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Guerino Mazzola U & ETH Zürich Internet Institute for Music Science Just and Well-tempered Modulation Theory Just and Well-tempered Modulation Theory

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model 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?

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model Space Ÿ 12 of pitch classes in 12-tempered tuning Twelve diatonic scales: C, F, B b, E b, A b, D b, G b, B, E, A, D, G Scale = part of Ÿ 12 C

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model I IVVIIIIIVIVII

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model I IV II VI V III VII Harmonic strip of diatonic scale

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model C (3) F (3) B b (3) E b (3) A b (3) D b (3) G b (3) B (3) E (3) A (3) D (3) G (3) Dia (3) triadiccoverings

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model S (3) Space of cadence parameters S (3) k 1 (S (3) ) = {II S, V S } S (3) k 2 (S (3) ) = {II S, III S } S (3) k 3 (S (3) ) = {III S, IV S } S (3) k 4 (S (3) ) = {IV S, V S } S (3) k 5 (S (3) ) = {VII S } k S (3) k(S (3) )

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model 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

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model S (3) T (3) kk A etet e t.A etet modulation S (3) T (3) = cadence + symmetry modulation S (3) T (3) = cadence + symmetry

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model S (3) T (3) kk Given a modulation k, g:S (3) (3) g M M A quantum for (k,g) is a set M of pitch classes such that: MMM the symmetry g is a symmetry of M, g(M) = M (3) M the degrees in k( (3) ) are contained in M MT M T is rigid, i.e., has no proper inner symmetries M M is minimal with the first two conditions

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model Modulation Theorem for 12-tempered Case S (3), (3) For any two (different) tonalities S (3), (3) there is a modulation (k,g) and M a quantum M for (k,g) Further: M S (3), (3) M, M (3) M M is the union of the degrees in S (3), (3) contained in M, and thereby defines the triadic covering M (3) of M (3) M (3) the common degrees of (3) and M (3) are called the modulation degrees of (k,g) the modulation (k,g) is uniquely determined by the modulation degrees.

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

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experiments Ludwig van Beethoven: op.130/Cavatina/ # 41 Inversion e b E b (3) B (3) Inversion e b : E b (3) B (3) 4:00 mi-b->si

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e be be be b E b (3) experiments b B (3) Inversion e b

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experiments Ludwig van Beethoven: op.106/Allegro/ # Inversion d b G (3) E b (3) Inversion d b : G (3) E b (3) dbdbdbdbgg # # :50 sol->mi b

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Ludwig van Beethoven: op.106/Allegro/ # CatastropheE b (3) D (3) ~ b (3) Catastrophe : E b (3) D (3) ~ b (3) experiments 6:00 mi b->re= Si min.

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experiments Theses of Erwin Ratz (1973) and Jürgen Uhde (1974) Ratz: The sphere of tonalities of op. 106 is polarized into a world centered around B-flat major, the principal tonality of this sonata, and a antiworld around B minor. Uhde: When we change Ratz worlds, an event happening twice in the Allegro movement, the modulation processes become dramatic. They are completely different from the other modulations, Uhde calls them catastrophes. B minor B-flat major

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C (3) B b (3) E b (3) D b (3) G b (3) E (3) A (3) G (3) experiments Thesis:The modulation structure of op. 106 is governed by the inner symmetries of the diminished seventh chord C # -7 = {c #, e, g, b b } in the role of the admitted modulation forces. F (3) A b (3) B (3) D (3) ~ b (3)

