Enzymind Labs

Music is a temporal sequence, and many of the elements are limited recurrent sequences. Individual instrumental sounds, musical phrases, and whole compositions typically have both periodic and aperiodic elements that can be modeled as finite recursive systems. Some music forms, like the fugue, have several voices that recursively (transformed elements reoccur) interact with each other, and within the theme. The fugue's contrapuntal components are independent in contour and rhythm, and interdependent in harmony. Whithin the rythm, recurrent elements -- temporal, pitch progression, or other features -- can occur.

Some music has been analyzed by examining the allowed recurrent rules; see Chord progression rewrite rules.

Peter Requadt (PSR) showed me a looping pedal, a digital recording device with which a rythm can recurr by overdubbing onto an initial rythm. Within a minute or two he created a repeating rythm and then superimposed a temporally scaled fascimile within half of the rythm's time frame. The structure was something like this:

where the tempo of A and B is the same, and the tempo of A' B' is geometrically similar to A B at twice the tempo (such that they occupied half the interval). The notes played in A B and A' B' are roughly the same, but could be shifted/mirrored in pitch or other dimension.

[ To Do: Audio example of this rythm structure. Would be nice to do a spectrogram as a function of time too.]

In terms of L-systems this system can be modeled as the sum (in the logical AND sense, all voices played simultaneously) of generations one and two (not the zeroth generation, or initial condition) of the aperiodic sequence:

b; a -> {Null}, b -> a' b' , where the prime (') indicates a tempo doubling

This could be extended to more generations, and the same simple composition can be modeled by other related systems.

The looping pedal enables easy composition of this structure, but also repeats (periodically) the short sequence indefinitely.

Fri, 2007-10-26 20:36 — PSR

 Also see:

Self−similar syncopations: Fibonacci, L−systems, limericks and ragtime

 by Kevin Jones
There was an Old Man with a beard,
Who said, "It is just as I feared! −
Two Owls and a Hen,
Four Larks and a Wren,
Have all built their nests in my beard!"
[Edward Lear]

Making its first appearance a little over a century ago, the lure of the limerick is such that it has grown to
become one of the world's most popular verse forms. There is something strangely appealing and intuitively
"natural" about its slightly skewed symmetry.
At about the same time − in fact precisely 100 years ago in 1899 − the music of ragtime was let loose on the
world with the publication of Scott Joplin's Maple Leaf Rag, a seminal blend of Western art music traditions
combined with African and Latin syncopations. It deserves to be be highlighted as one of the most significant
events in the history of music − a trailblazer to the explosive development of contemporary popular music,
which is almost certainly one of the most widely−shared cultural experiences on the planet.
There are interesting symmetries shared by the limerick and ragtime, which can be observed and heard in their
family groups of stressed and unstressed syllables, or beats, and which lie at the heart of what gives these
forms their characteristic structure or "feel". They possess self−similar qualities which are related to fractal
models used by contemporary scientists, and can provide a keen insight into some quite profound
inter−relationships between the arts and sciences.

 

COMPOSING WITH CHAOS

Applications of a New Science for Music

David Clark Little
Cornelis Springerstr. 14-2
1073 LJ AMSTERDAM

 Abstact: In this paper the author shows where concepts and mathematical models derived from the developing field of Chaos Science can be applied to electroacoustic and instrumental composition. Examples of non-linear dynamics include Lorenz's model of fluid behaviour, Verhulst's model of population growth, Hénon's analysis of the multiple celestial body problem, Barry Martin's Algorithm which produces quasi-organic forms, and the 'Baker' mixing function. Besides broadening the numerical techniques available for electronic music generation, concepts such as fractal structure, feedback process and iterative function can be applied to 'ordinary' composition as well. For example, in designing melodic curve, defining meter, planning instrumentation, manipulating symbols, creating ornamentation and elaboration, etc. Some suggestions as to mapping are made, the critical boundary between science and art. Musical examples are used from the following works by the author: Harpsi-Kord for harpsichordist and tape, Fractal Piano for computer-guided pianola, Brain-Wave for recorder-players, The Five Seasons for 6 percussionists and tape, Modi-Fications for marimba & tape, and Hyperion's Tumble for tape.

Godel, Escher, Bach, Douglas Hofstadter:

Regarding scale-free networks and emergence, I've been re-reading Godel, Escher, Bach. It's extraodinarily rich on this topic. It makes a lot more sense to me now that I've been thinking a lot about formal systems and the number theory of sequences.