A 7-limit best list

This is a first pass at a 7-limit best list. The first entry is the mapping matrix, the second period and generator, the third (unweighted) rms generator steps to consonances, the fourth (unweighted) rms error. Badness is not listed to save space, but is less than 300; generator steps are less than 40, and rms error less than 50 cents. The ordering is by badness, lowest to highest.

[[9, 1, 1, 12], [0, 2, 3, 2]] [133, 884] 16.0156 .13045

[[1, 3, 2, 3], [0, 22, -5, 3]] [1200, -77] 16.6933 .25334

[[1, 15, 4, 7], [0, 16, 2, 5]] [1200, -1006] 10.09125 .87536

[[1, 1, 3, 3], [0, 6, -7, -2]] [1200, 117] 7.60482 1.6374

[[1, 1, 7, 5], [0, 4, -32, -15]] [1200, 175] 23.13367 .18381

[[1, 0, 0, 2], [0, 2, 3, 1]] [1200, 929] 1.825742 34.5661

[[1, 21, 13, 13], [0, 40, 22, 21]] [1200, -583] 23.13367 .22219

[[2, 0, 3, 4], [0, 2, 1, 1]] [600, 951] 2.3094 23.945

[[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016

[[1, 0, -4, -13], [0, 1, 4, 10]] [1200, 1896] 6.36396 3.665

[[1, 0, 3, 1], [0, 7, -3, 8]] [1200, 271] 7.572 2.589

[[4, 0, 3, 5], [0, 1, 1, 1]] [300, 1885] 2.828 19.137

[[1, 1, -5, -1], [0, 2, 25, 13]] [1200, 352] 16.289 .585

[[19, 0, 14, -37], [0, 1, 1, 3]] [63, 1901] 33.811 .1402

[[1, 31, 0, 9], [0, 38, -3, 8]] [1200, -929] 26.517 .2287

[[1, 1, 2, 2], [0, 2, 1, 3]] [1200, 322] 1.8257 48.926

[[1, 0, 1, 2], [0, 6, 5, 3]] [1200, 317] 3.7416 12.2738

[[1, 1, 2, 3], [0, 9, 5, -3]] [1200, 78] 7.5167 3.0659

[[1, 27, 24, 20], [0, 34, 29, 23]] [1200, -897] 21.2446 .4048

[[1, 11, 42, 25], [0, 14, 59, 33]] [1200, -807] 36.1201 .1439

[[4, 0, 4, 7], [0, 6, 5, 4]] [300, 317] 14.8773 .8816

[[2, 0, 11, 12], [0, 1, -2, -2]] [600, 1908] 4.2427 10.903

[[2, 4, 5, 6], [0, 28, 12, 13]] [600, -18] 32.445 .1871

[[5, 8, 0, 14], [0, 0, 1, 0]] [240, 2789] 3.5356 15.8153

[[1, 0, 15, -59], [0, 1, -8, 39]] [1200, 1901] 29.7769 .22341

[[10, 0, 47, 36], [0, 2, -3, -1]] [120, 951] 29.439 .2289

[[3, 0, 45, 94], [0, 1, -8, -18]] [400, 1901] 37.370 .1469

[[1, 2, 2, 3], [0, 4, -3, 2]] [1200, -126] 4.223 12.1886

[[1, 0, 4, 6], [0, 1, -1, -2]] [1200, 1921] 1.826 65.953

[[1, 1, 1, 2], [0, 8, 18, 11]] [1200, 88] 10.5435 2.0643

[[3, 0, 7, 18], [0, 1, 0, -2]] [400, 1911] 5.3385 8.1007

[[1, 45, 39, 32], [0, 58, 49, 39]] [1200, -898] 36.1202 .18298

[[1, 0, 4, 2], [0, 2, -2, 1]] [1200, 981] 2.4152 41.5247

[[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.1323

[[2, 4, 7, 7], [0, 6, 17, 10]] [600, -83] 20.1825 .60032

[[1, 0, 2, -1], [0, 5, 1, 12]] [1200, 381] 7.7028 4.139

[[1, 1, 0, 3], [0, 3, 12, -1]] [1200, 232] 8.3667 3.579

[[1, 0, -4, 6], [0, 1, 4, -2]] [1200, 1902] 3.5356 20.163

[[3, 5, 7, 8], [0, 7, 1, -12]] [400, -14] 33.764 .2218

[[1, 0, 4, -2], [0, 1, -1, 3]] [1200, 1940] 2.415 43.6595

[[1, 27, 11, 40], [0, 41, 14, 60]] [1200, -744] 38.0416 .1869

[[1, 3, 6, -2], [0, 5, 13, -17]] [1200, -339] 17.9397 .84588

[[3, 0, 7, -1], [0, 1, 0, 2]] [400, 1889] 4.062 16.597

[[1, 4, 2, 2], [0, 15, -2, -5]] [1200, -193] 12.596 1.7312

[[3, 5, 7, 0], [0, 0, 0, 1]] [400, 3406] 2.1213 61.3125

[[3, 0, 7, 6], [0, 2, 0, 1]] [400, 956] 4.062 16.787

[[1, 0, 4, 1], [0, 1, -1, 1]] [1200, 2025] 1.354 154.263

[[1, 11, -3, 20], [0, 23, -13, 42]] [1200, -491] 34.506 .2393

[[1, 0, 15, 25], [0, 1, -8, -14]] [1200, 1902] 10.02497 2.859

