Re: [tuning] The elusive second skhisma

Keenan,

If you mean by "simpler", with smaller numbers, there is none
(if you don't count 1/1). Successively better approximations
to an irrational number will have higher and higher numerators
and denominators.

Manuel

Keenan P. wrote,

>Help me out; does anyone know the simplest 5-limit ratio, other than the
>32805/32768 skhisma, smaller than 2 cents?

According to http://www.kees.cc/tuning/s235.html), it is
274658203125/274877906944, or 1.38 cents. The simplest one smaller than that
is 7625597484987/7629394531250, or 0.86 cents.

>Correct me if I'm wrong, but
>doesn't there have to be one, since there's never a "best" rational
>approximation to an irrational number (there's always a better one)?

You're correct!

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>According to http://www.kees.cc/tuning/s235.html it is
>274658203125/274877906944, or 1.38 cents. The simplest one smaller than
that
>is 7625597484987/7629394531250, or 0.86 cents.

Thank you. I found the first one independently by fooling around in Scala.
5^15*3^2, or the difference between 15 thirds plus 2 fiths and 7 octaves.
Might I ask why your values are both "upside-down" (inverse)? Their
denominators are larger than their numerators, so they're both smaller than
one.

>You're correct!

Obviously. :) I was afraid I hadn't been clear enough until I read your
message. You understood and clarified for other people.

Stay Tuned,
Keenan P.