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generalization Modulation Theorem for 12-tempered 7-tone Scales S and triadic coverings S (3) (Muzzulini) q-modulation = quantized modulation (1) S (3) is rigid. For every such scale, there is at least one q-modulation. For every such scale, there is at least one q-modulation. The maximum of 226 q-modulations is achieved by the harmonic scale #54.1, the minimum of 53 q-modulations occurs for scale #41.1. The maximum of 226 q-modulations is achieved by the harmonic scale #54.1, the minimum of 53 q-modulations occurs for scale #41.1. (2) S (3) is not rigid. For scale #52 and #55, there are q-modulations except for t = 1, 11; for #38 and #62, there are q-modulations except for t = 5,7. All 6 other types have at least one quantized modulation. For scale #52 and #55, there are q-modulations except for t = 1, 11; for #38 and #62, there are q-modulations except for t = 5,7. All 6 other types have at least one quantized modulation. The maximum of 114 q-modulations occurs for the melodic minor scale #47.1. Among the scales with q-modulations for all t, the diatonic major scale #38.1 has a minimum of 26. The maximum of 114 q-modulations occurs for the melodic minor scale #47.1. Among the scales with q-modulations for all t, the diatonic major scale #38.1 has a minimum of 26.

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just theory Modulation theorem for 7-tone scales S and triadic coverings S (3) in just tuning (Hildegard Radl) f cg daeb log(5) log(3) ebebebeb abababab bbbbbbbb f#f#f#f# dbdbdbdb

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S (3) T (3) Just modulation: Same formal setup as for well-temperedtuning. just theory A etetetet e t.A

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Lemma: If the seven-element scale S is generating, a non-trivial automorphism A of S (3) is of order 2. Proof: The nerve automorphism Nerve(A) on Nerve(S (3) ) preserves the boundary circle of the Möbius strip and hence is in the dihedral group of the 7-angle. By Minkowskys theorem, the composed group homomorphism A> GL 2 ( Ÿ ) GL 2 ( Ÿ 3 ) is injective. Since #GL 2 ( Ÿ 3 ) = 48, the order is 2. Lemma: Let M = : S (3) T (3) be a modulator, with A = Ÿ 2, the -orbit is (x) = e Ÿ (1+R)t.x e Ÿ (1+R)t.M(x) Lemma: Let M = e t.A: S (3) T (3) be a modulator, with A = e a.R. For any x Ÿ 2, the -orbit is (x) = e Ÿ (1+R)t.x e Ÿ (1+R)t.M(x) just theory

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Just modulation: Target tonalities for the C-major scale. bbbbbbbb abababab ebebebeb bb*bb*bb*bb* dbdbdbdb fgd aeb db*db*db*db* just theory

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Just modulation: Target tonalities for the natural c-minor scale. bbbbbbbb abababab ebebebeb bb*bb*bb*bb* dbdbdbdb fgd aeb db*db*db*db* just theory

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Just modulation: Target major tonalities from the natural c-minor scale. bbbbbbbb abababab ebebebeb bb*bb*bb*bb* dbdbdbdb fgd just theory

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Just modulation: Target minor tonalities from the Natural c-major scale. bbbbbbbb fgd aeb db*db*db*db* just theory

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Just modulation: Target tonalities for the harmonic C-minor scale. bbbbbbbb abababab ebebebeb bb*bb*bb*bb* dbdbdbdb fgd aeb db*db*db*db* g#g#g#g# d#d#d#d# f#f#f#f# a* fbfbfbfb b bb gbgbgbgb eb*eb*eb*eb* just theory

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Just modulation: Target tonalities for the melodic C-minor scale. bbbbbbbb abababab ebebebeb fgd ae just theory

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f cg d dbdbdbdb abababab ebebebeb bbbbbbbbaeb f#f#f#f#

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no modulations infinite modulations limited modulations four modulations major, natural, harmonic, melodic minor just theory

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rhythmic modulation

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Classes of 3-element motives M Ÿ generic rhythmic modulation

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onset Percussion encoding 62^ Retro-gradeof62^ 62^ R(62^) rhythmic modulation

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3:18-5:48 rhythmic modulation 12/8 B.1-6 m1 m2 m1 m2 m3 m1 m2 m3 m4 m1 m2 m3 m4 m5 m1 m2 m3 m4 m5 m6,m7 B.7-12 m1 m2 m1 m2 m3 m1 m2 m3 m4 m1 m2 m3 m4 m5 m1 m2 m3 m4 m5 m6,m7 R B modulation pivots new tonic at 9/8 of bar 21 new bar system

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