Dave's 18 best 5-limit temperaments

Here's a list of temperaments with keenan badness less than 15, where
keenan badness is g exp(sqrt(r/7.4)) :

81/80

map [[0, -1, -4], [1, 2, 4]]

keenan 6.263263749 rms 4.217730124 g 2.943920288

generators 503.8351546 1200

15625/15552

map [[0, 6, 5], [1, 0, 1]]

keenan 6.601347654 rms 1.029625097 g 4.546060566

generators 317.0796754 1200

128/125

map [[0, -1, 0], [3, 6, 7]]

keenan 7.686514108 rms 9.677665980 g 2.449489743

generators 491.2018553 400

2048/2025

map [[0, -1, 2], [2, 4, 3]]

keenan 7.826993942 rms 2.612822498 g 4.320493799

generators 494.5534684 600

32805/32768

map [[0, -1, 8], [1, 2, -1]]

keenan 8.087460995 rms .1616904714 g 6.976149846

generators 498.2724869 1200

3125/3072

map [[0, 5, 1], [1, 0, 2]]

keenan 8.209877206 rms 4.569472316 g 3.741657387

generators 379.9679494 1200

393216/390625

map [[0, 8, 1], [1, -1, 2]]

keenan 9.019558680 rms 1.071950166 g 6.164414003

generators 387.8196732 1200

78732/78125

map [[0, 7, 9], [1, -1, -1]]

keenan 9.925545192 rms 1.157498146 g 6.683312553

generators 442.9792974 1200

250/243

map [[0, -3, -5], [1, 2, 3]]

keenan 10.05091489 rms 7.975800816 g 3.559026083

generators 162.9960265 1200

2109375/2097152

map [[0, 7, -3], [1, 0, 3]]

keenan 10.08322927 rms .8004099292 g 7.257180353

generators 271.5895996 1200

25/24

map [[0, 2, 1], [1, 1, 2]]

keenan 10.18726181 rms 28.85189698 g 1.414213562

generators 350.9775007 1200

648/625

map [[0, -1, -1], [4, 8, 11]]

keenan 11.09063733 rms 11.06006024 g 3.265986323

generators 505.8656425 300

20000/19683

map [[0, 4, 9], [1, 1, 1]]

keenan 11.40932735 rms 2.504205191 g 6.377042156

generators 176.2822703 1200

1600000/1594323

map [[0, -5, -13], [1, 3, 6]]

keenan 11.64300516 rms .3831037874 g 9.273618495

generators 339.5088258 1200

1990656/1953125

map [[0, 9, 5], [1, 1, 2]]

keenan 12.03289099 rms 2.983295872 g 6.377042156

generators 77.96498962 1200

16875/16384

map [[0, -4, 3], [1, 2, 2]]

keenan 12.16857021 rms 5.942562596 g 4.966554810

generators 126.2382718 1200

6561/6400

map [[0, -1, -4], [2, 4, 8]]

keenan 12.52652750 rms 4.217730124 g 5.887840578

generators 503.8351546 600

135/128

map [[0, -1, 3], [1, 2, 1]]

keenan 14.05153795 rms 18.07773298 g 2.943920288

generators 522.8623453 1200

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here's a list of temperaments with keenan badness less than 15,
where
> keenan badness is g exp(sqrt(r/7.4)) :
...

Thanks for that Gene. Much appreciated. Why no names?

That uncovered two more middle-of-the-road temperaments that didn't
appear in your list of 32 and that I didn't have in my spreadsheet.

6561/6400 (twin meantone)
1990656/1953125

But I don't understand why we lost the following that _were_ in your
list of 32:

semisuper
parakleismic
hemithird

I know why you didn't find the two half-meantones and the two
half-kleismics. Because you don't consider them to be 5-limit linear
temperaments. Can you point me to a definition of 5-limit or linear
temperament (or 5-limit linear temperament) that excludes them, other
than yours?.

But I don't understand why you didn't find twin kleismic
(244140625/241864704).

The fact that we lost some and didn't find twin kleismic, leads me to
suspect there may still be some we haven't seen yet, that have keenan
badness < 15.

Have you got too low a cutoff on complexity? Like 9 or 10 gens rms? Of
course, with my badness measure, you don't need any cutoff on
complexity or error, but I understand you might need them to limit the
amount of computation. We should go to at least 30 gens and 50 cents
to be sure it's ultimately only my badness cutting them off.

There should be _at_least_ 26 temperaments (or 22 without the
half-meantones and half-kleismics) on a list with keenan badness < 15.

By the way, if you were to set your cuttoffs to

smith badness < 861
rms complexity < 13.2 gens
rms error < 28.9 cents

you would include in your list, all those on my list (kb<15) so far,
with a minimum of what I consider junk.

