ghgonnet@watdaisy.UUCP (Gaston H Gonnet) (07/23/83)
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 ...
jdd@allegra.UUCP (07/25/83)
A truly neat hack if you have a computer-controlled synthesizer or other
instument that can generate arbitrary frequencies is dynamic detuning of
notes sounded in combination to force them to be at exactly small-integer
ratios. Any static detuning is going to be a compromise, with some
combinations sounding pretty good and some not that great, but dynamic
detuning can make them all sound perfact. It works, too.
BTW, this is patented, so don't get in trouble.
Cheers,
John ("My Voice Detune Every Time I Sing") DeTreville
Bell Labs, Murray Hill