V EXPERIMENTS V.2 (Thu Apr 19) Synthesis II: Symmetries

Poetical Production via Baudelaire 3rd Movement Melodic Germ 26 Motif Classes Poetical Production via Baudelaire 3rd Movement CORRESPONDENCE GLUING SYMMETRIES Melodic Germ Rhythms and Their Modulations 1st Movement SELECTION REFLECTIONS ORNAMENTS Variations via Messiaen Grids 2nd Movement DECOMPOSITION ALTERATIONS ORNAMENTS Fractal Refinement 4th Movement SELECTION DILATATION ORNAMENTS REFINEMENT Bass Licks All Movements SELECTION

classes of 3-element motives M Í Ÿ122 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 The classification of all possible concept types and the instances within one fiexed type is a ‚wild‘ problem. Let us just look at all isomorphism types of three-element motives with pitch modulo 12 and onset modulo 12 in the integer valued coordinate spaces. Here, isomorphism means that one is allowed to reflect, shift or rotate a given motive. The ‚orbits‘ of these isomorphism actions are the classes, there are 26 of them. I shall come back to this example below when discussing a jazz CD.

10 10 10

Here, we illustrate the theory on the jazz CD „Synthesis“ which I recorded in 1990. Its entire structure, in harmony, in rhythmics, and in melodics, was deduced and constructed by use of the composition software presto® (written for Atari computers, but now also working on Atari emulations on Mac OS X), and starting from the 26 classes of three-element motives. This composition (a grant from the cultural department of the city of Zurich) was not recognized as a computer-generated music by the jazz critics. Only the piano part was played by myself, the entire bass and percussion part was played by synthesizers, driven by the presto® application via MIDI messages. During the production of this composition (Synthesis is a four-part, 45-minute piece), I never felt inhibited in my piano playing, in the contrary, it was a great pleasure to collaborate with complex structures of rhythm or melody, objects of a complexity that human percussionists would never be able to play from a score.

Definition of Symmetry: Symmetry is correspondence of parts as an expression of a whole. Semiotically speaking: significant = form, transformation signification = relating/comparing parts significate = added meaning from relating parts ..... ..... ..... God ..... ..... ..... job added value: God = job?, etc.

Roman Jakobson‘s poetical function: projecting the pardigmatic axis onto the syntagmatic axis.

This is a widespread misunderstanding about Symmetries are a local principle: The whole work is extremely seldom built in a symmetric way. E.g.: Opera „Hin und zurück“ (Forth and Back) by P. Hindemith. https://www.youtube.com/watch?v=yv9f0gdhSt8 This is a widespread misunderstanding about symmetries as „principle of totalitarian ordering“

J.S. Bach Musikalisches Opfer, BWV 1079 retrograde canon

Dodecaphonism = strategy with local symmetries

Boulez

Limited transpositions by a minor third motivic zigzag

motivic zigzag in op.106 1 0 -2 1 ( ) b. 75-78

b. 79-80

L 3

a e b Hindemith turbidity for minor = Zarlino duality + inner symmetry f c a♭ e♭ b♭

Example of a serial transformation described by Eimert Example of a serial transformation described by Eimert. It results from a counter-clockwise rotation by 45, followed by dilatation by 2.

Why are all symmetries musically reasonable? inversion: ok. retrograde: ok. retrograde inversion? ok, because it is concatenation of two symmetries that are ok! Principle: Concatenation of musically reasonable symmetries is musically reasonable.

Theorem (concatenation theorem): Every symmetry is a concatenation of some of these musically reasonable symmetries: Translation by one unit (typically: transposition by one semitone). m-fold dilation (typically: m-fold augmentation in time) reflection in one coordinate (typically: pitch inversion) shearing of second coordinate in direction of first (typically in arpeggio) exchange of two parameters (typically: time and pitch)

OrnaMagic

grid of translations f1 f2