Non-Pythagorean scale

Anyone make any sense out of this?

http://en.wikipedia.org/wiki/Non-Pythagorean_scale

This is apparently what it sounds like:

http://www.youtube.com/watch?v=9TzZPsVMtYM
http://www.youtube.com/watch?v=sjGMji0Zzis

--- In tuning@yahoogroups.com, Mats Öljare <oljare@...> wrote:
>
> Anyone make any sense out of this?
>
> http://en.wikipedia.org/wiki/Non-Pythagorean_scale
>
> This is apparently what it sounds like:
>
> http://www.youtube.com/watch?v=9TzZPsVMtYM
> http://www.youtube.com/watch?v=sjGMji0Zzis

Looks like it's based on frequencies which are logs of integers, rather than integers as in the harmonic series.

One interesting consequence of this is that, for any note in the scale, it actually does contain all the harmonics of that note (in addition to a bunch of other notes). For example, if you start with log(2) as your 1/1, the notes of the scale are

1/1, log(3)/log(2), 2/1, log(5)/log(2), log(6)/log(2), log(7)/log(2), 3/1...

which will eventually hit 4/1, 5/1, and all other harmonics, but it has increasingly many irrational frequencies in between.

The official page, http://www.applesinstereo.com/pythagorean.php , has a number of errors in it:

* "Successive pitches in the logarithmic scale grow closer and closer to one another, and the number of distinct tones in each octave increases nearly exponentially, with each successive octave."

"Nearly exponentially" is quite an understatement. The number of tones in each octave grows *doubly* exponentially; that is, its asymptotic growth rate is O(exp(exp(N))) rather than O(exp(N)). The growth is exponential in the ordinary harmonic series, but in this scale the numbers of notes grows yet exponentially faster than *that*.

* "The C two octaves above middle C is tone 64*, so there are 48 tones in the octave beginning with tone 16*. The next C is tone 256*, so there are 192 tones in the octave beginning with 64*. The next octave of C is 1024*, and so on."

This is full of mistakes. The C two octaves above middle C is 256*, not 64*. 64* would be a G because log(64)/log(2) = log(2^6)/log(2) = 6/1. The next C above that is 2^16 = 65536*, and the next C above that is 2^32 = 4294967296*. The numbers grow very quickly because it's doubly exponential. 1024* would be an E because log(1024)/log(2) = 10/1.

I evaluated the idea that this paragraph was correct and the errors were in the rest of the page, but that doesn't seem possible because it's so consistent otherwise. If you assume this part is correct then there have to be many more mistakes elsewhere for it to make sense.

I would email Robert Schneider about this, but I can't find his email and suspect it might be hard to get his attention.

Gene: If you got this far, what do you call the smallest multiplicative group that contains all real numbers of the form log(A)/log(B) for integers A > 1 and B > 1? What's a minimal set of generators for it?

Keenan