From: mclaren
Subject: a new iterated function for
generating non-just non-equal-tempered
tunings
While boring holes for resonators in a
5-tone-equal percussion instrument to be
installed at the Exploratorium, a new
iteration function occurred to me. Since
these functions are a fertile breeding
ground for non-just non-equal-tempered
scales, this one might prove of interest
to forum subscribers.
Operating an industrial drill press is
extremely peaceful work--excellent for
mathematical contemplation.
The function is an alternating series:
start with a number, take the tan(x),
and whenever it drops below 1.0, take
ex.
The first 10 terms of the function are:
- i[1] = abs(tan(sqrt(2))) = 6.334119167...
- i[2] = abs(tan(6.334119167)) = 0.05097795...
- i[3] = abs(e(0.05097795)) = 1.05229964...
- i[4] = abs(tan(1.05229964)) = 1.752641506...
- i[5] = abs(tan(1.752641506)) = 5.438434336...
- i[6] = abs(tan(5.438434336)) = 1.126353452...
- i[7] = abs(tan(1.126353452)) = 2.099871982...
- i[8] = abs(tan(2.099871982)) = 1.710348942...
- i[9] = abs(tan(1.710348942)) = 7.1119178021...
- i[10] = abs(tan(7.1119178021)) = 1.106677438...
I believe but cannot prove that all of these
numbers are transcendental. Numbers > 2.0
when octave-reduced, and < 1.0 when added to 1.0,
produce a musical scale:
- p[1] = 795.7728117 cents
- p[2] = 86.07888146 cents
- p[3] = 88.25468083 cents
- p[4] = 971.4371163 cents
- p[5] = 531.8296507 cents
- p[6] = 205.991463 cents
- p[7] = 164.8027358 cents
- p[8] = 929.1488291 cents
- p[9] = 996.2863799 cents
- p[10] = 175.4817393 cents
As usual, there does not appear to be any
obvious pattern in these numbers.
Another number arises from this series:
1 + the number of successive iterations
required for switchover between ex
and abs(tan(x)), or vice versa. That
number is:
- 1.21311151312121913141313111313...
This number also appears to be
a transcendental. Can you prove it?
As a musical interval this equates to
a neutral third of 334.4597716 cents.
This is a third which has not to
my knowledge appeared previously
in the musical literature.
--mclaren
