Heuristics (Was: Hi Dave K.)

In-Reply-To: <a2g36a+7sl3@eGroups.com>
Gene:
> > I don't know. What I'd like to know is what a version of your
> >heuristic would be which applies to sets of commas--is this what you
> >are aiming at?

Paul:
> Eventually. It would probably involve some definition of the dot
> product of the commas in a tri-taxicab metric. But I like to start
> simple, and perhaps if we can formulate the right error measure in 5-
> limit, we can generalize it and use it for 7-limit even without
> knowing how one would apply the heuristic.

My experience of generating and sorting linear temperaments from the 5- to
the 21-limit is that the "right" error metric for one can be wildly
inappropriate for others.

One assumption behind the heuristic is that the error is proportional to
the size/complexity of the unison vector. If you measure complexity as
the number of consonant intervals, that's the best case of tempering it
out. Higher-limit linear temperaments tend not to be best cases, but the
proportionality might still work. At least if you can magically produce
orthogonal unison vectors. I'll have to look at lattice theory more.

The other assumption is that the octave-specific Tenney metric
approximates the number of consonant intervals a comma's composed of. I'm
not sure how closely this holds. The Tenney metric is a good match for
the first-order odd limit of small intervals. But extended limits can
behave differently.

For example, 2401:2400 works well in the 7-limit because the numerator
only involves 7, so it has a complexity of 4 despite being fairly complex
and superparticular. Whereas a comma involving 11**4, or 14641, still
only has a complexity of 4 in the 11-limit. So if you could get a
superparticular like that, it'd lead to a much smaller error.

It should follow that 5**4:(13*3*2**4) or 625:624 will be particularly
inefficient between the 13- and 23-limits relative to what the heuristic
would predict. It still has a complexity of 4, whereas 13**3 is already
2197 and 23**2 is 529. Yes, 12168:12167 is a 23-limit comma with a
complexity of 3. (8*9*13*13):(13**3).

I'd prefer to see a heuristic for how complex a temperament produced for a
set of unison vectors or pair of ETs will be. Or one for how small the
error will be when it's generated by ETs.

Graham

--- In tuning-math@y..., graham@m... wrote:

> My experience of generating and sorting linear temperaments from
the 5- to
> the 21-limit is that the "right" error metric for one can be wildly
> inappropriate for others.

Can you give an example?

> One assumption behind the heuristic is that the error is
proportional to
> the size/complexity of the unison vector.

You can call it an assumption, if you wish -- I've verified its
approximate correctness for all 10 (wildly different) temperaments
I've tried, against Gene's rms measures.

> If you measure complexity as
> the number of consonant intervals, that's the best case of
tempering it
> out.

What does that mean?

> Higher-limit linear temperaments tend not to be best cases, but the
> proportionality might still work. At least if you can magically
produce
> orthogonal unison vectors. I'll have to look at lattice theory
more.

Well, so far I've only considered the case where one unison vector is
tempered out.

> The other assumption is that the octave-specific Tenney metric
> approximates the number of consonant intervals a comma's composed
of. I'm
> not sure how closely this holds.

This is based on the Kees van Prooijen lattice metric, and again its
good approximation was verified relative to Gene's rms measure.

> The Tenney metric is a good match for
> the first-order odd limit of small intervals. But extended limits
can
> behave differently.
>
> For example, 2401:2400 works well in the 7-limit because the
numerator
> only involves 7, so it has a complexity of 4 despite being fairly
complex
> and superparticular.

This is only one possible complexity measure, not the one Gene's
currently using, which already showed a good match with the
heuristic. A better one awaits . . .

> Whereas a comma involving 11**4, or 14641, still
> only has a complexity of 4 in the 11-limit. So if you could get a
> superparticular like that, it'd lead to a much smaller error.

You're missing the lattice justification for the heuristic. No wonder
you're skeptical!

> It should follow that 5**4:(13*3*2**4) or 625:624 will be
particularly
> inefficient between the 13- and 23-limits relative to what the
heuristic
> would predict.

Why? Try a 2D system based on 5 and 13. The heuristics should work
fine, especially if you weight ratios of 13 as less important than
ratios of 5.

In-Reply-To: <a2jliq+9hrb@eGroups.com>
Me:
> > My experience of generating and sorting linear temperaments from
> the 5- to
> > the 21-limit is that the "right" error metric for one can be wildly
> > inappropriate for others.

