I think the premise of calculating the optimal tuning for a given error measure, for use with that error measure, may be flawed, in the context of my heuristic. The tempering process may be a non-optimizing, but still quite simple and reasonable, one . . . hopefully the argument by which I derived the heuristic will spark someone's mind . . .

Here it is again:

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The thinking goes as follows. The "length" of a unison vector

n/d, n~=d, in the Tenney lattice with taxicab metric, or van Prooijen

lattice with triangular-taxicab metric, is proportional to log(n) +

log(d) (hence approx. proportional to log(d)), and also to

the "number" (in some weighted sense) of consonant intervals making

up that unison vector. Thus, in order to temper this unison vector

out (assuming that other UVs being tempered out, if any, are

orthogonal to this one), one must temper each consonant interval

involved by an "average" amount proportional to w/log(d), where w is

the musical width of the unison vector.

w=log(n/d)

w~=n/d-1

w~=(n-d)/d

Hence the amount of tempering implied by the unison vector is approx.

proportional to

(n-d)/(d*log(d))

Yes?

*******************************************************************

--- In tuning-math@y..., paul@s... wrote:

> (n-d)/(d*log(d))

I get stuff like 160000/151263 and 204800/194481 when I look at twintone commas with a better heuristic than 64/63--something which apparently is not hard to achieve. This does not seem right.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

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

>

> > (n-d)/(d*log(d))

>

> I get stuff like 160000/151263 and 204800/194481 when I look at

>twintone commas with a better heuristic than 64/63--something which

>apparently is not hard to achieve. This does not seem right.

What exactly do you mean, Gene? This heuristic is intended to work

when _only one_ comma is tempered out -- but aren't you hiding a

50:49 up your sleeve? If you're trying to apply the heuristic to

different ways of defining a two-comma temperament, then please

recall I did say the UVs should be as near orthogonal as possible --

therefore, if 50:49 is already tempered out, then you have to express

the other UV of twintone as some JI vector orthogonal to 50:49. If

all these alternatives are nearly orthogonal to 50:49, then perhaps

it _is_ right -- can we figure out what versions of twintone are

actually "implied" (hopefully in the sense of the correct metric)?

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

> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

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

> >

> > > (n-d)/(d*log(d))

> >

> > I get stuff like 160000/151263 and 204800/194481 when I look at

> >twintone commas with a better heuristic than 64/63

Wait a minute -- this isn't even correct!

n=64, d=63 -> (n-d)/(d*log(d)) = 0.0038312

n=160000, d=151263 -> (n-d)/(d*log(d)) = 0.0048429 (WORSE THAN 64/63!)

n=204800, d=194481 -> (n-d)/(d*log(d)) = 0.0043569 (WORSE THAN 64/63!)

Remember, (n-d)/(d*log(d)) is a heuristic for the _amount of

tempering_ -- and I assume we agree that the less tempering, the

better!

So what are you on about, Gene?

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

> Wait a minute -- this isn't even correct!

>

> n=64, d=63 -> (n-d)/(d*log(d)) = 0.0038312

>

> n=160000, d=151263 -> (n-d)/(d*log(d)) = 0.0048429 (WORSE THAN 64/63!)

>

> n=204800, d=194481 -> (n-d)/(d*log(d)) = 0.0043569 (WORSE THAN 64/63!)

What's going on here is that what you are calling bad, I was calling good. The reason for that is that I was seeing if one could reduce a lattice basis wrt your heuristic, and to do that 50/49 should be better than 64/63, etc, or at least better than something much smaller. It remains the case that no matter what we call "bad" or "good" the intervals I gave are in the range between 50/49 and 64/63, which means I take it that they define an amount of tempering between that defined by 50/49 and 64/63? I'm not convinced this is making sense.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

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

>

> > Wait a minute -- this isn't even correct!

> >

> > n=64, d=63 -> (n-d)/(d*log(d)) = 0.0038312

> >

> > n=160000, d=151263 -> (n-d)/(d*log(d)) = 0.0048429 (WORSE THAN

64/63!)

> >

> > n=204800, d=194481 -> (n-d)/(d*log(d)) = 0.0043569 (WORSE THAN

64/63!)

>

> What's going on here is that what you are calling bad, I was

>calling good. The reason for that is that I was seeing if one could

>reduce a lattice basis wrt your heuristic, and to do that 50/49

>should be better than 64/63, etc,

Why? This is not the heuristic for complexity we're looking at here --

it's the heuristic for error! It seems like you're confused about

this . . .

>or at least better than something >much smaller.

I'm not following. And Gene, so far there's a heuristic only for the

case of one unison vector tempered out. Anything beyond that is your

invention, not mine.

>It remains the case that no matter what we call "bad" or "good" the

>intervals I gave are in the range between 50/49 and 64/63, which

>means I take it that they define an amount of tempering between that

>defined by 50/49 and 64/63?

Alone, when defining a planar temperament? Sure, why not?

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

> Why? This is not the heuristic for complexity we're looking at here --

> it's the heuristic for error!

So small values of your heuristuc applied to 5-limit commas should correspond to good 5-limit temperaments, and larger values to not-so-good?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

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

>

> > Why? This is not the heuristic for complexity we're looking at

here --

> > it's the heuristic for error!

>

> So small values of your heuristuc applied to 5-limit commas should

>correspond to good 5-limit temperaments, and larger values to not-so-

>good?

One heuristic, |n-d|/(d*log(d)), gives you the the "error" component

of goodness, while another, log(d), gives you the "gens"/"complexity"

component. Please look again at

You'll need to use "Expand Messages".