1. Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch www.encyclospace.org Les mathématiques de l‘interprétation musicale: Champs vectoriels d‘interprétation Les mathématiques de l‘interprétation musicale: Champs vectoriels d‘interprétation

2. The Topos of Music Geometric Logic of Concepts, Theory, and Performance www.encyclospace.org in collaboration with Carlos Agon, Moreno Andreatta, Gérard Assayag, Jan Beran, Chantal Buteau, Roberto Ferretti, Anja Fleischer, Harald Fripertinger, Jörg Garbers, Stefan Göller, Werner Hemmert, Michael Leyton, Mariana Montiel, Stefan Müller, Thomas Noll, Joachim Stange-Elbe, Oliver Zahorka October 2002 1300pp, ≈ 150 1300pp, ≈ 150 ¤

3. Fields

4. √ H E L E e pEpE pepe √E√E √ E (I 1 ) √ E (I k ) I1I1 IkIk X T(E) = (d √ E /dE) -1 [ q /sec]he l x = √ (X)

5. P-Cells Product fields: Tempo-Intonation field E H S(H) EH EH Z(E,H)=(T(E),S(H)) T(E)

6. (e(E),d(E,D) = e(E+D)-e(E)) T(E) P-Cells Parallel fields: Articulation field E D ED E Z(E,D) =  T(E,D) = (T(E),2T(E+D)  T(E))

7. Root Fundament P-Cells Work with Basis Basis parameters E, H, L, and corresponding fields T(E), S(H), I(L) Pianola Pianola parameters D, G, C cell hierarchy A cell hierarchy is a Diagram D in Cell such that there is exactly one root cell the diagram cell parameter sets are closed under union and non-empty intersection T S I I ¥ S T ¥ I T ¥ S  T ¥ S  T ¥ I T ¥ I ¥ S  T ¥ I ¥ S TT

8. Typology T TT T Z(  T, ) Stemma mother daughter granddaughter

9. Emotions, Gestures, Analyses Typology Big Problem: Describe typology of shaping operators! w(E,H,…) H E ???? ???? ??!! ??!!

10. Calculations RUBATO ® software: Calculations via Runge-Kutta-Fehlberg methods for numerical ODE solutions

11. Typology Tempo Operators T(E)w(E)T w (E) = w(E).T(E) Deformation of the articulation field hierarchy TT TwTw T TwTw  ww T TwTw? Q w (E,D) = w(E) 0 w(E+D)—w(E) w(E+D)  w = Q w (E,D).Z Q w = J( √ w ) -1 „w-tempo“

12. Typology RUBATO ® : Scalar operator Linear action Q w on ED-tangent bundle Direction of field changes Numerical integration control

13. Inverse Theory Lie type Restriction Affine transport

14. Inverse Theory Stefan Müller: EspressoRubette

15. Inverse Theory Lie type Restriction Restriction Sum Affine transport

16. Affinetransportparameters Lie operator parameters:weights,directions Output fields Z. fiber(Z.) Inverse Theory Roberto Ferretti

17. Inverse Theory