[[1, 5, 5, 5], [0, 14, 11, 9]] [1200, -293] 8.5245 4.007

[[1, 16, 32, -15], [0, 17, 35, -21]] [1200, -1017] 33.811 .2558

[[1, 0, 1, -3], [0, 6, 5, 22]] [1200, 317] 13.4846 1.61056

[[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453

[[1, 6, 8, 11], [0, 7, 9, 13]] [1200, -756] 7.692 5.053

[[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410

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

> [[1, 0, 4, 1], [0, 1, -1, 1]] [1200, 2025] 1.354 154.263

This one slipped by me--ignore it.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> This is a first pass at a 7-limit best list. The first entry is the
mapping matrix, the second period and generator, the third
(unweighted) rms generator steps to consonances, the fourth
(unweighted) rms error. Badness is not listed to save space, but is
less than 300; generator steps are less than 40, and rms error less
than 50 cents. The ordering is by badness, lowest to highest.
>

For future reference, the blank lines between temperaments were a
nuisance re importing to Excel. And I think we should have
at least one decimal place for the rms-optimum generators, since
we're sometimes stacking 10 or more of them. Everything else about the
format was just fine, thanks.

Gene, I'm puzzled. How come we didn't see the 5-limit versions of
these four in any of your earlier 5-limit lists.

> [[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016
> [[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453
> [[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410
> [[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.132

In lowest terms the generators are 310, 356, 490 and 274 cents
respectively. These are the only three of possible interest that I
don't have names for, except I tentatively call the 490 cent one
superpythagorean (since its generator can be considered to be a 710
cent fifth).

For your next pass, you can limit the complexity (preferably
odd-limit-weighted) to 17, and the rms error to 30 cents, but let your
badness go up to about 500. Some of those that barely made it onto
your list are reasonably high on mine.

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

> Gene, I'm puzzled. How come we didn't see the 5-limit versions of
> these four in any of your earlier 5-limit lists.
>
> > [[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016
> > [[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453
> > [[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410
> > [[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.132

That's because as 5-limit systems they are all pretty bad. On the other hand, here is something I should have included on my 7s list:

[19, 19, 57, 79, -37, -14] [[19, 0, 14, -37], [0, 1, 1, 3]]

badness 160.2710884 g 33.81074780 rms .1401992317

generators [63.15789474, 1901.874626]

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> This is a first pass at a 7-limit best list. The first entry is the
>mapping matrix, the second period and generator, the third
>(unweighted) rms generator steps to consonances,

why unweighted? i thought we had 'voted' that weighting was
beneficial in this context.

i'm not seeing the injera temperament here (generators should be 600,
694); that concerns me . . .

