Pythagorean tuning lattices

Pythagoras really liked threes. A lot. You probably remember this from school. What they don't tell you in grammer school is that the tuning ratio 3/2 is a perfect fifth. Pythagoras is creditted with discovering that. The story goes that he walked into a shop where metal smithing was going on and the smithing was very consonant and melodic sounding. He ordered the smiths to switch hammers and pieces of metal. In his experiments, he discovered that the tuning was not in the arms of the smiths or in the hammers, but in the strips of metal they were hitting. Somehow, he got from this discovery to 3/2, which is obviously completely perfect because it has a three in it and three is the most perfect of all numbers. Pythagoras really liked threes

Alas, it was downhill from there. You may recall the circle of fiths. It leads around to every note in the scale and is a nifty trick for remembering sharps and flats. So, theoretically, if you really liked fifths, you could tune every fifth from the fifth below it. The fraction for note N = 3/2 * (N-1). So the first note is 1/1. The second is 3/2. the third is 9/8. The fourth is 27/16. The fifth is 81/64. Every note is 3^^x/2^^y. This tuning was invented by Pythagoras and the list of notes you get from it it called a Pythagorean Tuning Lattice.

Not really a very good tuning system, but one used for hundreds of years. Anyway, Kyle Gann wrote a paper on tuning for beginners: http://home.earthlink.net/~kgann/tuning.html. In it, he says, "Equal temperament could be described as the musical equivalent to eating a lot of red meat and processed sugars and watching violent action films." I knew it!