Bach apparently encoded his "well-tempered" tuning formula into a squiggle of loops & knots on the cover page of his "Well-Tempered Clavier" music. The subsequent music in the book is itself is a "proof" that his tuning formula works, by running through the various chords & progressions. Bach's diagram/cryptogram was apparently overlooked as a mere ornament for the past nearly 300 years, until musician/software engineer Bradley Lehman decoded this tuning diagram. Yes, equal tempering has the advantage of removing the differences of different key signatures, but also the disadvantage (for the composer) of taking advantage of different keys for different stylistic effects (including removing boredom) -- following the practise of the medieval "modes". I'm sure some mathematician has studied music tuning formulas in terms of continued fractions -- does anyone know any references? Some interesting fractions from the paper: 531441/524288 = [1, 73, 3, 2, 1, 1, 1, 23, 2, 5] 81/80 7/11 = [0, 1, 1, 1, 3] 128/125 = [1, 41, 1, 2] 1/11 http://em.oxfordjournals.org/content/33/1.toc