source file: m1456.txt Date: Tue, 23 Jun 1998 18:17:18 -0400 Subject: Mathematical curiosity -- what does it mean? From: "Paul H. Erlich" This message is in MIME format. Since your mail reader does not understand this format, some or all of this message may not be legible. ------ =_NextPart_000_01BD9EF4.AF2A0A7E Content-Type: text/plain In TD 999, I posted a derivation of optimal tunings of the diatonic scale, and in for my paper I did the same for decatonic scales. These tunings were derived by minimizing the mean squared error of all intervals in the triad or tetrad, respectively. Both scales are generated by an approximation to the 3/2, with a fixed half-octave (square root of two) also required to generate the decatonic scale. The results (found also in my paper) give the optimal approximation of the 3/2 as a fraction of an octave: (a+b*log(3)+c*log(5)+d*log(7))/n octave where all logs are to the base 2. If we choose the smallest possible integer n such that a, b, c, and d are all integers as well, then the values are: diatonic: a=2, b=-2, c=7, d=0, n=26 decatonic: a=61, b=14, c=d=-10, n=54 Now the expression above can also be interpreted as giving a (non-integral) number of steps in n-equal temperament. So the diatonic 3/2 is given as a number (15.0836) of steps in 26-tET, and the decatonic 3/2 is given as a number (31.8966) of steps in 54-tET. What's interesting is that 26-tET and 54-tET are just barely "overtempered" tunings for the respective scales. In other words, the 3/2 in these tunings is far enough from just that bringing it closer to just will decrease the worst error without increasing the second-worst error in the tuning, until the two are equal, and continuing to bring it closer to just will result in the optimal tuning. However, neither 26-tET nor 54-tET are wholly unacceptable for the respective scales, the worst errors being 17.08 cents (5/4) and 17.69 cents (6/5), respectively. (Note that 54-tET has a better approximation of 6/5 than the one that occurs in the decatonic scale, making it potentially confusing to use in practice.) I can see no reason why n should turn out to be an equal temperament that is suited for the scale in question. 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