Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch www.encyclospace.org

1. Mathematical Music Theory — Status Quo 2000 Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch www.encyclospace.org

2. Time Table • The Concept Framework • Global Classification • Models and Methods • Towards Grand Unification Contents

3. 1978 1980 1981 1984 1985 1986 1988 1990 1992 1994 1995 1996 1998 1999 2000 2001 Music Theory Software Grants Status quo Kelvin Null Akroasis Gruppentheore-tische Methode in der Musik M(2,Z)\Z2 Karajan Time Gruppen und Kategorien in der Musik Depth-EEG for Consonances and Dissonances Presto® Synthesis Geometrie der Töne Immaculate Concept RUBATO Project Morphologie abendländischer Harmonik Kuriose Geschichte RUBATO® NeXT KiT-MaMuTh Project Mac OS X Kunst der Fuge Topos of Music

4. translation dilinear • Mod = category of modules + diaffine morphisms: • A = R-module, B = S-module • Dilin(A,B) = (l,f) f:A ® B additive, l:R ® S ring homomorphism f(r.a) = l(r).f(a) • eb(x) = b+x; translation on B • A@B = eB.Dilin(A,B) Concepts eb.f: A ® B ® B

5. Topos of presheaves over Mod Mod@ = {F: Mod ®Sets, contravariant} Example: representable presheaf @B: @B(A) = A@B F(A) =: A@F A = address Concepts Yoneda Lemma The functor @: Mod®Mod@ is fully faithfull. B ~> @B

6. K Í B B Concepts • Database Management Systems • require recursively stable object types! • k Î B • K Î 2B no module! • Need more general spaces F B @ 0Ÿ@B K Í A@B K Í 0Ÿ@B • A = Ÿn: sequences (b0,b1,…,bn) • A = B: self-addressed tones • Need general addresses A F = W@B A = 0Ÿ KÎA @F F = presheaf over Mod F = @B

7. >® Functor(F) Frame(√) F = Form name one of four „space types“ a diagramn √ in Mod@ Forms a monomorphism in Mod@id: Functor(F) >® Frame(√) Concepts • Frame(√)-space for type • simple √ = Æ~> @B simple(√) =@B • limit √ = Form-Name-Diagram ® Mod@ • limit(√) = lim(Form-Name-Diagram ® Mod@) • colimit √ = Form-Name-Diagram ® Mod@ • colimit(√) = colim(Form-Name-Diagram ® Mod@) • power √ = Form-Name F ~> Functor(F) • power(√) =WFunctor(F)

8. A address A K >® Functor(F) Frame(√) Denotators D = denotator name Concepts KÎA @ Functor(F) „A-valued point“ Form F

9. Satellites AnchorNote MakroNote Onset Pitch Loudness Duration STRG Ÿ – – MakroNote • Ornaments • Schenker Analysis Concepts

10. Java Classes for Modules, Forms, and Denotators L L S S RUBATO® Concepts

11. Fr F2 x3 xn F1 x2 x1 Galois Theory Form Theory Defining equation Defining diagram Concepts fS(X) = 0 id √(F) Field S Form System Mariana Montiel Hernandez, UNAM

12. objects a = affine morphism f, h = natural transformations morphisms specify „address change“ a Category Loc of local compositions Type = PowerF ~> Functor(F) = G Classification local composition K Î A@WG K Í @A´G generalizes K Í A@G „objective“ local compositions K Í @A´G @a´h f/a L Í @B´H

13. Embedding functor Trace functor ObLoc Loc Classification ObLocA LocA • Theorem • Loc is finitely complete (while ObLoc is not!) • On ObLocA andLocA Embedding and Trace • are an adjoint pair: • ObLocA(Embedding(K),L) @ LocA(K,Trace(L))

14. K A@Gi◊ Ki @A´Gi◊ Ki KtÍ A@Gt KtÍ @A´Gt ◊ ◊ Kit Kti local isomorphism/A Classification

15. Category Gl of global compositions Objects: KI = functor K which is covered by a finite atlas I = (Ki) of local compositions in LocA at address A Morphisms: KI at address A LJ at address B f/a: KI ® LJ f = natural transformation, a: A ® B = address change f induces local morphisms fij/a on the charts Classification

16. Have Grothendieck topology Cov(Gl) on Gl Covering families (fi/ai: KIi ® LJ)i are finite, generating families. Classification • Theorem • Cov(Gl) is subcanonical • The presheaf GF:KI ~> GF(KI) of global affine functions is a sheaf.

