Mathematical curiosity -- what does it mean?

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. It seems like another crazy
coincidence.



------ =_NextPart_000_01BD9EF4.AF2A0A7E
Content-Type: application/ms-tnef
Content-Transfer-Encoding: base64
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------ =_NextPart_000_01BD9EF4.AF2A0A7E--