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Gene, I'm puzzled. How come we didn't see the 5-limit versions of
> > these four in any of your earlier 5-limit lists.
> >
> > > [[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016
> > > [[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453
> > > [[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410
> > > [[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.132
>
> That's because as 5-limit systems they are all pretty bad.

Sure they wouldn't have made the final list, but they are way better
than many others on the combined list I made of all those you
generated back then.

> On the
> other hand, here is something I should have included on my 7s list:
> > [19, 19, 57, 79, -37, -14] [[19, 0, 14, -37], [0, 1, 1, 3]]
>
> badness 160.2710884 g 33.81074780 rms .1401992317
>
> generators [63.15789474, 1901.874626]

You did include it.

However your list does not include the following (from Graham's
catalog):

twin meantone (double diatonic) [[2 . . .] [0 1 4 4]]
shrutar (double diaschismic) [[2 . . .] [0 2 -4 7]]
porcupine [[1 . . .] [0 3 5 -6]]
diminished [[4 . . .] [0 1 1 4]]
diaschismic (15-limit variant) [[2 . . .] [0 1 -2 -8]]
diaschismic (56-ET variant) [[2 . . .] [0 1 -2 9]]

It seems that your current badness is too lenient on complexity and
too hard on error.

Can you explain why you used complexity^3 * error for 5-limit and are
now using complexity^2 * error for 7-limit?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Gene, I'm puzzled. How come we didn't see the 5-limit versions of
> > these four in any of your earlier 5-limit lists.
> >
> > > [[1, 9, 9, 8], [0, 10, 9, 7]] [1200, -890] 6.377 3.32016
> > > [[1, 1, 5, 4], [0, 2, -9, -4]] [1200, 356] 6.8678 6.2453
> > > [[1, 0, -12, 6], [0, 1, 9, -2]] [1200, 1910] 6.831 6.410
> > > [[2, 1, 3, 4], [0, 4, 3, 3]] [600, 326] 4.899 10.132
>
> That's because as 5-limit systems they are all pretty bad.

Having found their 5-limit rms optima, I can see that a cutoff on your
badness (= complexity^3 * error) would explain the non-appearance of
all of them except the superpythagorean one (the 3rd one above). It's
non-appearance can only be explained by such a cutoff if your badness
uses unweighted complexity. Using weighted complexity it has a badness
of 1202 which is better than

1220703125/1207959552 semiminorsixths badness 1352
48828125/47775744 quintaminorthirds badness 1902

Maybe these two were only generated when you were using unweighted
complexity and maybe you always had your badness limit set below 1202
once you started using weighted complexity.

In my own badness ranking, superpythagorean (where 19683/20480
vanishes) comes 23rd (after septathirds and parakleismic).

Gene, I afraid we might have missed other 5-limit temperaments that
were too bad by your measure, but not mine. Could you please rerun
your 5-limit temperament generator with an rms error cutoff of 35
cents, weighted rms complexity cutoff of 10 generators, and badness =
weighted_complexity^3*error cutoff of 1900.

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> i'm not seeing the injera temperament here (generators should be 600,
> 694); that concerns me . . .

The feedback I'm getting from both you and Dave is that my badness cutoff was set too low. If we go all the way out to ennealimmal with a higher cutoff, we'll have a lot of temperaments. Perhaps we need to do either one of Dave's rolloffs or two different lists.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > i'm not seeing the injera temperament here (generators should be
600,
> > 694); that concerns me . . .
>
> The feedback I'm getting from both you and Dave is that my badness
>cutoff was set too low. If we go all the way out to ennealimmal with
>a higher cutoff, we'll have a lot of temperaments. Perhaps we need
>to do either one of Dave's rolloffs or two different lists.

if you're happy doing one of dave's rolloffs, then by all means (and
for 5-limit too!) . . . but i don't agree with dave that temperaments
like Shrutar necessarily have to show up in the 7-limit list . . .

