Visualizing the Convergence of Harmony.

So I've been getting back into music lately, trying to learn some jazz chords (and how to use them) on guitar. Oddly, I stumbled upon something interesting recently, something mathematical that applies to music.

Its the Monoid formed by the operator (x*y) mod n. Simple enough, times tables reduced down by modulus to be self contained. By when I rendered it I saw a shape:



See the rippling spirals formed by the smaller numbers with one digit? I decided to render it, translating the numeric values to pixel darkness. It looked cool, and a pattern started to emerge as n got larger. Here it is for n = 913:



(This picture had to be shrunk for Blogger) What its interesting to look at is the contiguous points of light and darkness, the way they form "diamonds". They can be thought of as folding points. For instance, the point in the center corresponds to folding it in half, both ways (vertically and horizontally). The 4 diamonds around that correspond to folding in thirds both ways, into 9 sub-squares. There are diamonds for the fourth, fifth, sixth seventh folding points and so on, each getting smaller and less distinct.

The coolness of this is that it corresponds exactly to the principles of musical harmony. Folding a frequency in half (doubling it) gives the octave, 3 times the fifth interval, 5 times the major third, and so on. So each diamond in the picture corresponds to a point of harmony moving toward dissonance into the areas in between, at least in the old school tunings before equal temperament. I wonder what the 12-TET monoid would look like. (I mean I wonder where the 12-TET points fall on this)

The beauty of the age of computers is that you can make music in pretty much ANY mathematical system of harmony, you can break all the rules imposed by physical instruments. I wonder what it will sound like 50 years from now.