Guerino Mazzola U & ETH Zürich Internet Institute for Music Science Penser la musique dans la logique fonctorielle des topoi Penser la musique dans la logique fonctorielle des topoi

Topographie musicale Stratification sémiotique Models mathématiques Espaces de concepts Fiction et facticité Vérité et beauté Topographie musicale Stratification sémiotique Models mathématiques Espaces de concepts Fiction et facticité Vérité et beauté programme

topographieCommunication Neutral level Poiesis Aisthesis Realities Semiosis Content Physical Mental Psychic Signification Expression

topographieCommunication Aisthesis Neutral level Poiesis

√ H E h e T(E) = (d √ E /dE) -1 [ q /sec] Physical Mental topographie Psychic

Harmonic analysis Semiosis topographie Content Signification Expression G major  E b major Tonal modulation G major  E b major Score „surface“

stratification ExpressionSignificationContent meta system ExpressionSignificationContent ExpressionSignificationContent motivation ExpressionSignificationContent connotation Hjelmslev Stratification

stratification interpretation layer performance fields differential geometry denotator layerformstopoi score layer syn. & para. articulation connotation classical sheaves Stratification on the mental level – – OnsetOnsetLoudnessLoudnessDurationDurationPitchPitch MakroNoteMakroNote MakroNoteMakroNote SatellitesSatellitesAnchorNoteAnchorNote STRG Ÿ

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! Anthropic 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

concepts 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)

concepts F = Form name one of four „space types“ a name diagram √ 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  ) limit( √ ) = lim(Form-Name-Diagram  ) colimit √ = Form-Name-Diagram  colimit() = colim(Form-Name-Diagram  ) colimit( √ ) = colim(Form-Name-Diagram  ) power √ = Form-Name F  Functor(F) power() =  Functor(F) power( √ ) =  Functor(F) Frame( √ ) >> Functor(F) FormsForms

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

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

Os X  RUBATO ®    concepts Java Classes for Modules, Forms, and Denotators

Expressions Denotators Sig(Ex i Sig(Ex i ) (Textual) Predicates Sig Ex i D/Ex i Ex k Predicate Expressions facticit éD

What is facticity of predicate Ex at denotator D? D/Ex D/Ex :A.TRUTH(F)(d) TRUTH(F) = space of “subsets” of space F of ‘truth values’ d   F (A) d  x F The coordinate d of a truth denotator D/Ex is a ‚sieve‘ in  x F. The coordinate d of a truth denotator D/Ex is a ‚sieve‘ in  x F. facticit é d A F

Special case 1: I = 0-module Then F = final object = 1 in d   1 (A) =  (A) A = 0: topos-theoretic  (0) = Hom(1,  = set of topos-theoretic truth values = Special values: d =  =  F =  T  d = T facticit é

Special case 2: I = — / Ÿ = S = circle group, F d   F (A) means this: Take again special address A = 0, i.e., d fuzzy logic In particular, if d =  = [0,e[  S an interval, we have fuzzy logic defined by the truth quantity e in the closed unit interval. e S facticit é

The truth denotators D/Ex associated with a predicate Ex local compositions are local compositions at address A and in the truth space F. They generalize and unify the topos-theoretic and fuzzy logic values, and classical objects of music theory. Summary facticit é d A F

1. Arbitrary/Atomic Predicates: Mathematical Musical (Primavista) Deictic (Shifters) 2. Motivated/Compound Predicates: Logical Geometric Classification of Predicate Constructions D/Ex facticit é

beaut é HarmoRUBETTE ®

beaut é RieM D #,d (Chord(222)) =d =  = [0,e[  S RieM D #,d (Chord(222)) = d =  = [0,e[  S TON ={C, F, A #, D #, G #, C #, F #, B, E, A, D, G} TON = {C, F, A #, D #, G #, C #, F #, B, E, A, D, G} val= {T,S, D, t, s, d} S = — / Ÿ A = 0 D #,d T,v = D #,d

beaut é RieN T,v (Chord) = Chord.Ext 0 (M T,v ) RieN T,v (Chord) =  Chord.Ext 0 (M T,v ) M T,v = monoid of all endomorphisms of prototypical triadic chords Ext 0 (M T,v ) = {chords invariant under M T,v } = basic open set in the extension topology TON ={C, F, A #, D #, G #, C #, F #, B, E, A, D, G} TON = {C, F, A #, D #, G #, C #, F #, B, E, A, D, G} Val= {T,S, D, t, s, d, T*,S*, D*, t*, s*, d*} F = Chords ( Ÿ 12 ) TRUTH(F) = sets of chords in F