A Letter from David Feldman

I received the following mail from composer and mathematician
David Feldman - one of our lurkers - who has allowed me to forward
it to the list:

I had a tuning idea the other day which led me to some computations which
I would be happy to share with you. Here is the idea. Suppose
that you are primarily interested in two intervals. Classically
you might think of r11 and r22, but any pair will serve. Then
you might think of dividing one of these intevals into equal
parts so as to produce a good approximation of the second. To
decide how many parts leads you to the irrational quantity
qog r1)/(log r2) which you must then approximate by a rational
number. Continued fractions provide the best tool.

All that is well known of course. The new (?) idea which currently
intrigues me is working with four intervals, r1, r2, r3 and r4
with
(log r1)/(log r2) (log r3)/(log r4) (approximately)
With such a foursome you could transform a musical structure which
depended on the first two intervals into one which depended on
the last two. For example

log (2/1)/log (3/2) 1.70951...

while

log (15/8)/log (13/9) 1.70945...

so up to a point (5, 12, 41, 53, ...) subdivisions of the
octave that produce good fifths corresponds to subdivisions
of a 15/8 that produce good 13/9 's. I call the
pairs (2/1,3/2) and (15/8,13/9) (mutually) isotempered.

The table I used to learn this fact (and many others like it)
I made as follows. I computed qg(r1)/log(r2) for every
r1 and r2 (in lowest terms) with denominator up to 10
and numerator not more then 10 greater then denominator.
Then I sorted the table by the magnitude of q. Finally
I computed the difference between consecutive lines in order
to look for good isotempered pairs. The table includes
the continued fraction partial quotients of each q. I made
it all with a simple maple program.

The format is very wide (it all fits on my new wide monitor screen)
which may be inconvenient for you. Let me know if you need
it reformated. I will send the table under separate cover.

David



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