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Thursday, December 17, 2009
Tuesday, February 10, 2009
Discrete Group Theory, Finite State Automata and music.
Idea:
A finite state automaton accepts two initial values (or a single 2 letter word, however you want to look at it).
The initial values are 2 numbers between 1 and 128 (the numbers correspond of course to MIDI notes).
Then, then the automata uses the group operation (via a lookup matrix or whatever...) to reduce the 2 numbers resulting in a single number (the "product" of number 1 and number 2 via the group operation). The output is now the second number, the old second number becomes the new first number. This process then repeats ad infinitum in the same sort of way that fibinacci numbers are generated recursively/iteratively.
Some details to work out:
1) Are only notes mapped to the group elements? Or perhaps volumes or durations too? Would they be produced via seperate processes into a long list and then the ordinal values of the list would be combined into tuples? Or would each group element be assigned to a single note/volume/duration tuple? If the latter the group would need around 128 * 128 * 32 + 128 elements. (128 notes, 128 volumes, 32 types of note durations - I made 32 up, sue me - and 128 types of un-notes, i.e. rests - I also pulled the 128 un-notes out of my ass, but you get the idea.) This is 524416 elements. Ssubscript9 has 362880 elements. Too little. Ssubscript10 has 3628800, far too many. Not sure what group would work best. Some subgroup of Ssubscript10. Or you could up the amount of volume possibilities to be up to 256 or something. This would give a value of 1,048,704 elements. Asubscript10 has 1,814,400 elements which is getting closer... you get the idea.
2) What pair of initial values produces the longest melody before the melody repeats? I am assuming that since it is a finite group there will eventually be a repeat. My knowledge of group theory is novice at best so this assumption may not be true.
3) Which mapping between musical tuple elements and group elements will produce the most interesting melodies? Is there anyway to make educated guesses rather than poke around randomly?
4) Will melodies wander and then get trapped in subgroups? e.g. if there is a cyclic subgroup of order two contained in the more massive group (which I would imagine would not be uncommon) then if the automata stumbled into the subgroup the melody would eternally cycle between the two values of the subgroup. How to avoid this issue? Or at least have a method whereby the initial 2 seed values garauntee a looooong melody before the problem is encountered. Is this overall issue a solvable/decidable problem?
5) what algorithm can be used to reduce to elements given that groups of the order of x will require a matrix of x^2 elements - using a matrix/lookup table is not efficient when the group is Ssubscript10!
NOTE: googling around about this idea lead me to this very interesting looking book:
http://www.worldscibooks.com/mathematics/7107.html
(and also this one called the "Topos of Music" but I am not quoting it: http://www.springer.com/birkhauser/mathematics/book/978-3-7643-5731-3)
A finite state automaton accepts two initial values (or a single 2 letter word, however you want to look at it).
The initial values are 2 numbers between 1 and 128 (the numbers correspond of course to MIDI notes).
Then, then the automata uses the group operation (via a lookup matrix or whatever...) to reduce the 2 numbers resulting in a single number (the "product" of number 1 and number 2 via the group operation). The output is now the second number, the old second number becomes the new first number. This process then repeats ad infinitum in the same sort of way that fibinacci numbers are generated recursively/iteratively.
Some details to work out:
1) Are only notes mapped to the group elements? Or perhaps volumes or durations too? Would they be produced via seperate processes into a long list and then the ordinal values of the list would be combined into tuples? Or would each group element be assigned to a single note/volume/duration tuple? If the latter the group would need around 128 * 128 * 32 + 128 elements. (128 notes, 128 volumes, 32 types of note durations - I made 32 up, sue me - and 128 types of un-notes, i.e. rests - I also pulled the 128 un-notes out of my ass, but you get the idea.) This is 524416 elements. Ssubscript9 has 362880 elements. Too little. Ssubscript10 has 3628800, far too many. Not sure what group would work best. Some subgroup of Ssubscript10. Or you could up the amount of volume possibilities to be up to 256 or something. This would give a value of 1,048,704 elements. Asubscript10 has 1,814,400 elements which is getting closer... you get the idea.
2) What pair of initial values produces the longest melody before the melody repeats? I am assuming that since it is a finite group there will eventually be a repeat. My knowledge of group theory is novice at best so this assumption may not be true.
3) Which mapping between musical tuple elements and group elements will produce the most interesting melodies? Is there anyway to make educated guesses rather than poke around randomly?
4) Will melodies wander and then get trapped in subgroups? e.g. if there is a cyclic subgroup of order two contained in the more massive group (which I would imagine would not be uncommon) then if the automata stumbled into the subgroup the melody would eternally cycle between the two values of the subgroup. How to avoid this issue? Or at least have a method whereby the initial 2 seed values garauntee a looooong melody before the problem is encountered. Is this overall issue a solvable/decidable problem?
5) what algorithm can be used to reduce to elements given that groups of the order of x will require a matrix of x^2 elements - using a matrix/lookup table is not efficient when the group is Ssubscript10!
NOTE: googling around about this idea lead me to this very interesting looking book:
http://www.worldscibooks.com/mathematics/7107.html
(and also this one called the "Topos of Music" but I am not quoting it: http://www.springer.com/birkhauser/mathematics/book/978-3-7643-5731-3)
APPLICATIONS OF AUTOMATA THEORY AND ALGEBRA
Via the Mathematical Theory of Complexity to Biology, Physics, Psychology, Philosophy, and Games
by John L Rhodes (University of California at Berkeley, USA)
edited by Chrystopher L Nehaniv (University of Hertfordshire, UK)
foreword by Morris W Hirsch (University of California at Berkeley, USA)
This book was originally written in 1969 by Berkeley mathematician John Rhodes. It is the founding work in what is now called algebraic engineering, an emerging field created by using the unifying scheme of finite state machine models and their complexity to tie together many fields: finite group theory, semigroup theory, automata and sequential machine theory, finite phase space physics, metabolic and evolutionary biology, epistemology, mathematical theory of psychoanalysis, philosophy, and game theory. The author thus introduced a completely original algebraic approach to complexity and the understanding of finite systems. The unpublished manuscript, often referred to as "The Wild Book", became an underground classic, continually requested in manuscript form, and read by many leading researchers in mathematics, complex systems, artificial intelligence, and systems biology. Yet it has never been available in print until now.
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