Paul:
> Can you give an example?

The first run through of my temperament generator, when I was using
step-cents gave absurdly complex and accurate 5-limit temperaments. Using
only consistent ETs works well enough up to the 15-limit, but beyond that
optimal temperaments are missed. At least with the current metrics.

Me:
> > One assumption behind the heuristic is that the error is
> proportional to
> > the size/complexity of the unison vector.

Paul:
> You can call it an assumption, if you wish -- I've verified its
> approximate correctness for all 10 (wildly different) temperaments
> I've tried, against Gene's rms measures.

How many dimensions?

Me:
> > If you measure complexity as
> > the number of consonant intervals, that's the best case of
> tempering it
> > out.

Paul:
> What does that mean?

It's the microtemperament formula. Last time I mentioned it, you pointed
me to one of your own messages. For the minimax temperament, tempering
out one unison vector, the error is the size of the comma divided by the
number of consonances making it up. When you're tempering out more than
one comma, the result will typically be worse than the best case for any
of the commas on their own. But it can never be better than for only one
comma.

Paul:
> Well, so far I've only considered the case where one unison vector is
> tempered out.

I'm only questioning size/complexity as a heuristic when you have more
than one unison vector. It might still work then.

> > The other assumption is that the octave-specific Tenney metric
> > approximates the number of consonant intervals a comma's composed
> of. I'm
> > not sure how closely this holds.
>
> This is based on the Kees van Prooijen lattice metric, and again its
> good approximation was verified relative to Gene's rms measure.

>From the exposition I have, 'The "length" of a unison vector
... in the Tenney lattice with taxicab metric ... is proportional to ...
the "number" ... of consonant intervals making
up that unison vector.' That's what I'm disagreeing with. Why does the
KvP metric behave differently?

Me:
> > For example, 2401:2400 works well in the 7-limit because the
> numerator
> > only involves 7, so it has a complexity of 4 despite being fairly
> complex
> > and superparticular.

Paul:
> This is only one possible complexity measure, not the one Gene's
> currently using, which already showed a good match with the
> heuristic. A better one awaits . . .

It's a complexity measure based on

1) The odd limit

2) Minimax tuning

I thought we agreed that (1) was as good as any simple, all-purpose,
numerical dissonance metric. Also that it gave the same results as the
octave-specific Tenney metric (or product limit) for small intervals. I'm
not prepared to abandon this solely in order to make your heuristic work.

I use (2) because it's simple to find the rule, at least for only one
commatic unison vector. I expect RMS optimisation would give similar
results provided all consonances are treated equally. If this isn't the
case, I want a good reason why.

Me:
> > Whereas a comma involving 11**4, or 14641, still
> > only has a complexity of 4 in the 11-limit. So if you could get a
> > superparticular like that, it'd lead to a much smaller error.

Paul:
> You're missing the lattice justification for the heuristic. No wonder
> you're skeptical!

I'm working with the precise theory I already have. And you're asking me
to give it up for a heuristic?

Me:
> > It should follow that 5**4:(13*3*2**4) or 625:624 will be
> particularly
> > inefficient between the 13- and 23-limits relative to what the
> heuristic
> > would predict.

Paul:
> Why? Try a 2D system based on 5 and 13. The heuristics should work
> fine, especially if you weight ratios of 13 as less important than
> ratios of 5.

Yes, it'll work fine as long as you fudge the metric to get it to work.
Making ratios of 13 "less important" means allowing them to be more out of
tune. My experience is that the more complex intervals get, the more
accurately they have to be tuned to sound right.

A planar temperament optimising the minimax would give exactly
1200*log2(625/624)/4 = 0.7 cents for the worst interval. In fact, I think
that's with 13:8 just so 5:4 and 13:10 are both out by 0.7 cents. Other
methods are hardly likely to make anything badly out of tune, but that's
fourth-order, superparticular, planar temperaments for you.

How is one temperament supposed to work or not work according to a
heuristic that only states a proportionality? My rule gives exact
results, and it works. But only when one unison vector is being tempered
out. Actually, not always then. For example 9*3 would be two consonances
in the 9-limit but should be weighted as 1.5. But the difference between
1.5 and 2 is less than that between 5 and 23.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a2jliq+9hrb@e...>
> Me:
> > > My experience of generating and sorting linear temperaments
from
> > the 5- to
> > > the 21-limit is that the "right" error metric for one can be
wildly
> > > inappropriate for others.
>
> Paul:
> > Can you give an example?
>
> The first run through of my temperament generator, when I was using
> step-cents gave absurdly complex and accurate 5-limit
temperaments. Using
> only consistent ETs works well enough up to the 15-limit, but
beyond that
> optimal temperaments are missed. At least with the current metrics.

