Re: [tuning-math] Digest Number 2230--irrational numbers

Virtually any irrational number can be used for musical tuning with a little ingenuity:

Feigenbaum's alpha, 2.5029+ over 2 produces an interval of 388.326 cents, a pretty
good major third.

Feigenbaum's delta, 4.67692+ over 4 yields 267.81, close to 7/6.

Apery's constant, Zeta(3) or 1.20205+ is a good approximation to 6/5

The square root of 3, 1.732+ is about 950.98 cents, near 26/15.

(The irrationals above were found in an interactive table in Wikipedia, by following
the Euler-Mascheroni constant link on the irrational number page ref given by Charles
Lucy.)

Hence, exotic irrational numbers can generate scales audibly equivalent to JI, at least
over restricted ranges. Similarly, as McLaren has shown, unusual divisions of integers
and real numbers can generate scales audibly equivalent to familiar ET's.tu

--- In tuning-math@yahoogroups.com, John Chalmers <JHCHALMERS@...> wrote:
>
> Virtually any irrational number can be used for musical tuning
...
> and real numbers can generate scales audibly equivalent to familiar
> ET's.tu
that's due to the
http://en.wikipedia.org/wiki/Just_noticeable_difference
limit of the human ear in
http://en.wikipedia.org/wiki/Pitch_%28music%29
perception alike:
http://en.wikipedia.org/wiki/Absolute_threshold_of_hearing

Quest:
How many digits of
http://en.wikipedia.org/wiki/Pi
can you discriminate barely by ears alone?

For the purpose of hearing the
http://en.wikipedia.org/wiki/Numerical_approximations_of_%CF%80

22/7

appears to be sufficicient precisely
and can replace PI adaequate.

Yours Sincerely
A.S.