Guerino Mazzola U & ETH Zürich Guerino Mazzola U & ETH Zürich Extending Set Theory to Harmonic Topology and Topos Logic 1. Music objects 2. Why topoi? 3. Logic Extending Set Theory to Harmonic Topology and Topos Logic 1. Music objects 2. Why topoi? 3. Logic

The address question (ontology): What is an elementary musical object? The address question (ontology): What is an elementary musical object? music objects

x Ÿ 12 (space of pitch classes) p — EHLD  — 4 (space of note events) E H D L

F x: —  F affine x = e t.g, e t = translation, g = linear music objects—01 x A = R-module = „address“ = e F.Lin R (A, F) A = R = — = e F.Lin — ( —, F) ª F 2

R = Ÿ, A = Ÿ 11, F = Ÿ 12 = Ÿ Ÿ 12 S   Ÿ Ÿ 12 ª Ÿ S   Ÿ Ÿ 12 ª Ÿ Ÿ 12 S 011 Webern: Op. 28 Dodecaphonic Series music objects

 gesture H E L  score h e l Position Key E music objects

HarmonyandCounterpoint Grand Unification Perspectives of

S (3) T (3) kk A etet e t.A etet modulation S (3)  T (3) = „cadence + symmetry “ music objects

C (3) E b (3) M (3) VEbVEbVEbVEb VII E b II E b III E b VCVC IV C VII C II C music objects Schönberg‘s Modulation Degrees

Ÿ 12 c e g = {1, c, f, f.c, c.f, f 2.c, c 2.f,...}  Ÿ Ÿ 12 = {1, c, f, f.c, c.f, f 2.c, c 2.f,...}  Ÿ Ÿ 12 | | = {0, 4, 7} | | = {0, 4, 7} = {1, c, f, f.c, c.f, f 2.c, c 2.f,...}  Ÿ Ÿ 12 = {1, c, f, f.c, c.f, f 2.c, c 2.f,...}  Ÿ Ÿ 12 | | = {0, 4, 7} | | = {0, 4, 7} Circel chords (G. Mazzola, Geometrie der Töne) e 7.3 c = 0 f = e 7.3 {c, f(c), f 2 (c),...} = {0, 4, 7} = {c, e, g} = major triad music objects

Trans(Dt,Tc) =  Ÿ Ÿ 12 f Dt Dominant Triad {g, h, d} Tc Tonic Triad {c, e, g} Modeling Riemann Harmony (Th. Noll, PhD Thesis) music objects „relative consonances“

Ÿ 12  Ÿ 3  Ÿ 4 z ~> (z mod 3, -z mod4) 4.u+3.v <~ (u,v) music objects

Ÿ 12 Ÿ 12 [  ]= Ÿ 12 [X]/(X 2 ) c+ . Ÿ 12 c c+ .d music objects

  = Ÿ 12 +  = consonances D  = Ÿ 12 +  {1, 2, 5, 6, 10, 11} = dissonances e .2.5 music objects

Parallels of fifths are always forbidden music objects

Ÿ 12 Ÿ 12 [  ] Ÿ Ÿ 12 Ÿ 12 [  Ÿ 12 [  ] Trans(Dt,Tc) = Trans(K ,K  )| ƒ  ƒƒƒƒ ƒƒƒƒ ƒƒƒƒ add.ch add.ch Trans(Dt,Tc) Trans(K ,K  ) K , D  music objects

space F Prize for parametrization addresses: Parametrized objects need parametric evaluation! Prize for parametrization addresses: Parametrized objects need parametric evaluation! music objects address A

— — f 0 : 0 —  — : 0 ~> 0 — — f 1 : 0 —  — : 0 ~> 1 E H F = — EH ª — 2 K    music objects

Ÿ 12 series S   Ÿ Ÿ 12 S   Ÿ Ÿ 12 More general: set of k sequences of pitch classes of length t+1 K = {S 1,S 2,...,S k } Ÿ Ÿ 12 This is a „polyphonic“ local composition K  Ÿ Ÿ 12 S1S1 SkSk music objects Ÿ 12 S 0 11 Webern: Op. 28

Ÿ s  Ÿ t s ≤ t, define affine map 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 music objects

Gegenstand der Untersuchungen sind aber nicht die Töne selbst, denn deren Beschaffenheit spielt gar keine Rolle, sondern die Verknüpfungen und Verbindungen der Töne untereinander. Bach‘s „Art of Fugue“ (1924) Wolfgang Graeser (1908—1928)

Need recursive combination of constructions such as „sequences of sets of sets of curves of sets of chords“, etc. This leads to the theory of denotators, which we omit here. Need recursive combination of constructions such as „sequences of sets of sets of curves of sets of chords“, etc. This leads to the theory of denotators, which we omit here. music objects Eine kontrapunktische Form ist eine Menge von Mengen von Mengen (von Tönen) Bach‘s „Art of Fugue“ (1924) Wolfgang Graeser

F: Mod  Sets presheaves 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 why topoi?

