1. Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch www.encyclospace.org Construction of Sonata op. 3 by Mathematical Methods

2. composition Sonate für Klavier „Aut G (Messiaen III)\DIA (3) “ (1981) Gruppen und Kategorien in der Musik Heldermann, Berlin 1985 Construction on 58 pages 99 bars, 12/8 metrum, C-major Sonate pour piano „L‘Essence du Bleu“ (Acanthus 2002) CD: Patrizio Mazzola

3. C (3) B b (3) E b (3) D b (3) G b (3) E (3) A (3) G (3) op.106 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) Exposition Recapitulation Development Coda

4. separation 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) Aut Ÿ (C # -7 )

5. e -3 U g * U d/d # * U b b U a/a b e -3 modulators Modulators in op. 106 ExpositionRecapitulationDevelopment Coda Coda B b  G G  E b  D/b  B b  B b  G b  G  B b B b

6. Op. 106Op. 3 scheme Overall Scheme minor third  2 nd Messiaen scale „limited transposition“ major third  3 nd Messiaen scale „limited transposition“ Aut Ÿ (C # -7 ) = Aut Ÿ (C # -7 ) = Aut Ÿ (C # + ) = Aut Ÿ (C # + ) =

7. B b (3) A b (3) E (3) D (3) C (3) G b (3) moduations Thesis:The modulation structure of op. 3 is governed by the inner symmetries of the augmented triad chord C # + = {c #, f, a} in the role of the admitted modulation forces. G (3) B (3) E b (3) F (3) D b (3) A (3) Exposition Recapitulation Development Coda

8. C  B b  G b G b  A b  E E  F F  C U c # e -4 U a e -4 * e -4 * Modulators in op. 3 DevelopmentExpositionRecapitulation Coda Coda modulators

9. motivic principle Motivic Zig-Zag in op.106    Bars 75-78

10. Model Motivic Zig-Zag Scheme minor third  2 nd Messiaen scale „limited transposition“ major third  3 nd Messiaen scale „limited transposition“

11. m ö bius Motivic strip of Zig-Zag 6 7 4 1 9 8 2 5 3 (15) (15) (10) (11) (19) (19) (20) (2) (16)

12. main theme Main Theme CC C Bars 3-5

13. kernel Kernel of Development 6 7 4 1 9 8 2 5 3        ‘‘‘‘  U2U2U2U2 A B C D E F A‘ B‘ C‘ D‘ E‘ F‘                                                                          

14. dbdbdbdb Kernel Matrix 6 4 8 7 9 1 5 6 4 8 7 9 3 5 6 4 8 7 2 3 5 6 4 8 A B C D E F dbdbdbdb f a DrDrDrDr DlDlDlDl A‘ B‘ C‘ D‘ E‘ F‘ kernel

15. DrDrDrDr DlDlDlDl D  =  D r  D l kernel

16. kernel moduation DrDrDrDr DlDlDlDl Kernel Modulation U a : G b  A b UaUaUaUa Ua(Dl)Ua(Dl)Ua(Dl)Ua(Dl)