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> >
> > > i'm not seeing the injera temperament here (generators should be
> 600,
> > > 694); that concerns me . . .
> >
> > The feedback I'm getting from both you and Dave is that my badness
> >cutoff was set too low. If we go all the way out to ennealimmal
with
> >a higher cutoff, we'll have a lot of temperaments. Perhaps we need
> >to do either one of Dave's rolloffs or two different lists.
>
> if you're happy doing one of dave's rolloffs, then by all means (and
> for 5-limit too!) . . . but i don't agree with dave that
temperaments
> like Shrutar necessarily have to show up in the 7-limit list . . .

They don't have to show up in the final list, but the fact that they
are not showing up in Gene's wide-open list leads to fears that we are
missing other possibly interesting ones that we haven't seen before.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > i'm not seeing the injera temperament here (generators should be
600,
> > 694); that concerns me . . .
>
> The feedback I'm getting from both you and Dave is that my badness
cutoff was set too low. If we go all the way out to ennealimmal with a
higher cutoff, we'll have a lot of temperaments. Perhaps we need to do
either one of Dave's rolloffs or two different lists.

You haven't addressed my question: Why complexity^2 * error now, when
you used complexity^3 * error for 5-limit?

I think if you change that, and put cutoffs at 17 gens for weighted
complexity and 25 or 30 cents for error, the list shouldn't be too
big. My earlier suggestion of 2000 or so badness cutoff is only if you
insist on continuing to use complexity^2 * error.

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

> You haven't addressed my question: Why complexity^2 * error now,
when
> you used complexity^3 * error for 5-limit?

the reason is log-flat badness, as usual! the formula depends on the
number of independent intervals you're trying to approximation --
gene gave us the general formula, derived from diophantine
approximation theory, a while back.

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > You haven't addressed my question: Why complexity^2 * error now,
> when
> > you used complexity^3 * error for 5-limit?
>
> the reason is log-flat badness, as usual!

I really don't think so!

> the formula depends on the
> number of independent intervals you're trying to approximation --
> gene gave us the general formula, derived from diophantine
> approximation theory, a while back.

Yes. And I thought it involved fractional powers of complexity, and
they didn't decrease by one for every additional prime. What will we
have for 13-limit? complexity^0 ? And at the 17-limit will complexity
suddenly become a good thing!?

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

> You haven't addressed my question: Why complexity^2 * error now, when
> you used complexity^3 * error for 5-limit?

We've been over that--it's what is needed to make things log-flat.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > You haven't addressed my question: Why complexity^2 * error
now,
> > when
> > > you used complexity^3 * error for 5-limit?
> >
> > the reason is log-flat badness, as usual!
>
> I really don't think so!

care to put some money on it?

> > the formula depends on the
> > number of independent intervals you're trying to approximation --
> > gene gave us the general formula, derived from diophantine
> > approximation theory, a while back.
>
> Yes. And I thought it involved fractional powers of complexity,

in general, yes.

> and
> they didn't decrease by one for every additional prime.

only in this one instance, linear temperament, 5-limit to 7-limit.

> What will we
> have for 13-limit? complexity^0 ?

you're extrapolating the pattern as if it were linear in that
exponent. it isn't.

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> >
> > > gene gave us the general formula, derived from diophantine
> > > approximation theory, a while back.
> >
> > Yes. And I thought it involved fractional powers of complexity,
>
> in general, yes.

It seems Gene is the only one who understands this derivation. I don't
feel very comfortable with this when we are putting so much store by
it, and against our own experience and what we've heard of others
experience with actually playing and listening. What if Gene has made
a mistake either in the derivation or in its relevance? We all make
mistakes.

> > and
> > they didn't decrease by one for every additional prime.
>
> only in this one instance, linear temperament, 5-limit to 7-limit.
>
> > What will we
> > have for 13-limit? complexity^0 ?
>
> you're extrapolating the pattern as if it were linear in that
> exponent. it isn't.