Yoneda Lemma The functorial Mod  is fully faithfull. M = Hom(?,M) ≈ ≈ Yoneda Lemma The functorial Mod  is fully faithfull. M = Hom(?,M) ≈ ≈ why topoi? Const.Const. SetsSets

Functorial Local Compositions Are left with two important problems for local compositions K   The definition of a general evaluation procedure; The definition of a general evaluation procedure; There are no general fiber products for local compositions. There are no general fiber products for local compositions. Are left with two important problems for local compositions K   The definition of a general evaluation procedure; The definition of a general evaluation procedure; There are no general fiber products for local compositions. There are no general fiber products for local compositions. Solution:  F = {subfunctors a   F} „generalized sieves“ Kˆ   F = {(f:X  A, k.f), k K}    = {(f:X  A, k.f), k   K}    Kˆˇ = Id = K why topoi?

Classical logic: F = 0 = zero module subsets d  = = {0} Have two values: d = = T, “true” d =  =  F =  T, “false” Fuzzy logic: F = S = — / Ÿ = circle group subsets d = [0, e[  = This logic is known as the Gödel algebra, in fact a Heyting algebra defined by the topology of these subsets. logic e S Hugo Riemann: Logik ist in der Funktionstheorie ein fundamentaler, aber dunkler Begriff.

logic Have natural generalization! d  d = [0, e[  F = any space (functor) A = any address d  objective local composition d  Ffunctorial local composition In this context, local compositions d are structurally legitimate supports of logical values and their combinations (conjunction, disjunction, implication, negation).

The „functorial“ change K ~> Kˆ has dramatic consequences for the global theory! The „functorial“ change K ~> Kˆ has dramatic consequences for the global theory! I IV II VI V III VII I IV V II IIIVI VII Ÿ A = 0 Ÿ Ÿ 12 A = Ÿ 12 Ÿ 12 Ÿ Ÿ 12 X  Ÿ 12 ~> X* = End*(X)  Ÿ Ÿ 12 logic

I IV II VI V III VII I* IV* II* VI* V* III* VII* II* I* Ÿ Ÿ 12 I*  II* =  ToM, ch. 25 logic

ˆ I*ˆ ˆ Ÿ 12 I* 1 Ÿ 12 II* f = e 11.0: Ÿ 12  Ÿ 12 ˆ  e 0.4  e 11.3  e 8.0 e 0.4. e 11.0 = e e 11.0 = e 8.0 ˆ ˆˆ  logic

Extension Topology Fix a space functor F, End(F) = set of endomorphisms of F, and an address A. ExTop A (F) =  F = {a   F} Extension topology on ExTop A (F): Subsets M   End(F), Basic open sets: Ext A (M) = {a, M  End(a)} logic

Naturality of Extension Topologies Proposition: Fix a space functor F two addresses A, B, and a retraction  : A  B. Then we have this continuous map: ExTop B (F) ExTop A (F)     Id F a a  logic

Naturality of Heyting Logic of Open Sets ExTop B (F) ExTop A (F)  U  V = U  V U  V = U  V U  V =  W  U  V W  U = (-U) o   (U  V)  (   (U)    (V))   (U  V  (   (U)    (V))   (U  V)  (   (U)    (V))   (U  V)  (   (U)    (V))   (U  V  (   (U)    (V))   (U  V)  (   (U)    (V)) logic Open B (F) Open A (F)   is a logical homomorphism Proposition:

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 } TON = {C, F, B b, E b, A b, D b, G b, B, E, A, D, G} Val= {T, S, D, t, s, d, T*,S*, D*, t*, s*, d*} F = Chords ( Ÿ 12 ) = Ÿ 12 TRUTH(F) = sets of chords in F logic

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