[<<][math][>>][..]Wed Jun 10 18:57:46 CEST 2009

One of those ideas that disappeared into the background.. Let's try again. A finite field[1] or Galois field is a field[2] with p^n elements, where p is a prime number. There is only one GF(p^n) upto isomorphism. The field p^n is the field of polynomials over p, modulo a reducable polynomial of order n. For music applications, it is probably possible to look at Galois fields in the same way as one would use the complex number field to represent sum-of-sines. Multiplicative cycles can be used as different rhytmic/harmonic components, and each element of the field could be associated to an instrument, or a combination of. For music we need the numbers 2, 3, 5 and 7 in copious quantities. Larger prime numbers are less interesting from a harmony/rhythm point of view. Given that we're looking for a certain combination of small prime numbers in the factorization of the multiplicative group, we can start typing things into a calculator as see if we end up at prime powers. 2^a 3^b 5^c 7^d = p^n - 1 A more systematic way is probably in order, i.e. generating[5] all power of primes below a certain number and investigating the factors of the multiplicative group order. However, to illustrate the idea let's just pick a few starting from the factorization. The first candidate is GF(211) which fits nicely in 8 bits. 2^2 . 3 . 5 . 7 = 211 - 1 Another one that fits in 8 bits is GF(241). 2^4 . 3 . 5 = 241 - 1 The question is: how to make this interesting? We can generate periodic sequences of numbers using formulas like: x_n = \sum a_i g_i^n where g_i are generators of cyclic subgroups, and a_i are arbitrary field elements. The first question is: given a particular representation of a GF, how many such sequences exist? Suppose we associate a certain configuration of drum sounds to each element in the field. How many patterns can be generated? Are they at all interesting? What about maps from the elements to a smaller number of drums? Maybe combinations of drums can be associated to terms in a polynomial. Is there an interesting way to combine a non-prime field with cycles this way? Maybe the interesting stuff lies in the symmetries of a field[7]? I.e. start with a simple rhythm. Find a (small) field to embed it in, then find variations of this by looking at the field's automorphisms. How can a GF be visualized? The structure of the additive group is quite straightforward: for GF(p^n) it is n times p cycles. How can the multiplicative group be shown on top of this? There's a visualization of GF(3^2) here[3]. Visualizing the the group as actions on a set seems like a good idea. Looks like a nice application for graph visualization[4]. Intuitively it seems that groups with small p and n in the same order of magnitude, will contain a more interesting structure. Instead of a single flat space (p^1) the structure is a vector space where each dimension has a separate entitiy. It allows the representation of instruments/notes (q) that can sound together in chords(n). I.e. 5 tones, 3 note-chords gives 5^3=15 with a multiplicative group of 14 = 2 . 7 elements. More concisely: order factorization add mul ---------------------------- 15 5^3 2 . 7 21 7^3 2 . 2 . 5 As a side note, maybe Galois Theory[7] is interesting to study in its own right, as it is connected to polynomials which are next to matrices the bread and butter of numerical math. [1] http://en.wikipedia.org/wiki/Finite_field [2] http://en.wikipedia.org/wiki/Field_(mathematics) [3] http://finitegeometry.org/sc/9/3x3.html [4] http://en.wikipedia.org/wiki/Graph_drawing [5] http://www.research.att.com/~njas/sequences/A000961 [6] http://en.wikipedia.org/wiki/Cyclic_group [7] http://en.wikipedia.org/wiki/Galois_theory

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