George Secor:

> > It depends on what method you're using to optimize. The 5-limit

> > minimax generator is a 1/4-comma fifth, which would also favor 31.

> > In order to get the slow-beating minor third of 50-ET you also get

>a

> > slow-beating major third, plus a faster-beating fifth that is not

> > acceptable to some. So I always thought that 31 was the clear

>choice

> > for meantone.

Me too. But some prefer RMS error, and some minimax beating, both of which favour something closer to 50-ET in the 5-limit.

Perhaps what we should do is give upper and lower bounds on the size of generator for which a particular ET notation is valid. Perhaps we should propose standard notations that fully cover the spectrum of generator sizes from say 50 cents to 600 cents. This would be for single chain (octave period) temperaments. We'd have to repeat this for 2, 3, 4, 5 chain.

On second thoughts maybe the most popular temperaments would be enough.

> >

> > > For schismic is it 41 or 53?

> >

> > How about 94?

>

>Now that I've more time to look at this, I would say definitely 94,

>for two reasons:

>

>1) If you're including the 7th harmonic, then you might as well take

>this to the 11 limit (since the minimax generator for both is the

>same). The 11th harmonic occurs in the series of fifths in 41, 53,

>and 94 in the +23 position, but in 41 it is also closer -- in the -18

>position, which is not typical for the schismic family of

>temperaments. So I eliminate 41 as my choice.

>

>2) The 7 and 11-limit minimax generator is ~702.193c (7:9 being

>exact) giving a maximum error of ~4.331c (for 5:7 and 5:9). The 53-

>ET fifth is ~701.887c (max. error ~12.681c), but the 94-ET fifth is

>much closer to the ideal: ~702.178c (max. error ~4.722c), and the

>same may be said for the 13 and 15-limit minimax generator

>(~702.109c, 13:14 being exact). So I eliminate 53, leaving 94 as my

>choice.

OK. 94 sounds good.

> > > For kleismic is it 53 or 72?

> >

> > I'll have to taken a better look at this one.

>

>I've done that and I conclude that, unless you are sticking with a 5

>limit, the choice is clearly 72 over 53. It is best to put the

>figures in a table to show this:

>

>Generator Size Max. error Exact

>

>5-minimax ~316.993c ~1.351c 2:3

>53-ET ~316.981c ~1.408c

>72-ET ~316.667c ~2.980c

>125-ET ~316.800c ~2.314c

>

>7,9-minimax ~316.765c ~2.732c 4:7

>53-ET ~316.981c ~6.167c

>72-ET ~316.667c ~3.910c

>125-ET ~316.800c ~3.088c

>

>11-minimax ~316.745c ~2.976c 9:11

>53-ET ~316.981c ~12.681c

>72-ET ~316.667c ~3.910c

>125-ET ~316.800c ~4.892c

>

>I threw 125 in there also, since it does slightly better than 72 at

>the 7 and 9 limit (and also at the 13 and 15 limit, which has the

>same minimax generator as for the 7 and 9 limit). But since the 11

>limit is the highest you can go while keeping the max error under 4

>cents, that's the limit I would use, and 72 has the advantage.

>

>Something else I noticed about the choice of 72 as the notation for

>both the Miracle and kleismic temperaments: the progression of

>sagittal symbols for a 72-ET panchromatic scale (one passing through

>all the tones) is the same as that for a sequence of tones differing

>by the generating interval in both temperaments. To illustrate:

>

>72-ET: C C\! C!) C\!/ B|) B/| B

>Miracle: C B\! Bb!) A\!/ G|) F#/| F

>kleismic: C A\! F#!) Eb\!/ B|) G#/| F

>

>The pattern then repeats. I believe that this is a useful property

>that provides a further justification for basing the kleismic

>notation on 72.

OK. 72 is good.

-- Dave Keenan

Brisbane, Australia

http://dkeenan.com

--- In tuning-math@yahoogroups.com, David C Keenan <d.keenan@u...>

wrote:

> George Secor:

>

> > > It depends on what method you're using to optimize. The 5-limit

> > > minimax generator is a 1/4-comma fifth, which would also favor

31.

> > > In order to get the slow-beating minor third of 50-ET you also

get

> >a

> > > slow-beating major third, plus a faster-beating fifth that is

not

> > > acceptable to some. So I always thought that 31 was the clear

> >choice

> > > for meantone.

>

> Me too. But some prefer RMS error, and some minimax beating, both

of which

> favour something closer to 50-ET in the 5-limit.

Since the 5-limit optimal generator size depends on the optimizing

method, then it's a tossup between 31 and 50.

I believe that the choice of the division that most closely

approximates the characterstics of meantone should then be determined

by the one that does the 7 limit best. The 7-limit minimax generator

is also a fifth narrow by 1/4-comma, and 31-ET wins hands down no

matter which method you use.

> Perhaps what we should do is give upper and lower bounds on the

size of

> generator for which a particular ET notation is valid. Perhaps we

> should propose standard notations that fully cover the spectrum of

> generator sizes from say 50 cents to 600 cents. This would be for

single

> chain (octave period) temperaments. We'd have to repeat this for 2,

3, 4, 5

> chain.

>

> On second thoughts maybe the most popular temperaments would be

enough.

If it comes down to a popularity contest, I would say that 31 takes

it on that basis, too.

Any other ideas that might favor 50?

--George

--- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:

> Since the 5-limit optimal generator size depends on the optimizing

> method, then it's a tossup between 31 and 50.

One rather rough and ready rule along these lines would be to pick the last division which occurs as a denominator of a convergent for both the rms and minimax generator. This is rough and ready, since we can see from my examples nothing stops two opitmized values from

being the same even though the rest cover a range; one could add

p=1 and p=4 (both not hard to compute) or simply p=4 to the mix to help fix that.

For Miracle, we get 72 in both the 7 and 11 limits, and for meantone,

we get 31 in both the 5 and 7 limits (11 limit too, obviously; I didn't bother to compute it since it was clear what the result would be.) For the 7-limit schismic, we have 94, but for 5-limit, it runs all the way up to 289. I suppose 53, 118, 171 and 289, all very similar from a 5-limit schismic point of view, would look completely different when notated sagitally?

> Any other ideas that might favor 50?

No, but it might make sense to say that for meantones that are on the

1/3-comma side of golden meantone (or some such boundary) a

50-ET-based notation may be preferable.

--- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:

> Any other ideas that might favor 50?

Here's one that comes close, but instead decides that the 81-et is right for the 5-limit. Has anyone even proposed using it?

For this definition, we compute the optimal rms, minimax and least fourth powers generators. We then take the smallest et which gives us a generator in the interval from the minimum to the maximum of the above values. In the case of 5-limit meantone, this is 81, though 50 comes close--very very close if we toss in the least cubes value as well, since 50 is pretty well identical to my world-famous

(7+sqrt(11))/38-comma meantone (which everyone must and shall use) in practice.

We could increase the size of the interval by some percentage if we wanted, of course.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:

> For this definition, we compute the optimal rms, minimax and least fourth powers generators. We then take the smallest et which gives us a generator in the interval from the minimum to the maximum of the above values.

This method works fine as a way of choosing an et for a temperament for computer music purposes, but is a little over the top as a way to define a notation et to associate to that temperament. In the case of miracle, it gives 175 for 7-limit miracle, which I've already been using for computer scores. For the 11-limit, it pulls the 401 et from out of left field, despite the fact that 72 in practice is more or less the same (the difference between 39/401 and 7/72 being all of

50/1203 = 0.04156 cents.)