zooming out of Paris

Look at http://www.egroups.com/files/harmonic_entropy/triad2p40000.jpg --
here the range is expanded to two octaves (to keep the number of points
manageable I had to reduce the product limit to 40000).

In-Reply-To: <CE80F17667E4D211AE530090274662729C54FA@...>
Paul Erlich wrote:

> Look at http://www.egroups.com/files/harmonic_entropy/triad2p40000.jpg
> --
> here the range is expanded to two octaves (to keep the number of points
> manageable I had to reduce the product limit to 40000).

Looking at this picture, I thought there seemed to be some self-similarity
in it. In particular, the triangle with corners 1:2:2, 1:2:3 and 1:3:3
looks like a smaller version of the 1:1:1, 1:1:2, 1:2:2 triangle.

So, is this exact? Well, moving from the large to small triangle means
transforming a:b:c into a:(a+b):(a+c). There will be a one-to-one mapping
if a lowest-number limit is enforced.

To check this, I altered my program to produce ratios within such a limit.
There isn't a problem with returning an infinite set, because the
restriction on the span sorts that out.

So yes, it looks like exact self-similarity if you choose the right limit.
Otherwise, it will only be approximate.

I don't know if this can be used to optimize the calculation, but it is
quite interesting.

Graham

Graham wrote,

>Looking at this picture, I thought there seemed to be some self-similarity
>in it. In particular, the triangle with corners 1:2:2, 1:2:3 and 1:3:3
>looks like a smaller version of the 1:1:1, 1:1:2, 1:2:2 triangle.

Thanks for taking the time to really look. I don't know if that's
self-similarity, but it sure is similarity.

>So, is this exact? Well, moving from the large to small triangle means
>transforming a:b:c into a:(a+b):(a+c). There will be a one-to-one mapping
>if a lowest-number limit is enforced.

Let's go back to diadic harmonic entropy. By analogy, you're saying there
should be a one-to-one mapping from the interval from 1:1 - 1:2 to the
interval from 1:2 - 1:3, by transforming a:b to a:(a+b) (this dyadic stuff
appears on the central "spines" of the triadic chart). This results in
similarity because if a:b is in lowest terms, a:(a+b) is in lowest terms --
and vice versa. And indeed, if you look at a harmonic entropy curve like the
ones I've been making, you do see some degree of similarity between the 1:1
- 1:2, 1:2 - 1:3, and 1:3 - 1:4 segments (though the curves don't look quite
the same since s is presumed to be a constant (logarithmic) interval rather
than a constant frequency difference).