**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 EnvelopeThe 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:

- random walks
- sieves
- clouds of sounds (stochastic distributions)

**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:

- encode the problem as a string
- generate a random initial population over possible strings
- evaluate the firness of each string using an objective fitness function
- repeat
- choose strings for reproduction based on their fitness

**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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