I've updated my spreadsheet to include the new discoveries.

http://dkeenan.com/Music/5LimitTemp.xls.zip

The names of my top 26 5-limit temperaments are (in order of keenan
badness):
meantone
kleismic
augmented
diaschismic
schismic
small diesic
wuerschmidt
tiny diesic
porcupine
orwell
neutral thirds
diminished
minimal diesic
amt
semisuper
1990656/1953125 ?
16875/16384 ?
twin meantone
half meantone-fifth
half meantone-fourth
twin kleismic
half kleismic-minor-third
half kleismic-major-sixth
parakleismic
pelogic
hemithird

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

> Thanks for that Gene. Much appreciated. Why no names?

Because I just gave computer output.

> But I don't understand why we lost the following that _were_ in your
> list of 32:
>
> semisuper
> parakleismic
> hemithird

I suspect it's because the computation broke down in the middle and I restarted it from where it seemed to have gotten; I'll try again and see what happens.

> I know why you didn't find the two half-meantones and the two
> half-kleismics. Because you don't consider them to be 5-limit linear
> temperaments.

Not this time--I made no attempt to exclude them, but it found
(81/80)^2 instead of the half-meantones. That's because they all share the same wedgie.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
>
> > Thanks for that Gene. Much appreciated. Why no names?
>
> Because I just gave computer output.
>
> > But I don't understand why we lost the following that _were_
in your
> > list of 32:
> >
> > semisuper
> > parakleismic
> > hemithird
>
> I suspect it's because the computation broke down in the
middle and I restarted it from where it seemed to have gotten; I'll
try again and see what happens.
>
> > I know why you didn't find the two half-meantones and the
two
> > half-kleismics. Because you don't consider them to be 5-limit
linear
> > temperaments.
>
> Not this time--I made no attempt to exclude them, but it found
> (81/80)^2 instead of the half-meantones.

hmm? i thought dave said 6561:6400 was half-meantone.
someone needs to write a gentle introduction to contortion, just
to list all the issues in one place.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> hmm? i thought dave said 6561:6400 was half-meantone.

No I called that twin meantone.

> someone needs to write a gentle introduction to contortion, just
> to list all the issues in one place.

I guess you haven't looked at my spreadsheet yet.
http://dkeenan.com/Music/5LimitTemp.xls.zip

I named them like this

Comma Name Mapping Gen Period
-------------------------------------------------------------
6561/6400 twin meantone [0,-1,-4] [2,4,8] 503.8 600
81/80 half meantone-fifth [0,2,8] [1,2,4] 348.1 1200
81/80 half meantone-fourth [0,-2,-8] [1,2,4] 251.9 1200

The same pattern repeats for kleismic.

If Gene or anyone already has a different system for naming these, I'd
like to learn it.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Thanks for that Gene. Much appreciated. Why no names?
>
> Because I just gave computer output.

Sorry. I thought you had the name lookup automated.

> > But I don't understand why we lost the following that _were_ in
your
> > list of 32:
> >
> > semisuper
> > parakleismic
> > hemithird
>
> I suspect it's because the computation broke down in the middle and
I restarted it from where it seemed to have gotten; I'll try again and
see what happens.

Thanks.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > hmm? i thought dave said 6561:6400 was half-meantone.
>
> No I called that twin meantone.
>
> > someone needs to write a gentle introduction to contortion, just
> > to list all the issues in one place.
>
> I guess you haven't looked at my spreadsheet yet.
> http://dkeenan.com/Music/5LimitTemp.xls.zip
>
> I named them like this
>
> Comma Name Mapping Gen Period
> -------------------------------------------------------------
> 6561/6400 twin meantone [0,-1,-4] [2,4,8] 503.8 600

isn't this just meantone itself? isn't this an instance of torsion?

> 81/80 half meantone-fifth [0,2,8] [1,2,4] 348.1 1200
> 81/80 half meantone-fourth [0,-2,-8] [1,2,4] 251.9 1200

these are the valid instances of contortion i was talking about. they
are not temperaments, but they are quite interesting. since
their 'badness' should simply be 4 or 8 times that of meantone, such
cases can be handled with an introductory passage and never mentioned
again -- it would be easy enough for the reader to supply them for
any given badness cutoff.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Comma Name Mapping Gen Period
> > -------------------------------------------------------------
> > 6561/6400 twin meantone [0,-1,-4] [2,4,8] 503.8 600
>
> isn't this just meantone itself? isn't this an instance of torsion?

I have no idea what torsion is, but of course this is not simply
meantone. It has a half-octave period. And thanks to Graham's list
I've corrected its (lowest-terms) generator now to 96.2 cents.

> > 81/80 half meantone-fifth [0,2,8] [1,2,4] 348.1 1200
> > 81/80 half meantone-fourth [0,-2,-8] [1,2,4] 251.9 1200
>
> these are the valid instances of contortion i was talking about.
they
> are not temperaments, but they are quite interesting. since
> their 'badness' should simply be 4 or 8 times that of meantone, such
> cases can be handled with an introductory passage and never
mentioned
> again -- it would be easy enough for the reader to supply them for
> any given badness cutoff.

This assumes the reader is a mathematician. Are we planning to publish
this in a math journal or a music journal? I don't see a problem with
listing them in the appropriate place in the list, based on their
complexity.

