jgk@osc.COM (Joe Keane) (03/13/91)
In article <30870@usc> alves@calvin.usc.edu (William Alves) writes: >The theoretical possibility of temperament (irrational ratios in tuning) >goes at least all the way back to Aristoxeneus (3rd cent BC), but the >question was about the *mathematical details* of equal temperament. Appa- >rently, fretted instrument players were commonly using equal temperament >back to the fifteenth century. The most common method of deriving equal >temperament practically was to tune to an 18/17 semitone (about 99 cents). The `rule of 18' is an empirical rule which says that each fret should be about 1/18 closer to the nut than the previous one. People knew empirically that this procedure didn't exactly make an octave, and it's easy to show mathematically that (18^17)^12 != 2. I believe that they compensated by making each interval some fixed amount bigger than the `rule of 18' would suggest. Given this, i'd say they were shooting for equal temperament. Although they may not have had the mathematical concept of the twelfth root of two, they could approximate that ratio to the necessary accuracy. [stuff about correct mathematical explanation deleted] >Marpurg insisted that J.S. Bach had used equal temperament and accused >Kirnberger, a student of Bach's, of abandoning his teachings. Quite the >contrary, said Kirnberger, Bach advocated unequal temperaments and C.P.E. >Bach wrote a testimonial backing up Kirnberger in this regard. It's unfortunate that the advocates of equal temperament cited Bach as a strong supporter. Even today we hear people claiming that his "Well-Tempered Clavier" is a demonstration of equal temperament. This is not true. >Though of course we do not have a mention of any tuning system from Bach's >own mouth, the fact that he wrote preludes and fugues in every key does not >mean he advocated or used equal temperament. In fact, he calls it the WELL- >Tempered Clavier because the temperament is good enough to be played in >every key, but, I believe, unequal enough that the different keys still >retain some differences in the intervals, and, hence, the mood and >affectation. In unequal temperament, each key has slightly different intervals in the various positions. This is similar to the seven modes but at a higher level of detail. I think that people who were used to unequal temperament learned to associate these differences with the actual key. Therefore when they heard music in equal temperament they still heard those differences which weren't actually there. Those of us who grew up on equal temperament have no idea what they were talking about. This is all confused by the fact that modern instruments have a mixture of temperaments. Here are the ones i can think of off the top of my head: free form: voice, fretless strings (e.g. violin) fixed equal temparement: keyed strings (e.g. piano) equal temperament but flexible: fretted strings (e.g. guitar) harmonic but flexible: simple wind instruments (e.g. bugle) none of the above: valved wind instruments (e.g. trumpet) If you know how a trumpet is really tuned, then you know more than most people. One thing i can tell you is that it's not equal temperament. -- Joe Keane, amateur mathematician