Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10 5/3/83; site watdaisy.UUCP
Path: utzoo!watmath!watdaisy!ghgonnet
From: ghgonnet@watdaisy.UUCP (Gaston H Gonnet)
Newsgroups: net.math
Subject: tempered scale
Message-ID: <236@watdaisy.UUCP>
Date: Fri, 22-Jul-83 19:18:19 EDT
Article-I.D.: watdaisy.236
Posted: Fri Jul 22 19:18:19 1983
Date-Received: Sat, 23-Jul-83 00:26:09 EDT
Organization: U of Waterloo, Ontario
Lines: 18

The ability to produce a tempered scale lies on the approximation
of small fractions between 1 and 1/2.  As mentioned in the net the
most important are 2/3, 3/4, then 3/5 and 4/5 and so on.
If we can represent the log2 of these fractions as an approximate
rational, the denominator of the rational is an appropriate order
of the scale.

	It turns out to be that the log2 that matters the most is
log2(3), second is log2(5) and so on.
The convergents of log2(3) are: 2, 3/2, 8/5, 19/12, 65/41, 84/53, ...
and those for log2(5) are: 2, 7/3, 65/28, ...
After this, it is rather obvious that the best choice is 12 (given the
particularly "lucky" circumstance that 7/3 is a good approx of log2(5)).

	Five notes are also possible, a *very* simplified scale, maybe
the dream of many children.  The next best choice after 12, which does
better on the log2(5) would be lcm(12,28)=84.  To do better on the simple
harmonics we have to go to 41 notes, then 53, then 306, then 665 then 15601 ...