Deverloper Rob Fielding demonstrates his implementation of Pythagorean Tuning on his Pythagoras synthesizer, now in development for the iPad.
Pythagorean tuning is based on the idea of going around the circle of fifths, tuning intervals in perfect fifths. While this creates pleasing fifths, things get interesting as you go all the way around the circle of perfect fifths and octaves aren’t in tune. This has traditionally been accommodated by including one interval, the “wolf interval” that is left out of tune.
Here’s what Fielding has to say about Pythagorean Tuning in Pythagoras:
When I first drew out the just circle of fifths, I noticed that this was extremely close to a 53 equal note per octave tuning. This allows the strings to be tuned in exact Just fourths to each other and have the tuning be close (to the pixel) with the harmonic series.
The point of tuning systems is to closely match harmonic series, of which Pythagorean tunings (circle of Just 5ths) is the simplest and most ancient. This allows you to play fretlessly without a lot of inharmonic slop, because the fret width is about the same size as the inaccuracy of the touch itself. In Mugician, fretlessness was to the pixel which has inaccuracies of its own. In Pythagoras, the tuning is sub-pixel so that each spot is some reasonable ratio with fifths and octaves as prime factors (ie: 2^n * 3^m).
Using the iPad as a multi-touch software instrument allows for dynamic pitch assignment, which introduces both dynamic scale and dynamic tuning possibilities, not possible on traditional instruments.