What does any of this have to do with the validity of the heuristics?
You seem to be talking about goodness/badness, as well as the
generating-from-ETs-missed-some issue, neither of which have anything
to do with the validity of the heuristics. Of course, once you define
a goodness/badness measure, you should be able to use the heuristic
for step/complexity, combined with the heuristic for cents/error, to
approximate that goodness/badness measure.

> Me:
> > > One assumption behind the heuristic is that the error is
> > proportional to
> > > the size/complexity of the unison vector.
>
> Paul:
> > You can call it an assumption, if you wish -- I've verified its
> > approximate correctness for all 10 (wildly different)
temperaments
> > I've tried, against Gene's rms measures.
>
> How many dimensions?

These were all 5-limit linear temperaments.

>
> I'm only questioning size/complexity as a heuristic

Hmm . . . you may be misunderstanding something. Can you clarify what
you mean by this?

> when you have more
> than one unison vector.

More than one tempered out? Why don't we focus on the case of just
one tempered out first.

> > > The other assumption is that the octave-specific Tenney metric
> > > approximates the number of consonant intervals a comma's
composed
> > of. I'm
> > > not sure how closely this holds.
> >
> > This is based on the Kees van Prooijen lattice metric, and again
its
> > good approximation was verified relative to Gene's rms measure.
>
> >From the exposition I have, 'The "length" of a unison vector
> ... in the Tenney lattice with taxicab metric ... is proportional
to ...
> the "number" ... of consonant intervals making
> up that unison vector.' That's what I'm disagreeing with.

Why are you disagreeing? Note that "number" is _weighted_ -- more
complex consonances are longer and count as "fewer" consonances.

?
>
> Me:
> > > For example, 2401:2400 works well in the 7-limit because the
> > numerator
> > > only involves 7, so it has a complexity of 4 despite being
fairly
> > complex
> > > and superparticular.
>
> Paul:
> > This is only one possible complexity measure, not the one Gene's
> > currently using, which already showed a good match with the
> > heuristic. A better one awaits . . .
>
> It's a complexity measure based on
>
> 1) The odd limit
>
> 2) Minimax tuning
>
> I thought we agreed that (1) was as good as any simple, all-
purpose,
> numerical dissonance metric. Also that it gave the same results as
the
> octave-specific Tenney metric (or product limit) for small
intervals. I'm
> not prepared to abandon this solely in order to make your heuristic
work.

My heuristic says the complexity is proportional to log(d), where d
is either the numerator or denominator (since they're close). Since
either n or d is the odd limit, my heuristic is equivalent to (1). So
you wouldn't be abandoning anything.

> I use (2) because it's simple to find the rule, at least for only
one
> commatic unison vector. I expect RMS optimisation would give
similar
> results provided all consonances are treated equally.

RMS optimiziation gave similar results to my heuristic -- so what's
the problem?
>
> Yes, it'll work fine as long as you fudge the metric to get it to
work.
> Making ratios of 13 "less important" means allowing them to be more
out of
> tune. My experience is that the more complex intervals get, the
more
> accurately they have to be tuned to sound right.

Well, this is the age-old question. It depends what you mean
by "sounds right". We've spent so much time discussing this in the
past, how this could go either way . . . personally, I don't find
ratios of 13 to be "meaningful" as isolated dyads, and in the context
of big otonalities, you can notice mistuning in the most consonant
ratios more easily than mistuning in the 13 identity.

> How is one temperament supposed to work or not work according to a
> heuristic that only states a proportionality?

I calculated the constants of proportionality for both heuristics for
10 vastly different temperaments, using Gene's rms optimized results,
and the constants of proportionality for each heuristic were all
within a factor of 2 of one another. Did you miss that post?

--- In tuning-math@y..., graham@m... wrote:

> My experience is that the more complex intervals get, the more
> accurately they have to be tuned to sound right.

So perhaps you'd like to temper the octaves most of all?