So what _is_ the pattern. I guess you don't know either, or you would
have said.

Gene,

We probably have gone over it before (although I think that was for
ETs, not linear temperaments), but I don't remember ^3 for 5-limit
and ^2 for 7-limit, I remember (improper)fractional powers. Can you
please point me to the earlier posts where you went over it? Or
explain it again? Or at least tell us what function generates the
sequence of exponents.

Ok. I found it. I last asked questions about this in
/tuning-math/message/1836

It seems the sequence of exponents must be (n+1)/(n-1) where n is the
number of odd primes being approximated.

Reading the thread following this post, It looks like I either failed
to understand Gene's answer to my question and gave up on empirically
verifying the claims, or got distracted by something else.

In
/tuning-math/message/1853
Gene wrote:

"When you measure the size of an et n by log(n), and are at the
critical exponent, the ets less than a certain fixed badness
are evenly distributed on average; if you plotted numbers of ets
less than the limit up to n versus log(n), it should be a rough line.
If you go over the critical exponent, you should get a finite list. If
you go under, it is weighted in favor of large ets, in terms of the
log of the size."

I think I understand this now, and could test it empirically given a
big enough list of 7-limit linear temperaments with no additional
cutoffs apart from badness. Correct me if I'm wrong, but I could do it
by sorting them into bins according to complexity, where each bin
corresponds to a doubling of complexity. i.e. bin 0 contains all those
with complexity between 1 and 2, bin 1 has all those with complexity
between 2 and 4, bin 2 between 4 and 8, and so on. I should find
roughly an equal number of temperaments in every bin.

As Paul observed, this is bound to fail for the lowest bins, and
indeed there could be bin -1 having complexities between 0.5 and 1,
and so on for increasingly negative bins, which will eventually all be
empty.

It also seems (from the to and fro between Gene and Paul in that
thread) that the only justifications for using _log_-flat (and not
something stronger) are that
(a) it's easy to deal with mathematically, and
(b) Gene likes it.

Sorry to be difficult about this.

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

> It seems Gene is the only one who understands this derivation. I don't
> feel very comfortable with this when we are putting so much store by
> it, and against our own experience and what we've heard of others
> experience with actually playing and listening. What if Gene has made
> a mistake either in the derivation or in its relevance? We all make
> mistakes.

I don't see we are setting much store by it, and the fact is it seems to work, giving us an infinite list of possibilities, but one which does not choke us when we go to higher complexities. The only bad thing about it is that it might not be a good idea to stick the derivation into a paper.

> So what _is_ the pattern. I guess you don't know either, or you would
> have said.

It's (n+1)/(n-1), where n is the number of odd primes; so for the 7-limit, n=3 and (3+1)/(3-1)=2.

> Gene,
>
> We probably have gone over it before (although I think that was for
> ETs, not linear temperaments), but I don't remember ^3 for 5-limit
> and ^2 for 7-limit, I remember (improper)fractional powers.

Yikes, I thought we went over it endlessly.

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

> It also seems (from the to and fro between Gene and Paul in that
> thread) that the only justifications for using _log_-flat (and not
> something stronger) are that
> (a) it's easy to deal with mathematically, and
> (b) Gene likes it.

(a) It has a rational basis; what else does?
(b) I tried g^3, leading to the grooviest 7-limit thread. I thought it showed a decided bias in favor of low complexities.
(c) It works.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > It also seems (from the to and fro between Gene and Paul in that
> > thread) that the only justifications for using _log_-flat (and not
> > something stronger) are that
> > (a) it's easy to deal with mathematically, and
> > (b) Gene likes it.
>
> (a) It has a rational basis; what else does?

Plenty else. Merely having a rational basis is no guarantee of
anything. It might be the _wrong_ rational basis.

I think what we really want is a human perceptual/cognitive/motor
basis; empirical, not purely mathematical. What's the point
otherwise? To be done properly that would have to be done
statistically by surveying lots of musicians who have tried different
temperaments. But we don't have that luxury and must rely on the
experience of the few of us on this list, and what we've learnt by
listening to others. To me, that seems quite rational too.