I don't see how you can disqualify them as 5-limit temperaments simply
because all the 5-limit intervals are approximated by an even number
of generators. What kind of a definition of temperament would disallow
that?

Isn't a (octave-equivanlent) 5-limit temperament simply any scale or
tuning system that approximates ratios of 1,3 and 5 and their octave
equivalents?

There can be no argument that they are not _linear_. They have a
single generator operating withing a whole-number fraction of an
octave.

I think, Paul, that maybe you're being blinded by your "hypothesis",
since here we have the same comma involved in different temperaments
with different complexities. How about you modify your hypothesis to
take care of that, rather than try to deny that these are
temperaments.

And lest you are tempted to now claim as Gene has, that these are
simply meantone, I claim that a temperament is fundamentally defined
by its mapping of generators and periods to primes (with generator and
period in lowest terms), not by the commas that vanish. different
mapping = different temperament. Seems obvious enough to me.

If your algorithm fails to find certain kinds of temperament that
other algorithms do find, you shouldn't try to deny their existence,
you should fix your algorithm.

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

> If your algorithm fails to find certain kinds of temperament that
> other algorithms do find, you shouldn't try to deny their existence,
> you should fix your algorithm.

I would fix it, except I see not finding these things to be a positive virtue. I don't want them; I think they are merely confusing the issue.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > If your algorithm fails to find certain kinds of temperament that
> > other algorithms do find, you shouldn't try to deny their
existence,
> > you should fix your algorithm.
>
> I would fix it, except I see not finding these things to be a
positive virtue. I don't want them;

There's plenty of stuff in there that I don't want, but I'm willing to
accept that someone else might want them.

> I think they are merely confusing the issue.

What issue is that? How are they confusing it?

You could eliminate them (you have schismic and neutral-thirds-related
ones as well, with your badness measure) by setting your badness
cutoff between 386 and 439, but that would also eliminate pelogic,
minimal diesic, 16875/16384, and 1990656/1953125. Paul would be upset
to lose pelogic and I think minimal diesic is of as much interest as
many others that would still be in there.

You could eliminate the schismic-related ones by setting your
complexity cutoff between 13.2 and 13.9 gens. Actually you could set
it as low as 11.1 and hemithird would fall off the list too. I
wouldn't mind since it's on the bottom of my list. And then you could
eliminate the rest by setting your badness cutoff between 650 and 652.
That will leave pelogic and minimal diesic but will eliminate
16875/16384, and 1990656/1953125, which is probably no skin of
anyone's nose. But it seems awfully contrived, if it's just because
you don't like them. And it will get pretty tiresome trying to get rid
of them by such means, and make our two lists agree, for every
combination of odd-harmonics.

And it still doesn't make me happy because, according to my badness,
if we eliminate the half and twin meantones we should also eliminate
parakleismic, pelog and hemithird. That would be OK by me, but I
understand pelogic is non-negotiable for Paul.

It seems we don't have a list we all agree on yet.

Twin meantone, half meantone-fourth and half meantone-fifth are all in
Graham's list too. They come just before pelogic and 16875/16384.

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

> I don't see how you can disqualify them as 5-limit temperaments simply
> because all the 5-limit intervals are approximated by an even number
> of generators. What kind of a definition of temperament would disallow
> that?

the natural one, of deforming the ji lattice so that it meets itself and reduces the number of dimensions of infinite extent.

> Isn't a (octave-equivanlent) 5-limit temperament simply any scale or
> tuning system that approximates ratios of 1,3 and 5 and their octave
> equivalents?

no. they're cool, though. but these tuning systems _begin_ as ji, and evolve.
>
> There can be no argument that they are not _linear_. They have a
> single generator operating withing a whole-number fraction of an
> octave.

so what? that's true of diaschismic and augmented and diminished too, isn't it?

> I think, Paul, that maybe you're being blinded by your "hypothesis",
> since here we have the same comma involved in different temperaments
> with different complexities. How about you modify your hypothesis to
> take care of that, rather than try to deny that these are
> temperaments.

they are tuning systems, not temperaments.

> And lest you are tempted to now claim as Gene has, that these are
> simply meantone, I claim that a temperament is fundamentally defined
> by its mapping of generators and periods to primes (with generator and
> period in lowest terms), not by the commas that vanish. different
> mapping = different temperament. Seems obvious enough to me.

you're talking about linear tuning systems, not linear temperaments. tempering is always a tempering of ji. at least if you read barbour, blackwood, mandelbaum, and most of the reputable sources.

> If your algorithm fails to find certain kinds of temperament that
> other algorithms do find, you shouldn't try to deny their existence,
> you should fix your algorithm.

ok, they should be in the paper, since no one should be expected to multiply or divide by 4 or 8. but they're not temperaments -- the steel cage of just intonation doesn't turn into one of the contortional cases just by tempering the metal. it may be just as good as a tuning system, but it's a different beast.

having a period that is a fraction of an octave is already way off the map for most of the potential audience, btw. i'm glad we seem to agree that *that* should be changed . . .