I sent this post out on Valentine's Day, 2011. The referenced article is "Bach's extraordinary temperament: our Rosetta Stone—1" by Bradley Lehman. Currently, this article is hidden behind a paywall, but I do recall reading the article, so I was able to get it somehow. http://em.oxfordjournals.org/content/33/1.toc One of the basic theses of this article is that instead of complaining about the inconsistencies of various tuning methods for the piano, Bach developed a particular tuning method, and then wrote "The Well-Tempered Clavier" to demonstrate -- through his excellent music -- the advantages of his tuning method. Bach says, in other words, "don't complain about the tuning method; embrace it, and write music that sounds good for it". Here are some additional links for Lehman's work: http://www.larips.com/ http://www-personal.umich.edu/~bpl/larips/outline.html https://www.youtube.com/watch?v=__tbvLNH6FI https://www.youtube.com/watch?v=UT5B1nvxC8A http://en.wikipedia.org/wiki/The_Well-Tempered_Clavier http://www.juilliard.edu/journal/ongoing-quest-bachs-temperament?destination... http://www.hpschd.nu/tech/tmp/lehman.html

Date: Mon, 14 Feb 2011 12:30:44 -0800 To: math-fun@mailman.xmission.com From: Henry Baker <hbaker1@pipeline.com> Subject: JSBach as hacker: "Well" Tempered /= "Equal Tempered

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