17. res KI ADn* Have universal construction of a „resolution of KI“ res:ADn*® KI It is determined only by the KIaddress A and the nerve n* of the covering atlas I. Classification

18. Theorem (global addressed geometric classification) • Let A = locally free of finie rank over commutative ring R • Consider the objective global compositions KI at A with (*): • locally free chart modules R.Ki • the function modules GF(Ki) are projective • (i) Then KI can be reconstructed from the coefficient system of • retracted functions • res*F(KI) Í F(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

19. Applications of classification: • String Quartet Theory: Why four strings? • Composition: Generic compositional material • Performance Theory: Why deformation? Classification

20. Mazzola Mazzola/Noll Noll Ferretti Nestke Noll • There are models for these musicological topics • Tonal modulation in well-tempered and just intonation and general scales • Classical Fuxian counterpoint rules • Harmonic function theory • String quartet theory • Performance theory • Melody and motive theory • Metrical and rhythmical structures • Canons • Large forms (e.g. sonata scheme) • Enharmonic identification Models

21. What is a mathematical model of a musical phenomenon? Precise Concept Framework Instance specification Formal process restatement Proof of structure theorems Mathematics Music • Field of Concepts • Material Selection • Process Type • Grown rules for process • construction and • analysis Models Deduction of rules from structure theorems Why this material, these rules, relations? Generalization! Anthropomorphic Principle!

22. Arnold Schönberg: Harmonielehre (1911) Old Tonality Neutral Degrees (IC,VIC) Modulation Degrees (IIF, IVF, VIIF) New Tonality Cadence Degrees (IIF & VF) Models • What is the considered set of tonalities? • What is a degree? • What is a cadence? • What is the modulation mechanism? • How do these structures determine the modulation degrees?

23. II III IV V VI VII I Models

24. g graviton gluon W+ electromagnetic force strong force weak force gravitation quantum = set of pitch classes = M S(3) T(3) force = symmetry between S(3) and T(3) k k Models

25. IVC IIEb VIIEb M(3) IIC VEb VIIC IIIEb VC Eb(3) C(3) Models

26. Ÿ12[e] ƒ1 e e.2.5 Ÿ12 @Ÿ3 x Ÿ4 Unification K = Ÿ12 +e.{0,3,4,7,8,9} = consonances D = Ÿ12 +e.{1,2,5,6,10,11} = dissonances

27. Rules of CounterpointFollowing J.J. Fux C/D Symmetry inHuman Depth-EEG Extension to ExoticInterval Dichotomies Unification

28. 0 Ÿ12 @Ÿ12 ◊ ◊ X = { } Trans(X,X) Ÿ12@ 0@Ÿ12

29. ƒ1 Z12[e] Z12 Trans(D, T) = Trans(K,K)|ƒe Z12 [e] @ Z12 [e] Z12 @ Z12 ƒe D = C-dominant triad T = C-tonic triad K

30. The Topos of Music Geometric Logic of Concepts, Theory, and Performance in collaboration with Moreno Andreatta, Jan Beran, Chantal Buteau, Karlheinz Essl, Roberto Ferretti, Anja Fleischer, Harald Fripertinger, Jörg Garbers, Stefan Göller, Werner Hemmert, Mariana Montiel, Andreas Nestke, Thomas Noll, Joachim Stange-Elbe, Oliver Zahorka www.encylospace.org