Dave's 23 best 5-limit temperaments

81/80 meantone

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

generators 503.8351546 1200

keenan 6.263263749 rms 4.217730124 g 2.943920288

15625/15552 kleismic

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

generators 317.0796754 1200

keenan 6.601347654 rms 1.029625097 g 4.546060566

128/125 augmented

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

generators 491.2018553 400

keenan 7.686514108 rms 9.677665980 g 2.449489743

2048/2025 diaschismic

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

generators 494.5534684 600

keenan 7.826993942 rms 2.612822498 g 4.320493799

32805/32768 shismic

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

generators 498.2724869 1200

keenan 8.087460995 rms .1616904714 g 6.976149846

3125/3072 small diesic

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

generators 379.9679494 1200

keenan 8.209877206 rms 4.569472316 g 3.741657387

393216/390625 wuerschmidt

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

generators 387.8196732 1200

keenan 9.019558680 rms 1.071950166 g 6.164414003

78732/78125 tiny diesic

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

generators 442.9792974 1200

keenan 9.925545192 rms 1.157498146 g 6.683312553

250/243 porcupine

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

generators 162.9960265 1200

keenan 10.05091489 rms 7.975800816 g 3.559026083

2109375/2097152 orwell

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

generators 271.5895996 1200

keenan 10.08322927 rms .8004099292 g 7.257180353

25/24 neutral thirds

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

generators 350.9775007 1200

keenan 10.18726181 rms 28.85189698 g 1.414213562

648/625 diminished

map [[0, 1, 1], [4, 5, 8]]

generators 394.1343571 300

keenan 11.09063733 rms 11.06006024 g 3.265986323

20000/19683 minimal diesic

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

generators 176.2822703 1200

keenan 11.40932735 rms 2.504205191 g 6.377042156

1600000/1594323 amt

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

generators 339.5088258 1200

keenan 11.64300516 rms .3831037874 g 9.273618495

6115295232/6103515625 semisuper

map [[0, 7, 3], [2, -3, 2]]

generators 528.8539366 600

keenan 11.67903530 rms .1940181460 g 9.933109620

1990656/1953125 Extends to the 1029/1024^126/125 =
[9,5,-3,-21,30,-13] system, and needs a name.)

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

generators 77.96498962 1200

keenan 12.03289099 rms 2.983295872 g 6.377042156

16875/16384 (Extends to the 225/224^49/48 = [4,-3,2,13,-8,-14] system, and needs a name.)

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

generators 126.2382718 1200

keenan 12.16857021 rms 5.942562596 g 4.966554810

1224440064/1220703125 parakleismic

map [[0, -13, -14], [1, 5, 6]]

generators 315.2509133 1200

keenan 13.40122787 rms .2766026501 g 11.04536102

135/128 pelogic

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

generators 522.8623453 1200

keenan 14.05153795 rms 18.07773298 g 2.943920288

274877906944/274658203125 hemithird

map [[0, -15, 2], [1, 4, 2]]

generators 193.1996149 1200

keenan 14.38723787 rms .6082244804e-1 g 13.14026890

48828125/47775744

map [[0, 11, 6], [1, 1, 2]]

generators 63.83293258 1200

keenan 14.40230680 rms 2.796054904 g 7.788880963

1220703125/1207959552

map [[0, -13, -2], [1, 6, 3]]

generators 407.5847053 1200

keenan 14.45277274 rms 1.059594779 g 9.899494934

4294967296/4271484375

map [[0, -9, 7], [1, 2, 2]]

generators 55.27549315 1200

keenan 14.64533251 rms .4831084292 g 11.34313302

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

> 1600000/1594323 amt
>
> map [[0, -5, -13], [1, 3, 6]]
>
> generators 339.5088258 1200
>
> keenan 11.64300516 rms .3831037874 g 9.273618495

This extends to the 7-limit system 5120/5103^4375/4374 =
[5,13,-17,-76,41,9], with a 28/99 generator. This could be called "amt" also, unless someone has a better idea for a name--which probably would not be hard.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> 1990656/1953125 Extends to the 1029/1024^126/125 =
> [9,5,-3,-21,30,-13] system, and needs a name.)
>
> map [[0, 9, 5], [1, 1, 2]]
>
> generators 77.96498962 1200
>
> keenan 12.03289099 rms 2.983295872 g 6.377042156

How about "quarter major thirds"?

> 16875/16384 (Extends to the 225/224^49/48 = [4,-3,2,13,-8,-14]
system, and needs a name.)
>
> map [[0, -4, 3], [1, 2, 2]]
>
> generators 126.2382718 1200
>
> keenan 12.16857021 rms 5.942562596 g 4.966554810

I've called it "quarter fourths" in the past, but it could also be
"third of major thirds".