> (b) I tried g^3, leading to the grooviest 7-limit thread. I thought
it showed a decided bias in favor of low complexities.

This is a valid consideration. It's good that you expressed it _as_ an
opinion and I suppose your opinion should carry as much weight as
anyone else's.

> (c) It works.

That also is a matter of opinion.

I must apologise for not remembering, and not checking the archives,
regarding the use of complexity^3 for 5-limit and complexity^2 for
7-limit.

OK. Continue using complexity^2 for 7-limit, but _please_ can we have
a list with a higher badness cutoff and _weighted_ complexity.

I expect you dislike using weighted complexity because the
justification for it is not purely mathematical?

Badness should be high enough to include:

twin meantone (injera) [[2 . . .] [0 1 4 4]]
shrutar (double diaschismic) [[2 . . .] [0 2 -4 7]]
porcupine [[1 . . .] [0 3 5 -6]]
diminished [[4 . . .] [0 1 1 4]]
diaschismic (15-limit variant) [[2 . . .] [0 1 -2 -8]]
diaschismic (56-ET variant) [[2 . . .] [0 1 -2 9]]

One per line, no blank lines between please.

Surely this list won't be too big if you use a weighted complexity
cutoff of 17 generators and an error cutoff of 25 cents.

This is in no way intended to be the final list, this is just to make
sure we've seen everything worth seeing before deciding on the final
cutoffs.

Regards,
-- Dave Keenan

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

> Badness should be high enough to include:
>
> twin meantone (injera) [[2 . . .] [0 1 4 4]]

yup.

> shrutar (double diaschismic) [[2 . . .] [0 2 -4 7]]
> porcupine [[1 . . .] [0 3 5 -6]]
> diminished [[4 . . .] [0 1 1 4]]
> diaschismic (15-limit variant) [[2 . . .] [0 1 -2 -8]]
> diaschismic (56-ET variant) [[2 . . .] [0 1 -2 9]]

i'm not sure about the rest of these. maybe diminished, but the rest
have never been advocated as 7-limit temperaments.

> This is in no way intended to be the final list, this is just to
make
> sure we've seen everything worth seeing before deciding on the
final
> cutoffs.

gotcha. i'm just worried that there will be way too much to consider
here. why not just insist that injera makes it in and go from there?

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Badness should be high enough to include:
> >
> > twin meantone (injera) [[2 . . .] [0 1 4 4]]
>
> yup.
>
> > shrutar (double diaschismic) [[2 . . .] [0 2 -4 7]]
> > porcupine [[1 . . .] [0 3 5 -6]]
> > diminished [[4 . . .] [0 1 1 4]]
> > diaschismic (15-limit variant) [[2 . . .] [0 1 -2 -8]]
> > diaschismic (56-ET variant) [[2 . . .] [0 1 -2 9]]
>
> i'm not sure about the rest of these. maybe diminished, but the rest
> have never been advocated as 7-limit temperaments.

Apparently you mean they have only been advocated at _higher_limits
than 7. But that doesn't disqualify them from consideration at the
7-limit.

So I've been forced to find their minimum rms errors myself, so I can
calculate Gene's badness (and my own) for them.

In my opinion Porcupine is of far more interest at the 7-limit than
double meantone. Shrutar and the [1, -2, -8] diaschismic are of about
equal interest, just slightly less interest than double-meantone, but
are at (or just off) the bottom of the list. Diminished and the other
diashismic don't make it.

> gotcha. i'm just worried that there will be way too much to consider
> here. why not just insist that injera makes it in and go from there?

Well hey we could at least take a look at how many are on such a list
instead of refusing to look 'cause we're worried. I don't understand
yours or Gene's attitude in this regard.

