Mathematical Tools in Music II

Group Theory. This very abstract branch of Mathematics was used by Xenakis in Nomos Alpha, for solo cello. Musical "variations" based on a rather limited number of elements are produced by the rotation of two cubes whose vertices represent a combined value of three sound parameters (the meeting edges). Two such cubes rotating inside each other, bring the total number of parameters to six. The rotations (or transformations) themselves are described by a group structure.

Foot note. Sometimes considered an abstract speculation without practical consequences, group theory proved to be the key to solving the puzzle of the ever growing number of sub-atomic particles which were being discovered 30-40 years ago. On a more philosophical level, groups describe symmetries or invariances under certain transformations. Today too, physicists are looking for deeper symmetries which will help us to understand the fundamental structure of matter.

Sound parameters. After WWII some composers attempted to control in their compositions all sound parameters. The most obvious sound qualities or parameters are:

        Duration                Time interval
        Pitch                   Frequency
        Dynamics (loudness)     Amplitude
        Timbre                  Waveform
        Articulations           Envelope
The first column above includes terms used by musicians; the second, their correspondent description in acoustics. Although widely accepted, this chart is not without flaw since it represents a rather crude description of the qualities of a sound as it was proposed in late 19th century by Helmholtz. More about this to follow soon.

It is interesting to observe that, in the Western musical culture, pitch and rhythm were practically the only ingredients in the music written before the 16-17th centuries. The Baroque period introduced dynamics as an important element of music while only in the 19th century composers started paying serious attention to timbre (orchestration). A strong interest in composing the way articulations are applied is very recent. Historically, the list of parameters has increased from 2 to at least 5 in the last millenium.

Game Theory. Two participants agree to play, one move at a time, according to a matrix of payements which determines what the second player gains or looses after each participant moves once. Linaia Agon, based on the mythological musical duel between king Linos and Apollo, uses such matrices to direct the improvisation of the performers. It is Xenakis' answer to "aleatory music", a trend in the 60s, asking the musicians to improvise or make decisions during the performance based on rules given by the composer.

Random walks. Similar to Brownian motion, they describe random movements which might have a general direction, but usuallt are not easy to detect. Mists for solo piano by Xenakis uses:

Fractals. French mathematician Benoit Mandelbrot proposed the term fractal geometry for geometries which deal with spaces with fractional numbers of dimenssions. The concept has to do with the degree of detail we see and with the degree of "roughness" of a surface of volume. If one "zooms in" on such objects, one discovers shapes which never repeat themselves but, at different levels, are self-similar. This property was exploited by musicians in creating melodic lines which bring back similar sucession of intervals but never repeat exactly.

Chaos theory (Complex Dynamic Systems). Phenomena like whirlpools in a river or turbulence were considered "chaotic", and unpredictable until not too long ago. This theory finds general patterns which seem to govern such seemingly random occurences in a loose way. Nonlinear equations (involving terms squared or at higher powers) usually describe such processes. A number of composers have been interested in applying this in their music.

Genetic algorithms. Related to Cellular Automata and, to some extent, even to Markov chains, they:

Genetic algorithms have been used in composition mostly to generate melodies (strings of sounds) but also spectra (timbres) which evolve in less predictable ways thus creating more interesting and rich such timbres.

Catastrophe Theory. A theory first proposed by French mathematician Rene Thom which deals with sudden, abrupt, and discontinuous changes. Composers Aurel Stroe and David Rosenboom have used it in their works.

Information Theory. Born out of the desire to find out what is the most efficient way of transmitting messages, it is used in music either to analyze stylistic features of existing works or to controll the pacing of events in a piece of music. The latter is a permanent concern of composers of all times and aesthetic persuations.

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