31, 112 and 11-limit Meantone

We get a good version of 11-limit Meantone by using the mapping the 31-et gives us. This has

[1, 4, 10, -13, 4, 13, -24, 12, -44, -71]

for a wedgie and

[[1, 1, 0, -3, 11], [0, 1, 4, 10, -13]]

for a mapping to primes. I *still* cannot get the 31-et to be poptimal using this! In fact, the best poptimal for it seems to be our old pal, 112-equal Meantone, a hitherto unknown star in the Meantone firmament.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> We get a good version of 11-limit Meantone by using the mapping the
31-et gives us. This has
>
> [1, 4, 10, -13, 4, 13, -24, 12, -44, -71]
>
> for a wedgie and
>
> [[1, 1, 0, -3, 11], [0, 1, 4, 10, -13]]
>
> for a mapping to primes. I *still* cannot get the 31-et to be
poptimal using this! In fact, the best poptimal for it seems to be
our old pal, 112-equal Meantone, a hitherto unknown star in the
Meantone firmament.

What this says to me is that p-optimality is an interesting guide in
many cases but we shouldn't let it take over from commonsense. In the
case of meantone it says to me that the range of generators is too
wide and we may need two ETs to cover it.

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:

> What this says to me is that p-optimality is an interesting guide in
> many cases but we shouldn't let it take over from commonsense.

Sounds right, but I think it should be considered.

In the
> case of meantone it says to me that the range of generators is too
> wide and we may need two ETs to cover it.

Too wide? The problem has been that it is too damned narrow, being
stuck in the neighborhood of 1.53 to 1.58 or thereabouts in terms of Blackwood's constant. We can't seem to get down to 1.5, for 31, and only for 5-limit do we get up to 1.6, for 81.

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

> Too wide? The problem has been that it is too damned narrow, being
> stuck in the neighborhood of 1.53 to 1.58 or thereabouts in terms of Blackwood's constant. We can't seem to get down to 1.5, for 31, and only for 5-limit do we get up to 1.6, for 81.

At least t = pi/2 is in the range, so if you are bored with Lucy tuning (t = (2*pi-5)/(7-2*pi)) you could try this instead, and tune to fifths of 2400*((2*pi+3)/(7*pi+10)) cents. It's poptimal.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> Too wide? The problem has been that it is too damned narrow, being
> stuck in the neighborhood of 1.53 to 1.58 or thereabouts in terms
of Blackwood's constant. We can't seem to get down to 1.5, for 31,
and only for 5-limit do we get up to 1.6, for 81.

I guess I don't really understand this poptimal stuff after all.
Isn't there rather a big difference between the RMS and the max-
absolute optimum generators for meantone, at least at the 5 limit.
I'm sorry I don't know what Blackwood's constant is.

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"

> I'm sorry I don't know what Blackwood's constant is.

I didn't use the right one anyway, but let me define something more general.

Let 0 < p/q < 1 be a fraction in lowest terms representing a generator as a fractional part of an interval of repetition. Let p1/q1 and p2/q2 be the fractions on either side of p/q in the qth row of the Farey sequence, with p1/q1 on the side with the better values of the generator, and p2/q2 on the side with the values going in the wrong direction, so to speak. We now define the linear fractional transformation

BI(z, p/q) = (p1*z + p2)/(q1*z + q2)

This has the property that BI(0,p/q)=p2/q2,
BI(1, p/q)=(p1+p2)/(q1+q2)=p/q,
BI(infinity,p/q)=p1/q1. The value of x=BI(z,p/q) for z>=1 is the generator which gives a ratio between the large and small steps of the q-step MOS; we may call BI the inverse Blackwood transformation. If we solve for z in terms of x, we get

BW(x, p/q) = -(q2*x - p2)/(q1*x - p1),

the Blackwood transformation which takes a generator x and associates it to a Blackwood-style constant z. The transformation I used was BW(x, 7/12), whereas Blackwood used BW(x, 4/7). (If there is confusion, one could use BI(z, p/q, +) for the case where p1/q1>p/q and BI(z, p/q, -) if p1/q1<p/q.)

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:

> I guess I don't really understand this poptimal stuff after all.
> Isn't there rather a big difference between the RMS and the max-
> absolute optimum generators for meantone, at least at the 5 limit.

The rms (and least-fourth power) value is 7/26-comma, and the minimax value is 1/4-comma, which I would call a small difference.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> The rms (and least-fourth power) value is 7/26-comma, and the
minimax value is 1/4-comma, which I would call a small difference.

OK.

With regard to sagittal notation, 112-ET fails the following test.

(b) The best fifth (approx 2:3) in the ET must be the same as the fifth
calculated by applying the temperament's mapping to the best
approximation of the generator (and period) in that ET.

The best fifth in 112-ET is 66 steps. The best approximation of the
meantone fifth generator in 112-ET is 65 steps.

I agree with George ("At last!", he says) that sagittal notation of
open meantone chains of up to 30 notes should be based on 31-ET notation.

31-ET passes Gene's earlier test, of being the highest denominator of
a convergent for both RMS and max-absolute generators.