card(A) = m, a „m-chord“, then Mult  (A‘, A‘,k) = Mult  (A,A,k) Complement Theorem (for n=12): Let A,B:0@ pc-setspace n (...) be two pcsets and A‘, B‘ their complements in the aggregate U n. Then we have Mult  (A‘, B‘,k) = Mult  (A,B,k) + n - (card(A)+card(B)) (Hexachord Theorem) In particular, if n=2m, and if card(A) = m, a „m-chord“, then Mult  (A‘, A‘,k) = Mult  (A,A,k) Lemma: Mult  (B,A,k) = Mult  (A,B,-k) (i)Mult  (B,A,k) = Mult  (A,B,-k) (ii)Mult  (A,B,k) + Mult  (A‘,B,k) = card(B) Proof: Observe that Mult  (A,B,k) = card(e k (A)  B)!

36. PC-Spaces x  TTO  Ÿ 12 @ Ÿ 12 corresponds to x: Ÿ 12 @ pc-space 12 (x) X: Ÿ 12 @ pc-setspace 12 (x,y,z,...) = a set of TTO operators TTO Sets and Strings „Strings of TTO operators“ would be serial motives like above, i.e., ( Ÿ k-1 -addressed) sequences x 0, x 1,...,x k-1 in Ÿ 12 @ pc-space 12 But we have already have „wasted“ the address by Ÿ 12...

37. PC-Spaces If A is an address (a R-module), and F is a form, we have the new form A † @F, the address A killing form, with B@A † @F =(B  A)@F B  A is the affine tensor product. Since 0 R  B =B, we have B  A is the affine tensor product. Since 0 R  B =B, we have 0 R @A † @F = A@F Take Ÿ 12 † @pc-space 12. Then 0 Ÿ @ Ÿ 12 † @pc-space 12 = Ÿ 12 @pc-space 12 0 Ÿ @ Ÿ 12 † @pc-space 12 = Ÿ 12 @pc-space 12 This implies Ÿ k-1 @ Ÿ 12 † @pc-space 12 = ( Ÿ 12 @pc-space 12 ) k A string of TTO operators is a Ÿ k-1 -addressed denotator: Ÿ k-1 @ Ÿ 12 † @pc-space 12 = ( Ÿ 12 @pc-space 12 ) k A string of TTO operators is a Ÿ k-1 -addressed denotator: Ÿ k-1 @ Ÿ 12 † @pc-space 12 = ( Ÿ 12 @pc-space 12 ) k

38. PC-Spaces Two-dimensional arrays = abstract score simulation Rows = voices Columns = measure No duration specified... Two-dimensional arrays = abstract score simulation Rows = voices Columns = measure No duration specified... 45A9 7 design principles: coherence closure saturation hierarchy heterarchy design principles: coherence closure saturation hierarchy heterarchy AST does not have higher-dimensional modules... or global compositions

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