RE: [tuning] Re: moving on to "scales"; well-temperament wazoo

John deLaubenfels wrote,

>Paul, are you considering all pairs of notes in your "relaxing" process,
>or just fifths and thirds (major and minor), or what?

All pairs of notes!

>>Let's say fifths were all that mattered (not a bad approximation).

>I'm having trouble making sense of this statement. Tuning would be SO
>much easier if this were true!

It's not a bad approximation because by far the strongest feature in the
harmonic entropy curve is the dip at the fifth.

>>There's a pattern to all the well-temperaments that deviate
>>significantly from 12-tET -- can anyone spot it?

>Yeah, sure: all the >700 cent fifths are bunched together, followed by
>(or preceded by) all the <700 cent fifths.

Well, that's essentially the same pattern that Ed Foote mentioned, but there
is another, subtler feature that I was referring to.

>It's not clear to me what value this process has, when slight deviations
>in the starting conditions cause the relaxation to go in so many
>different directions, including collapsing into 11 notes.

I was trying to get a better sense of the "landscape", since all I knew was
that my program was claiming to find a local minimum at 12-tET, and another
local minimum at a particular well-temperament. Well, after this experiment,
it became clear that there is a whole range of well-temperaments, ranging
from 12-tET to tunings where the worst fifth is upwards of 710 cents, that
come out as local minima. Since (I did not mention this yet, so I'm glad you
asked) the total discordance of these scales was greater the farther they
were from 12-tET, it is clear that 12-tET is the only true local minimum
here. I can't be too upset at my optimizer for not always converging to
12-tET, since 11-dimensional space can be hard to navigate. However, there
may be some significance to the fact that all the well-temperaments
significantly different from 12-tET that the optimizer got stuck at share a
common pattern, beyond the obvious one you mentioned -- anyone see it?

John deLaubenfels wrote,

>Paul, I must be misunderstanding you, because I seem to hear you saying
>two contradictory things:

> . Because the biggest dips in the harmonic entropy curve are at
> 3/2, it's "not a bad approximation" to say that this is all that
> matters. You further emphasize this by saying that, between
> major and minor seconds, the curve is "featureless".

> . But, when I suggest you reduce the strength of minor seconds and
> eliminate the unison dip to keep the scales from collapsing into
> others with fewer notes, you say you don't want to "bias the
> results."

John, I think what you're missing is, the 12 tones (well, effectively only
11) are free to move aroung wherever they want, and there's nothing (in
principle) to stop two adjacent tones from ending up, say, a minor third
apart. But if I start out reducing the weight on the intervals between
adjacent tones, because of an assumption that they're minor seconds and
therefore don't matter, I'll be biasing the program against finding a
solution where adjacent tones are a minor third apart.

>You say you're looking for "interesting new scales". Could you better
>define what makes a scale "interesting" in your view?

Lots of possible things, but one of them is being a local minimum of
discordance.

>How do you test for convergence in a particular run of your optimization
>program? The results you posted have precision to .01 cents in some
>values, but only to integer cents in others - what does this mean?

I have used parameters of 10^-8 as the convergence tolerance for both the
notes and the discordance value. The accuracy, though, is effectively
limited to 1 cent because I only calculated the harmonic entropy curve in 1
cent intervals, and use linear interpolation to make it a continuous
function. Maybe that explains the "sticky" well-temperaments?