In-Reply-To: <a2k66t+5ccs@eGroups.com>
paulerlich wrote:

> --- In tuning-math@y..., graham@m... wrote:
>
> > My experience is that the more complex intervals get, the more
> > accurately they have to be tuned to sound right.
>
> So perhaps you'd like to temper the octaves most of all?

I've mostly used octave-equivalent systems so far, so the opportunity
doesn't present itself. Making the octaves worse would also make the most
complex intervals worse, and some instruments don't allow you to do so
anyway. Besides, I treat octaves and fifths as special cases. But this
is certainly something to look at in the future.

Oh, and if we're getting into specifics, a system with 11:8, 9:7, 9:8 and
11:7 tends to leave 8 on a par with 7, 11 and 9. So 2 would only end up
with a third the error of 7 and 11.

Graham

In-Reply-To: <a2k64u+qmc3@eGroups.com>
paulerlich wrote:

> What does any of this have to do with the validity of the heuristics?
> You seem to be talking about goodness/badness, as well as the
> generating-from-ETs-missed-some issue, neither of which have anything
> to do with the validity of the heuristics. Of course, once you define
> a goodness/badness measure, you should be able to use the heuristic
> for step/complexity, combined with the heuristic for cents/error, to
> approximate that goodness/badness measure.

I'm saying that one dimensional results don't usually generalise well to
more complex cases.

Me:
> > I'm only questioning size/complexity as a heuristic

Paul:
> Hmm . . . you may be misunderstanding something. Can you clarify what
> you mean by this?

Me:
> > when you have more
> > than one unison vector.

Paul:
> More than one tempered out? Why don't we focus on the case of just
> one tempered out first.

Yes, that's fine, we agree on that case. That's why I said I wasn't
questioning it.

Me:
> > >From the exposition I have, 'The "length" of a unison vector
> > ... in the Tenney lattice with taxicab metric ... is proportional
> to ...
> > the "number" ... of consonant intervals making
> > up that unison vector.' That's what I'm disagreeing with.

Paul:
> Why are you disagreeing? Note that "number" is _weighted_ -- more
> complex consonances are longer and count as "fewer" consonances.

I was assuming that numbers were numbers and didn't carry weights. I
thought that was the difference between numbers and amounts.

> My heuristic says the complexity is proportional to log(d), where d
> is either the numerator or denominator (since they're close). Since
> either n or d is the odd limit, my heuristic is equivalent to (1). So
> you wouldn't be abandoning anything.

The examples I gave before show that the Tenney length of a unison vector
isn't a good predictor of the smallest number of intervals within a given
odd limit that make it up. The numerator and denominator being close
simply mean that the Tenney metric will be a predictor of the odd limit
*for that interval*. It works differently when you look at combinations
of intervals.

> RMS optimiziation gave similar results to my heuristic -- so what's
> the problem?

I assume there's no problem with RMS as opposed to minimax. If your
results agree, there's no problem.

> Well, this is the age-old question. It depends what you mean
> by "sounds right". We've spent so much time discussing this in the
> past, how this could go either way . . . personally, I don't find
> ratios of 13 to be "meaningful" as isolated dyads, and in the context
> of big otonalities, you can notice mistuning in the most consonant
> ratios more easily than mistuning in the 13 identity.

Yes, it's not something I would normally pursue. I work on the
simplest-case metric that all intervals deemed consonant are treated
equally in tuning. I can then do the fine tuning by ear. But if you're
suggesting something that will only work with the opposite weighting to
what I now prefer, I'll disagree with it.

> > How is one temperament supposed to work or not work according to a
> > heuristic that only states a proportionality?
>
> I calculated the constants of proportionality for both heuristics for
> 10 vastly different temperaments, using Gene's rms optimized results,
> and the constants of proportionality for each heuristic were all
> within a factor of 2 of one another. Did you miss that post?

I obviously didn't pay much attention to it. Do you have a rough figure
for "Erlich's constant" then? A factor of 2 still sounds a bit wayward.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a2k66t+5ccs@e...>
> paulerlich wrote:
>
> > --- In tuning-math@y..., graham@m... wrote:
> >
> > > My experience is that the more complex intervals get, the more
> > > accurately they have to be tuned to sound right.
> >
> > So perhaps you'd like to temper the octaves most of all?
>
> I've mostly used octave-equivalent systems so far, so the
opportunity
> doesn't present itself. Making the octaves worse would also make
the most
> complex intervals worse,

Huh? And if this is true, why wouldn't this be true of, say, ratios
of 5?

