Re: moving on to "scales"; well-temperament wazoo

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

[Paul Erlich, archive 11800 (TD 756.5):]
>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!

[Paul, archive 11813 (TD 756.18):]
>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.

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.

JdL

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

[Paul Erlich:]
>All pairs of notes!

OK. In what relative strength? You don't say so, but I'm guessing
that you consider minor seconds as strongly as fifths. In real-life
music, they are of course much more rare, and I think your process would
yield better results if you reflected that.

Also, the giant dip in entropy at unison would be better off eliminated
for purposes of this relaxation process, IMHO, since it's often pulling
two notes into each other, which is clearly not wanted in a scale.

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

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

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

BUT, if you considered only fifths, it's clear to me you'd fall back
into 12-tET in a hurry! I suspect the same thing would happen if you
considered only fifths and thirds.

[Paul:]
>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.

Well, given a requirement to optimize for all keys equally, 12-tET
undoubtedly IS the best possible tuning, and that result would fall out
by any number of optimization processes.

>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.

Yes, I do appreciate that!

>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?

Why don't you just share it with us? I suspect that what's making your
process converge in so many different ways is the major and minor
seconds, which in real-life aren't what have pressing need to be
fine-tuned.

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

> OK. In what relative strength? You don't say so, but I'm guessing
> that you consider minor seconds as strongly as fifths. In real-life
> music, they are of course much more rare, and I think your process
would
> yield better results if you reflected that.

I don't want to bias the results by considering music that has been
made with tunings only in a very small region of the 11-dimensional
space! I'm looking for new scales. You already know how to find the
best tuning for a particular piece of 12-tET-ish music -- that's not
the point of this.
>
> Also, the giant dip in entropy at unison would be better off
eliminated
> for purposes of this relaxation process, IMHO, since it's often
pulling
> two notes into each other, which is clearly not wanted in a scale.

But I won't find any meaningful new local minima by doing that, so
I'd rather not do that and just choose starting points appropriately.

> >It's not a bad approximation because by far the strongest feature
in
> >the harmonic entropy curve is the dip at the fifth.
>
> BUT, if you considered only fifths, it's clear to me you'd fall back
> into 12-tET in a hurry! I suspect the same thing would happen if
you
> considered only fifths and thirds.

Which is pretty much all that matters, since those intervals (and
their octave-equivalents) account for the lion's share of the
features of the harmonic entropy curve), and yet the optimizer, which
should be falling back into 12-tET, usually gets stuck on one of
those interesting well-temperaments.

> Why don't you just share it with us? I suspect that what's making
your
> process converge in so many different ways is the major and minor
> seconds, which in real-life aren't what have pressing need to be
> fine-tuned.

John, remember, the harmonic entropy curve is featureless around
minor seconds, major seconds, and everything in-between. So this
can't be right. Anyway, I'll hold out some more in case anyone can
spot it.

[response to Paul Erlich, TD 760.24 (archive 11903)]

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."

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

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?

JdL