RE: A program for the optimization of 12 tone tunings

Jeff Welty writes:

> Anyone interested?

Sure. Maybe you can tell us how your program works, what is the
expression you are optimising. Is it linear or nonlinear?
What are the 12 note weights you mentioned?

Manuel Op de Coul coul@ezh.nl

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Message written at 20 Dec 1995 21:16:51 +0000
In-reply-to: <0099B1E361BED53A.4D16@ezh.nl> (message from Manuel Op de Coul on
Tue, 19 Dec 1995 12:48:28 -0800)

Manuel has covered most of the points, but I though I would add that
my friend Yasamasa Kanada holds the record for calculating teh digits
of pi, and teh apparently random nature extends all the way to the 4
billion digits he has calculated and verified. (actually 4,294,960,000
digits are certainly correct)

Frequency distribution for pi-3 up to 3,000,000,000 decimal places:
'0' : 299999143; '1' : 299995932; '2' : 299989126; '3' : 299992290
'4' : 300002257; '5' : 299979016; '6' : 300025447; '7' : 299975510
'8' : 300016550; '9' : 300024729; Chi square = 9.24

Frequency distribution for 1/pi up to 3,000,000,000 decimal places:
'0' : 300009029; '1' : 300002431; '2' : 299992729; '3' : 299998519
'4' : 299970883; '5' : 299980175; '6' : 300010123; '7' : 300025696
'8' : 300008447; '9' : 300001968; Chi square = 7.40

Yasamasa tells me he has gne way beyond this record but the details
are embargoed until January, but the distributions seem the same.

What is always odd is that the other well-known transcendental number
e is very un-random. Strange. My father always used to ask me to
give him a transcendental which was not pi, e or multiples; after all
there are very many of them. I agree that e+pi is not decided --
there is a theorem by W.S.Brown which is significant for computer
algebra based on a conjecture which has e+pi being transcendental as a
specal case, and I have not heard that it was decided. My PhD depends
in part on Gelfond-Schneider theorem, and I used to search this stuff.

Oh well, enough memory lane for tonight.

==John ff

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