> and some instruments don't allow you to do so
> anyway.

?

--- In tuning-math@y..., graham@m... wrote:

> Me:
> > > >From the exposition I have, 'The "length" of a unison vector
> > > ... in the Tenney lattice with taxicab metric ... is
proportional
> > to ...
> > > the "number" ... of consonant intervals making
> > > up that unison vector.' That's what I'm disagreeing with.
>
> Paul:
> > Why are you disagreeing? Note that "number" is _weighted_ -- more
> > complex consonances are longer and count as "fewer" consonances.
>
> I was assuming that numbers were numbers and didn't carry weights.

Note that I put "number" in quotes.

> > My heuristic says the complexity is proportional to log(d), where
d
> > is either the numerator or denominator (since they're close).
Since
> > either n or d is the odd limit, my heuristic is equivalent to
(1). So
> > you wouldn't be abandoning anything.
>
> The examples I gave before show that the Tenney length of a unison
vector
> isn't a good predictor of the smallest number of intervals within a
given
> odd limit that make it up.

But I've always argued that the complexity measure should be
weighted. It's easier to hear progressions by 3/2 than progressions
by 5/4 . . .

> The numerator and denominator being close
> simply mean that the Tenney metric will be a predictor of the odd
limit
> *for that interval*. It works differently when you look at
combinations
> of intervals.

Again, just focusing on linear temperaments from a "two-dimensional"
just lattice for now . . .

> Yes, it's not something I would normally pursue. I work on the
> simplest-case metric that all intervals deemed consonant are
treated
> equally in tuning. I can then do the fine tuning by ear. But if
you're
> suggesting something that will only work with the opposite
weighting to
> what I now prefer, I'll disagree with it.

Maybe we each have to write our own paper, then. I'm hoping someone
will help me with the math for mine . . .

> > > How is one temperament supposed to work or not work according
to a
> > > heuristic that only states a proportionality?
> >
> > I calculated the constants of proportionality for both heuristics
for
> > 10 vastly different temperaments, using Gene's rms optimized
results,
> > and the constants of proportionality for each heuristic were all
> > within a factor of 2 of one another. Did you miss that post?
>
> I obviously didn't pay much attention to it.

/tuning-math/message/2491
"Expand Messages" as usual.

> A factor of 2 still sounds a bit wayward.

Not bad at all considering the wide range of complexities of these
temperaments . . . but I'm still hunting for the "natural" set of
definitions of "error" and "complexity" that will make the heuristic
work real real good. Once you've found a good temperament, changing
its error function is not going to change its goodness very much. So
why not look for a mathematically pretty way to find good
temperaments? That's something I'm interested in, at any rate.

In-Reply-To: <a2k8io+j4al@eGroups.com>
Me:
> > I've mostly used octave-equivalent systems so far, so the
> opportunity
> > doesn't present itself. Making the octaves worse would also make
> the most
> > complex intervals worse,

Paul:
> Huh? And if this is true, why wouldn't this be true of, say, ratios
> of 5?

It would be true in a 25-limit system. Also, where 3 is involved, in a
15-limit system, although I'm not sure how the maths work out for that.
So far, I've only tuned up 11-limit systems.

> > and some instruments don't allow you to do so
> > anyway.
>
> ?

I have a Korg X5D (currently being repaired) which only supports octave
based tuning tables. If I want to use it, everything else has to fall in
line.

Graham

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Maybe we each have to write our own paper, then. I'm hoping someone
> will help me with the math for mine . . .

We could make a deal--we each help the other guy with what they need help on the most. :)

So
> why not look for a mathematically pretty way to find good
> temperaments? That's something I'm interested in, at any rate.

I'd like to see a quick and easy estimate which inputs a wedgie and outputs a badness measure; a second pass could then refine that.

Can you guys please drop the "(Was: Hi Dave K.)" from the title of
this thread. I have so little time to spend on tuning at the moment
that I'm only reading posts that have my name in them. (in the body
text is fine). But the search finds it whether in the body or the
title and this thread is driving me crazy.

Thanks.

By the way, I didn't have a clue what "taxicab error" was. I'm glad if
Gene found a way to give it meaning. :-)

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I'm glad if
> Gene found a way to give it meaning.

Haven't understood it, as of yet . . .

:-)