a class of non-octave scales

This is a brief note about a certain family of scales which divide just
intervals into equal number of steps. I was interested in these scales:

5:4 divided into 5 equal steps of 77.26 cents
7:4 divided into 7 equal steps of 138.40 cents
9:4 divided into 9 equal steps of 155.99 cents
11:4 divided into 11 equal steps of 159.21 cents
13:4 divided into 13 equal steps of 156.96 cents

etc, some of which sound good to me.

The reason for the post to tuning-math is that I noticed the family has a
maximum step-size at (11:4)/11, and I wondered if there was a reason.
Expanding the family to include

(x:4)/x

where x is not necessarily an integer, I noted the step-size increases
with

(x/4)^(1/x),

and setting the derivative of this equal to zero I found the exact maximum
occurs at

x = 4e (where e is Euler's number, the base of natural logarithms etc)...

I'm not sure what an "equal division" into an irrational number of parts
really means, but if you can swallow it, there it is.

cheers --jon

--- In tuning-math@y..., jon wild <wild@f...> wrote:

> I'm not sure what an "equal division" into an irrational number of
parts
> really means, but if you can swallow it, there it is.

What it means is more easily appreciated if you don't look at it in
terms of divisions but in terms of generators. Then the maximum
generator of this kind is e^(1/4e) = 159.22 cents. One can certainly
use this as a scale without reference to a period of repition, but if
you want one, you may choose any such period you like; the result
will be the same.