I guess all I can say is, I won't be at all confident that we haven't
missed something unless we look at everything with complexity
(weighted or otherwise) less than 17 gens and rms error less than 25
cents and Gene's badness less than 500 if unweighted rms complexity is
used or less than 350 if weighted rms complexity is used. Of course
weighted should be used.

This is only slightly more than is required to ensure that
double-meantone appears. Double meantone has a Gene's-badness of 320.
The [1, -2, -8] diaschismic is 343. These are using weighted
complexity.

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

> In my opinion Porcupine is of far more interest at the 7-limit than
> double meantone.

far more interest? how do you figure?

> Diminished and the other
> diashismic don't make it.

none of the ways of extending diminished to 7-limit suggested by
herman miller here:

http://www.io.com/~hmiller/music/temp-diminished.html

make it? can you detail why that is?

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > In my opinion Porcupine is of far more interest at the 7-limit
than
> > double meantone.
>
> far more interest? how do you figure?

Porcupine has about the same weighted complexity but a significantly
smaller error.

Porcupine [3,5,-6], 5.84 gens, 6.81 cents
Twin meantone [1,4,4] with half-octave period, 5.34 gens, 11.22 cents

> > Diminished and the other
> > diashismic don't make it.
>
> none of the ways of extending diminished to 7-limit suggested by
> herman miller here:
>
> http://www.io.com/~hmiller/music/temp-diminished.html
>
> make it? can you detail why that is?

I didn't say that. I was only referring to the one in my message, in
my list of temperaments that weren't in Gene's list.

The [1,1,1] mapping (which was in Gene's list) will make it. The
[1,1,4] mapping won't because of its combination of 8.26 gens and 9.15
cents.

Herman's page referred to above, only gives the [1,1,1] mapping. Are
there really any others worth considering?

Pending Gene's next list with badness up to 350, based on weighted
complexity, here's a list of 27 we might agree on. It's given in order of
increasing complexity.

It consists of all those I _know_of_ with

weighted-complexity < 13 gens,
error < 23 cents,
Gene's-weighted-badness < 343,

but there may well be some we don't know about.

So are there any others that anyone thinks should be included?

Periods Gens per Gen Wtd Compl RMS err
Name per oct 1:3 1:5 1:7 (cents) (gens) (cents)
----------------------------------------------------------------------
dominant 7th 1 1 4 -2 498 3.13 20.16
(meantone with inaccurate 7s)
diminished 4 1 1 1 85 3.14 19.14
quintuple thirds 5 0 1 0 91 3.29 15.82
(Blackwood's decatonic)
augmented 3 1 0 2 111 3.68 16.60
with inaccurate 7s
diaschismic (pajara) 2 1 -2 -2 108 3.94 10.90
tertiathirds 1 4 -3 2 126 4.31 12.19
(Negri's system)
kleismic 1 6 5 3 317 4.37 12.27
with inaccurate 7s
augmented 3 1 0 -2 89 4.63 8.10
meantone 1 1 4 10 504 5.32 3.67
twin meantone 2 1 4 4 94 5.34 11.22
(injera)
twin wide 2 4 3 3 274 5.75 10.13
subminor thirds
porcupine 1 3 5 -6 162 5.84 6.81
superpythagorean 1 1 9 -2 490 6.21 6.41
neutral thirds 1 2 -9 -4 356 6.52 6.25
with complex 5s
magic 1 5 1 12 381 6.79 4.14
narrow minor thirds 1 10 9 7 310 7.40 3.32
semisixths 1 7 9 13 444 7.42 5.05
(tiny diesic)
subminor thirds 1 7 -3 8 271 7.42 2.59
(orwell)
miracle 1 6 -7 -2 117 7.61 1.64
quartaminorthirds 1 9 5 -3 78 7.66 3.07
supermajor seconds 1 3 12 -1 232 7.74 3.58
schismic 1 1 -8 -14 498 8.61 2.86
diaschismic 2 1 -2 -8 104 9.47 3.82
(15-limit variant)
octafifths 1 8 18 11 88 10.50 2.06
(semi minimal-diesic)
shrutar 2 2 -4 7 53 11.19 2.25
(double diaschismic)
kleismic 1 6 5 22 317 11.41 1.61
with complex 7s (catakleismic)
half wuerschmidt- 1 16 2 5 194 11.75 0.88
thirds
-------------------------------------------------------------------

I tried hard to get a list that included ennealimmal, and that could be
made to agree between our two methods. This is not because I think
ennealimmal is particularly interesting, but because I know Gene does.

