The chromatic scale is composed of 12 notes. The most recent technical innovation of western music is the even-tempered tuning system, which uses logarithms[2^(1/12)] to establish even spacings between all intervals. While this system is only capable of approximating the "pure" Pythagorean integral ratios to within a certain error tolerance(the Pythagorean comma of [3/2]^12:2^7, or about 1.4%), it allows songwriters and composers greater freedom to change keys within a song without having to re-tune the instruments used or add cumbersome extra notes(G# AND Ab, for instance).


The DoveSong Foundation -- Don Robertson's Duochord -- -- - The next level
Ancient Greek Origins of the Western Musical Scale
Tuning Systems
Black Belt Guitar

Tuning Lattices: Advanced Theory

The lattice diagram
Explanation of John Chalmers's Lattice Diagrams

Try the Flash cadence plotter