1. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists Modeling Tonal Modulation

2. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists Schoenberg (Harmonielehre): There is, for example, a very popular harmony treatise, in which moduations are nearly exclusively made using the dominant seventh or diminished seventh chord. And the author only demonstrates that after each major or minor tirad any of those two chords can be played, and thereby go to any tonality. If I wanted that, I could have finished even earlier. In fact I am capable to show (using „gauged“ examples from liteature) that you may use any triad after any other triad. So if that reaches every tonality and thereby modulation has been realized, the procedure would even be simpler. But if somebody, to tell a story, makes a journey, he would not choose the air line. The shortest path is the worst. The bird‘s perspective is the perspective of a bird‘s brain. If everything is blurred, everything is possible. Differences disappear. And it is then irrelevant if I have made a moduation with a dominant or diminished seventh chord. The essential of a moduation is not the target, but the trajectory.

3. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists old tonality neutral/pivot degrees (I C, VI C ) fundamental degrees (II F, IV F, VII F ) new tonality cadence degrees (II F & V F ) Arnold Schönberg: Harmonielehre (1911) What is the set of tonalities?What is the set of tonalities? What is a degree?What is a degree? What is a cadence?What is a cadence? Which is the modulation mechanism?Which is the modulation mechanism? How do these structures determine the fundamental degrees?How do these structures determine the fundamental degrees?

4. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists pitch class space Ÿ 12 for 12-tempered tuning 0 1 2 3 4 5 6 7 8 9 10 11 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

5. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists I IVVIIIIIVIVII

6. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists I IV II VI V III VII Harmonic band of major scale C (3)

7. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists 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) triadic interpretations

8. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists S (3) space of cadence parameters k 1 (S (3) ) = {II S, V S } k 2 (S (3) ) = {II S, III S } k 3 (S (3) ) = {III S, IV S } k 4 (S (3) ) = {IV S, V S } k 5 (S (3) ) = {VII S } k k(S (3) )

9. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists S (3) T (3) gluon strong force W+W+ weak force  electromagnetic force graviton gravitation force = symmetry between S (3) and T (3) S (3) and T (3) quantum = set of pitch classes = M kk

10. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists S (3) T (3) kk A TtTt T t.A TtTt modulation S (3)  T (3) = „cadence + symmetry “

11. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists S (3) T (3) kk Given a modulation k, g:S (3)  (3) g M a quantum for the modulation (k,g) is a set M of pitch classes such that: the symmetry g is a symmery of M, g(M) = M the symmetry g is a symmery of M, g(M) = M the degrees in k(  (3) ) are contained in M the degrees in k(  (3) ) are contained in M M  T is rigid, i.e., has no non-trivial symmetries M  T is rigid, i.e., has no non-trivial symmetries M is minimal with the first two conditions M is minimal with the first two conditions

12. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists Modulation theorem for 12-tempered tuning For two different tonalities S (3),  (3) there exist a modulation (k,g) and a modulation (k,g) and a quantum M for (k,g) a quantum M for (k,g) (= quantized modulation) Moreover: M is the union of the degrees in S (3),  (3) contained in M which thereby define the triadic interpretation M (3) of M M is the union of the degrees in S (3),  (3) contained in M which thereby define the triadic interpretation M (3) of M the common degrees of  (3) and M (3) are called the modulation degrees of (k,g) 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. the modulation (k,g) is uniquely determined by the modulation degrees.

13. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists C (3) E b (3) M (3) VEbVEbVEbVEb VII E b II E b III E b VCVC IV C VII C II C

14. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists # fourths

15. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists

16. Modulation theorem (12-tempered case) for the 7-tone scales S and triadic interpretations S (3) (Daniel Muzzulini: Musical Modulation by Symmetries. J. for Music Theory 1995) q-modulation = quantized modulation (1) S (3) is rigid. For such a scale, there is at least one q-modulation. For such a scale, there is at least one q-modulation. The maximum of 226 q-modulations is reached for the harmonic minor scale #54.1, the minimum of 53 q-modulations happens for the scale #41.1. The maximum of 226 q-modulations is reached for the harmonic minor scale #54.1, the minimum of 53 q-modulations happens for the scale #41.1. (2) S (3) isn‘t rigid. For the scales #52 and #55, there are q-modulations except for transposition t = 1, 11; for #38 and #62, there are q-modulations except for t = 5,7. All the 6 other types have at least one q-modulation. For the scales #52 and #55, there are q-modulations except for transposition t = 1, 11; for #38 and #62, there are q-modulations except for t = 5,7. All the 6 other types have at least one q-modulation. The maximum of 114 q-modulations happens for the melodic minor scale #47.1. Among the scales with q-modulations for all t, the major scale #38.1 has a minimum of 26. The maximum of 114 q-modulations happens for the melodic minor scale #47.1. Among the scales with q-modulations for all t, the major scale #38.1 has a minimum of 26.

17. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists Sonata scheme for the allegro movement in Beethoven‘s op.106 according to Erwin Ratz: Einführung in die musikalische Formenlehre. Universal Edition, Wien 1973 !! !

18. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists 7:06 Fast normal modulation: B b  G b

19. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists 4:50 Normal modulation: G  E b

20. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists Modulation of „catastrophe“ typeE b (3)  D (3) ~ b (3) Modulation of „catastrophe“ type: E b (3)  D (3) ~ b (3) 6:00

21. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists Erwin Ratz‘ (1973) and Jürgen Uhde‘s (1974) theses Ratz: The sphere of tonalities of op. 106 is polarized in a“world“ centered around B-flat major, the main tonality of the sonata, and an „antiworld“ centered around b-minor. Uhde*: When one changes between the worlds of Ratz— an event that happens twice in the allegro movement—then the modulation processes become dramatic. They are completely different from other modulations, and Uhde calls them „catastrophes“. b minor B-flat major * Uhde J: Beethovens Klaviermusik III. Reclam, Stuttgart 1974

22. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists C (3) B b (3) E b (3) D b (3) G b (3) E (3) A (3) G (3) Thesis:The modulation structure of op. 106 is governed by the symmetries of the diminished seventh chord C # -7 = {c #, e, g, b b } carrying the admitted modulation forces. F (3) A b (3) B (3) D (3) ~ b (3) Exposition Recapitulation Development Coda

23. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists 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) Sym(C # -7 ) = max. symmetry group separating D from B Sym(C # -7 ) = max. symmetry group separating D from B

24. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists C (3) B b (3) E b (3) D b (3) G b (3) E (3) A (3) G (3) F (3) A b (3) B (3) D (3) ~ b (3)

25. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists e -3 I g * I d/d # * I b b I a/a b e 3 Modulators in op. 106/allegro Exposition Exposition Recapitulation Recapitulation Development Development Coda Coda B b  G G  E b  D/b  B b  B b  G b  G  B b B b B b  G G  E b  D/b  B b  B b  G b  G  B b B b symmetries of transposition!

26. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists 7:06 Fast normal modulation: B b  G b IBbIBbIBbIBb VI G b III G b  V G b Inversion b b I IV II VI V III VII a-flat in II  V  VII b-flat in III  VI  I f in III  V  VII

27. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists 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 oppressive

28. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists e be be be b E b (3) b B (3) Inversion e b Transposition T 4

29. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists Inversion d b G (3)  E b (3) Inversion d b : G (3)  E b (3) dbdbdbdbgg #124 - 125 #126 - 127 4:50

30. Guerino Mazzola (Spring 2015 © ): Math for Music Theorists Transposition -3 B b (3)  G (3) Transposition -3 : B b (3)  G (3) VII G