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Comments

• [1] js from princeton October 12, 2009 - 06:40PM

  I appreciated your comment that "the math majors our there" would like to know that well-tempered tuning was based on the 12 root of 2, which you regarded as quite esoteric.

  I've always wondered how well tempered tuning works, but your hint provided a huge leap forward. Musical intervals are naturally multiplicative (an octave represents a doubling of the frequency), so taking roots represents a way of defining natural interval. Since we have twelve steps to a western harmonic scale, the twelfth root of a doubling represents the smoothest increment between two adjacent notes (a multiple of about 1.05946).

  By the way, I thought I heard you say that well tempered tuning has been around for about a "century and a half". Bach would seem to contradict that, and according to Wikipedia, the mathematics of even-tempered scales was known more than a century before Bach. Bach was not even the first composer to call for well-tempered instruments in his compositions.

  Whether Bach himself played a "well tempered clavier" remains a matter of conjecture, but Bach did tune his own instruments, and expressed disatisfaction with other tunings.

  There is a well known Irish folk group, the Wolftones, who derive their name from an anomaly in early tuning. Prior to the well tempered age, certain intervals were dissonant and jarring, particularly the sixth, which was known as the wolf tone. And so this group, which has championed the cause of Northern Irish Catholics, declared themselved to be out of tune.

  
  
• [2] John Schaefer from WNYC October 13, 2009 - 11:27AM

  JS, thanks for the comment - and I never knew that was where the Wolftones got their name from... BUT, you are perilously close to opening a huge can of tempered worms by using the term "well tempered" instead of "equal tempered." Equal temperament has been the norm since about the mid 19th century, but Bach and those after him often used "well tempered" scales, which are not quite the same thing. In well-tempered scales, all the keys are usable but there are subtle differences in some of the tunings, especially as you get further from the key of C major. In equal-temperament, all the scales have exactly the same shape - no differences at all. Equal-temperament is the one based on the 12th root of 2.

  You CAN say that equal temperament is a type of well-tempered scale, but you can't say the reverse.

  There. I've just made my own eyes glaze over.