Guerino Mazzola U & ETH Zürich Internet Institute for Music Science Mathematical Music Theory — Status Quo 2000 Mathematical Music Theory — Status Quo 2000

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

Status quo TimeTheoryMusicGrants Kelvin Null Akroasis ImmaculateConcept SynthesisSoftware Presto ® RUBATO ® NeXT Mac OS X M(2,Z)\Z 2 Karajan Gruppentheore- tische Methode in der Musik Gruppen und Kategorien in der Musik Geometrie der Töne Topos of Music MorphologieabendländischerHarmonik Kunst der Fuge Depth-EEG for Consonances and Dissonances RUBATOProject KiT-MaMuThProject Kuriose Geschichte

Concepts Mod = category of modules + diaffine morphisms Mod = category of modules + diaffine morphisms: A = R-module, B = S-module Dilin(A,B) = (,f) f:A  B additive, :R  S ring homomorphism f(r.a) = (r).f(a) e b (x) = b+x; translation on B = e B.Dilin(A,B) e b.f: A  B  B dilinear translation

Concepts presheaves Topos of presheaves over Mod = {F: Mod   Sets, contravariant} representable Example: = address F(A) =: A = address Yoneda Lemma The Mod  is fully faithfull. B Yoneda Lemma The Mod  is fully faithfull. B

 Ÿ K   K Concepts K  B B Database Management Systems require recursively stable object types! k  no module! K  B no module! Need more general spaces F B  A = Ÿ n : sequences (b 0,b 1,…,b n ) A = B: self-addressed tones Need general addresses A K K F = presheaf over Mod F = A = A =  Ÿ F

Concepts F = Form name one of four „space types“ a diagramn √ in Frame( √ ) a monomorphism in id: Functor(F) >  Frame( √ ) Frame( √ )-space for type simple( √ ) simple √ =  simple( √ ) limit √ = Form-Name-Diagram  limit( √ ) = lim(Form-Name-Diagram  ) colimit √ = Form-Name-Diagram  colimit( √ ) = colim(Form-Name-Diagram  ) power √ = Form-Name F  Functor(F) power( √ ) =  Functor(F) Frame( √ ) >> Functor(F) FormsForms

Concepts Frame( √ ) >> Functor(F) Form F DenotatorsDenotators K K Functor(F) „A-valued point“ D = denotator name A address A K

Concepts – – OnsetOnsetLoudnessLoudnessDurationDurationPitchPitch MakroNoteMakroNote MakroNoteMakroNote SatellitesSatellites AnchorNoteAnchorNote STRG Ÿ Ornaments Ornaments Schenker Analysis Schenker Analysis

 RUBATO ®    Concepts Java Classes for Modules, Forms, and Denotators

Concepts Galois Theory Field S f S (X) = 0 Form Theory Form System id √ (F) Defining equation Defining diagram x2x2x2x2 x1x1x1x1 xnxnxnxn x3x3x3x3 F2F2F2F2 FrFrFrFr F1F1F1F1 Mariana Montiel Hernandez, UNAM

Classification Category Loc of local compositions Type = Power F  Functor(F) = G K  G L  H f/   = affine morphism f, h = natural transformations local composition local composition K  G K  G objective generalizes K „objective“ local compositions objects morphisms specify „address change“ 

Classification ObLoc ObLoc A Loc Loc A Tracefunctor Embeddingfunctor Theorem Loc is finitely complete (while ObLoc is not!) Loc is finitely complete (while ObLoc is not!) On ObLoc A and Loc A Embedding and Trace On ObLoc A and Loc A Embedding and Trace are an adjoint pair: ObLoc A (Embedding(K),L)  Loc A (K,Trace(L)) Theorem Loc is finitely complete (while ObLoc is not!) Loc is finitely complete (while ObLoc is not!) On ObLoc A and Loc A Embedding and Trace On ObLoc A and Loc A Embedding and Trace are an adjoint pair: ObLoc A (Embedding(K),L)  Loc A (K,Trace(L))

K  K t t  K t  G t i  K i  G i  K i local isomorphism/A K it  K ti  Classification

Category Gl of global compositions Objects: K I = functor K which is covered by a finite atlas I = (K i ) of local compositions in Loc A at address A Morphisms: K I at address A L J at address B f/  : K I  L J f = natural transformation,  = address change f induces local morphisms f ij /  on the charts Classification

Have Grothendieck topology Cov(Gl) on Gl Covering families (f i /  i : K I i  L J ) i are finite, generating families. Classification Theorem Cov(Gl) is subcanonical Cov(Gl) is subcanonical The presheaf  F : K I  F (K I ) of global affine functions is a sheaf. The presheaf  F : K I  F (K I ) of global affine functions is a sheaf.Theorem Cov(Gl) is subcanonical Cov(Gl) is subcanonical The presheaf  F : K I  F (K I ) of global affine functions is a sheaf. The presheaf  F : K I  F (K I ) of global affine functions is a sheaf.

Classification 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

Classification Theorem (global addressed geometric classification) Let A = locally free of finie rank over commutative ring R Consider the objective global compositions K I at A with (*): locally free chart modules R.K i the function modules  F (K i ) are projective the function modules  F (K i ) are projective (i) Then K I can be reconstructed from the coefficient system of retracted functions res* F (K I )  F ( 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 finie rank over commutative ring R Consider the objective global compositions K I at A with (*): locally free chart modules R.K i the function modules  F (K i ) are projective the function modules  F (K i ) are projective (i) Then K I can be reconstructed from the coefficient system of retracted functions res* F (K I )  F ( 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 (*).

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

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

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

Old Tonality Neutral Degrees (I C, VI C ) Modulation Degrees (II F, IV F, VII F ) New Tonality Cadence Degrees (II F & V F ) Arnold Schönberg: Harmonielehre (1911) 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? Models

I IVVIIIIIVIVII Models

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

C (3) E b (3) M (3) VEbVEbVEbVEb VII E b II E b III E b VCVC IV C VII C II C Models

Unification  = Ÿ 12 +  = consonances  = Ÿ 12 +  {1,2,5,6,10,11} = dissonances Ÿ 12  Ÿ 3 x Ÿ 4 e .2.5 ƒƒƒƒ Ÿ 12 [  ]

Unification C/  Symmetry in Human Depth-EEG Extension to Exotic Interval Dichotomies Rules of Counterpoint Following J.J. Fux

Ÿ 12  Ÿ 12 0 Ÿ Ÿ 12 X = { }  Trans(X,X) 

Z 12 Z 12 [  ] Z Z 12 Z 12 [  Z 12 [  ] Trans(D, T) = Trans(K,K)| ƒ  ƒƒƒƒ ƒƒƒƒ ƒƒƒƒ D = C-dominant triad T = C-tonic triad K

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