Guerino Mazzola U & ETH Zürich Internet Institute for Music Science Mathematical Models of Tonal Modulation and Application to Beethoven‘s op. 106 Mathematical Models of Tonal Modulation and Application to Beethoven‘s op. 106

Contents A Modulation Model Experiments with Beethoven Generalizations Open Questions A Modulation Model Experiments with Beethoven Generalizations Open Questions

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?

Model Space Z 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 Z 12 C

Model I IVVIIIIIVIVII

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

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

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) )

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

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

Model S (3) T (3) kk Given a modulation k, g:S (3)  (3) 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

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.

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

Experiments Ludwig van Beethoven: op.130/Cavatina/ # 41 Inversion e b E b (3)  B (3) Inversion e b : E b (3)  B (3)

e be be be b E b (3) Experiments Ludwig van Beethoven: op.130/Cavatina/ # 41 Inversion e b E b (3)  B (3) Inversion e b : E b (3)  B (3)b B (3) Inversion e b b bb bb bb b g a ba ba ba b f

Experiments Ludwig van Beethoven: op.106/Allegro/ # Inversion d b G (3)  E b (3) Inversion d b : G (3)  E b (3) dbdbdbdbgg # #

Generalization Ludwig van Beethoven: op.106/Allegro/ # CatastropheE b (3)  D (3) ~ b (3) Catastrophe : E b (3)  D (3) ~ b (3)

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

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)

Generalization All 7-scales in well-tempered pitch classes -> Daniel Muzzulini/Hans Straub All 7-scales in well-tempered pitch classes -> Daniel Muzzulini/Hans Straub Diatonic, melodic, and harmonic scales in just tuning -> Hildegard Radl Diatonic, melodic, and harmonic scales in just tuning -> Hildegard Radl Applications to rhythmic modulation -> Guerino Mazzola Applications to rhythmic modulation -> Guerino Mazzola Ongoing research: Modulation for generalized tones -> Thomas Noll Ongoing research: Modulation for generalized tones -> Thomas Noll

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.

Questions Develop methodology and software for systematic experiments on given corpora of musical compositions (recognize tonalities and modulations thereof!) Develop methodology and software for systematic experiments on given corpora of musical compositions (recognize tonalities and modulations thereof!) Modulation in other musical dimensions, such as motive, rhythm, and global object spaces. Modulation in other musical dimensions, such as motive, rhythm, and global object spaces. Modulation in other concept spaces for tonality and harmony, e.g., self-addressed pitch, or harmonic topologies following Riemann. Modulation in other concept spaces for tonality and harmony, e.g., self-addressed pitch, or harmonic topologies following Riemann. Generalizing the quantum/force analogy in the modulation model, develop a general theory of musical dynamics, i.e., the theory of musical interaction forces between general structures. Generalizing the quantum/force analogy in the modulation model, develop a general theory of musical dynamics, i.e., the theory of musical interaction forces between general structures.