So far, no matter how I tinker with my two rolloff parameters and Gene's
complexity and error cutoffs, I can't make two lists that agree while
including ennealimmal, even if I'm willing to include stuff that I would
normally consider junk.
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > In my opinion Porcupine is of far more interest at the 7-limit
> than
> > > double meantone.
> >
> > far more interest? how do you figure?
>
> Porcupine has about the same weighted complexity but a
significantly
> smaller error.
>
> Porcupine [3,5,-6], 5.84 gens, 6.81 cents
> Twin meantone [1,4,4] with half-octave period, 5.34 gens, 11.22
cents

thanks for that. somehow i didn't think of the errors as being so
different. still, i wouldn't say 'far more interest', given that the
complexity is actually lower for injera.

> > > Diminished and the other
> > > diashismic don't make it.
> >
> > none of the ways of extending diminished to 7-limit suggested by
> > herman miller here:
> >
> > http://www.io.com/~hmiller/music/temp-diminished.html
> >
> > make it? can you detail why that is?
>
> I didn't say that. I was only referring to the one in my message,
in
> my list of temperaments that weren't in Gene's list.
>
> The [1,1,1] mapping (which was in Gene's list) will make it.

whew!

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

Could you put this list either into a full period matrix format or give the wedgie? I'm finding the conversion aggravating, and I think temperaments should *always* be given in one of the two forms above.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
>
> Could you put this list either into a full period matrix format or
give the wedgie? I'm finding the conversion aggravating, and I think
temperaments should *always* be given in one of the two forms above.

I'm sorry. I just gave enough info for anyone to recognise and
distinguish them, and assumed you would have your software _generate_
the more detailed list, using the cutoffs I gave. I eagerly await your
next list. I fear it will still contain some temperaments that are of
no real interest to musicians.

My spreadsheet is at
/tuning-math/files/Dave/7LimTemp.xls

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

> I'm sorry. I just gave enough info for anyone to recognise and
> distinguish them, and assumed you would have your software _generate_
> the more detailed list, using the cutoffs I gave.

I wanted the list to find the cutoffs, but it's all done now.

I eagerly await your
> next list. I fear it will still contain some temperaments that are of
> no real interest to musicians.

We've got people who sniff suspiciously at anything remotely resembling an audible deviation from RI.

>We've got people who sniff suspiciously at anything remotely resembling an
>audible deviation from RI.

We do? Then I argue they would have little interest in temperament.

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> >We've got people who sniff suspiciously at anything remotely resembling an
> >audible deviation from RI.

> We do? Then I argue they would have little interest in temperament.

I did say audible. Over on the main list, they are sniffing at the scale I gave in 612-et, but logically, should they?

>I did say audible. Over on the main list, they are sniffing at the scale I
>gave in 612-et, but logically, should they?

Most probably not. I haven't seen that thread yet, but it doeesn't sound
like there's a reason for us to sniff about giving it in JI, either.

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Most probably not. I haven't seen that thread yet, but it doeesn't sound
> like there's a reason for us to sniff about giving it in JI, either.

Just what I was planning--give some JI scales, and point out the
2401/2400 appromimations also.

>Just what I was planning--give some JI scales, and point out the
>2401/2400 appromimations also.

Cool. Even I am wondering how such a small comma can add many
consonances in an 11-tone scale. Actually, if you really want
to win friends and influence people, you'll have Maple draw a
picture of the block in the lattice. Some of the naysayers might
not even know what you mean by "increases consonances", that one
can leave everything in JI and still ignore the wolf.

-Carl