Question for Gene & more news about linear BP

Hi again.

Gene may say I'm responding to something that has already been answered.
Well, this is just because I've still not stopped examining the interesting
properties of BP.

On March 12th, Gene wrote:

> The linear temperament I associate to BP is the "bohpier" temperament;
> the linear temperament obtained by tempering out 3125/3087 and
> 245/243, with wedgie <<13 19 23 0 0 0||. If the octave is a generator,
> the other generator will be a sharp 27/25 of size about 3^(1/13); if
> we want to use 3 as an equivalence instead, we get a period of
> 3^(1/13) as a generator, and can use the octave as the other
> generator, hence we get the same generators looked at two different
> ways. If we get rid of 2 altogether, we don't have a linear
> temperament, but an equal temperament, obtained by dividing 3 into 13
> equal parts.

1. could you enlighten me a bit more about "bohpier"? I'm not sure if I got
the idea.

2. I'm not sure about where you got octaves. As far as I could find out,
none of the commas which can be tempered out come from 2/1.

3. Actually, I managed to find three possible generators usable for the
linear tempering of BP (leaving the period at 3/1, of course). First (and I
think this is the most "powerful" version), to temper out 245/243, the
optimum generator is "(15/7)^(1/3)" which flattens both 7/3 and 5/3 by 1/3
of that, allowing a pure 7/5. Then, to temper out 3125/3087, the optimum
generator is "(7/3)^(1/5)" which sharpens the 7/5 and flattens the 5/3 by
1/5 of that, allowing a pure 7/3. And finally, to temper out 16875/16807,
the optimum generator is "(27/5)^(1/5)" which makes both 7/3 and 7/5 sharper
by 1/5 of that, allowing a pure 5/3. -- Interestingly, if you choose any of
these as a generator (and 3/1 as a period), the resulting scales look all
very similar if you stack 12 of them (i.e. make a 13-tone scale). Even
better, they all meet the condition of "two intervals per class". They only
differ in the numbers of large and small intervals.

Petr