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.