Manifolds and Stemmata in Musical Time Guerino Mazzola

Manifolds and Stemmata in Musical Time Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola. ch www. encyclospace. org/talks

program • The Performance Transformation • Normative vs. Immanent Analysis • Local and Global Meters • Nerves and Weights • Metro. Rubette®: Examples • Czerny‘s Tempi and Stemmata • Analytical Performance on Rubato®

transformations H h √ e E e T(E) = (d√E/d. E)-1 [q /sec] √E E

analysis ? (symbolic) time maximal local meter no local meter

meters n/16 0 a b c d e 2 3 4 6 8 10 12

nerves c 3 0 a 4 6 e 12 2 10 d nerve of the covering {a, b, c, d, e} x dominates y iff simplex(y) Í simplex(x) b

nerves weight(x) = S z Œ simplex(x), m 22 20 18 16 14 12 10 8 6 4 p length (z) £ length(z) m=p=2

examples Java Classes for Modules, Forms, and Denotators RUBATO® L L S S Os X

examples

stemmata 120 110 100 M. M. 90 80 70 60 bar 1 bar 2 bar 3 bar 4

stemmata presto® curvetta 4 mamma curva 1 curvetta 3 curva 2

stemmata Chopin: Impromptu op. 29 tr tr tr

performance Mother m d. T Daughter T Granddaughter Z(d. T, m) Tm

performance X X 0 √ F Z(X) = J(√ )(X)-1 D performance field, field defined on cube F = the frame of Z X 0 Œ I = initial set X 0 = ÚXZ(t) ÚXZ = integral curve through X x= √(X) x 0 D = (1, 1, …, 1) = Const. x 0 = √I(X 0) initial performance x = x 0 - t. D

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

Performnce Big Problem: Describe Typology of shaping operators! Emotions, Gestures, Analyses w(E, H, …) H E

performance J. S. Bach: Die Kunst der Fuge — Contrapunctus III Joachim Stange-Elbe Metrical and Motivic Weights act on agogics, dynamics, and articulation sopran score alt sum of all tenor bass

The Topos of Music Geometric Logic of Concepts, Theory, and Performance in collaboration with Moreno Andreatta, Jan Beran, Chantal Buteau, Karlheinz Essl, Roberto Ferretti, Anja Fleischer, Harald Fripertinger, Jörg Garbers, Stefan Göller, Werner Hemmert, Mariana Montiel, Andreas Nestke, Thomas Noll, Joachim Stange-Elbe, Oliver Zahorka www. encyclospace. org