I can certainly live without those 4 beyond pelogic, i.e. hemithird
and the three new unnamed ones (which can probably remain unnamed). So
that would leave 19.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> > 1600000/1594323 amt
> >
> > map [[0, -5, -13], [1, 3, 6]]
> >
> > generators 339.5088258 1200
> >
> > keenan 11.64300516 rms .3831037874 g 9.273618495
>
> This extends to the 7-limit system 5120/5103^4375/4374 =
> [5,13,-17,-76,41,9], with a 28/99 generator. This could be called
"amt" also, unless someone has a better idea for a name--which
probably would not be hard.

How was the name amt arrived at. Is it an abbreviation for something?
It could be called "fifth of eleventh".

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

> How was the name amt arrived at. Is it an abbreviation for something?

It's an acronym for "acute minor third", from its generator.

> It could be called "fifth of eleventh".

Sounds like a borg.

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

> > 1990656/1953125 Extends to the 1029/1024^126/125 =
> > [9,5,-3,-21,30,-13] system, and needs a name.)
> >
> > map [[0, 9, 5], [1, 1, 2]]
> >
> > generators 77.96498962 1200
> >
> > keenan 12.03289099 rms 2.983295872 g 6.377042156
>
> How about "quarter major thirds"?

Not accurate; I think "chromic" would be a good name, since the generator is 21/20~25/24, the chromatic semitone or chroma.

> > 16875/16384 (Extends to the 225/224^49/48 = [4,-3,2,13,-8,-14]
> system, and needs a name.)
> >
> > map [[0, -4, 3], [1, 2, 2]]
> >
> > generators 126.2382718 1200
> >
> > keenan 12.16857021 rms 5.942562596 g 4.966554810
>
> I've called it "quarter fourths" in the past, but it could also be
> "third of major thirds".

I think quadrafourths would be fine.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
>
> > > 1990656/1953125 Extends to the 1029/1024^126/125 =
> > > [9,5,-3,-21,30,-13] system, and needs a name.)
> > >
> > > map [[0, 9, 5], [1, 1, 2]]
> > >
> > > generators 77.96498962 1200
> > >
> > > keenan 12.03289099 rms 2.983295872 g 6.377042156
> >
> > How about "quarter major thirds"?
>
> Not accurate; I think "chromic" would be a good name, since the
generator is 21/20~25/24, the chromatic semitone or chroma.

what about chromatic unison vectors?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > How was the name amt arrived at. Is it an abbreviation for
something?
>
> It's an acronym for "acute minor third", from its generator.

Lets call it that then, since AMT isn't pronouncable and why save one
syllable just to make it more obscure.

> > It could be called "fifth of eleventh".
>
> Sounds like a borg.

Tee hee. Yeah. One with a lisp. A cute minor.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
>
> > > 1990656/1953125 Extends to the 1029/1024^126/125 =
> > > [9,5,-3,-21,30,-13] system, and needs a name.)
> > >
> > > map [[0, 9, 5], [1, 1, 2]]
> > >
> > > generators 77.96498962 1200
> > >
> > > keenan 12.03289099 rms 2.983295872 g 6.377042156
> >
> > How about "quarter major thirds"?
>
> Not accurate; I think "chromic" would be a good name, since the
generator is 21/20~25/24, the chromatic semitone or chroma.
>
> > > 16875/16384 (Extends to the 225/224^49/48 = [4,-3,2,13,-8,-14]
> > system, and needs a name.)
> > >
> > > map [[0, -4, 3], [1, 2, 2]]
> > >
> > > generators 126.2382718 1200
> > >
> > > keenan 12.16857021 rms 5.942562596 g 4.966554810
> >
> > I've called it "quarter fourths" in the past, but it could also be
> > "third of major thirds".
>
> I think quadrafourths would be fine.

I'm happy with both of those. The only worry: there are lots of things
called chromas in Manuel's intnam.par. Would "chromatic" be better, or
would that tend to suggest something else?

so 7625597484987:7629394531250 would be the first major one we're
cutting off? it's of course important as the 5-limit aspect of
ennealimmal. you might argue that ennealimmal is a higher-limit
consideration. but will the current criteria dave is using preclude
its inclusion when we get to higher limits? i certainly hope not. we
don't want to go through this same subjective process over again for
every 'limit'.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > How was the name amt arrived at. Is it an abbreviation for
> something?
> >
> > It's an acronym for "acute minor third", from its generator.
>
> Lets call it that then, since AMT isn't pronouncable and why save
one
> syllable just to make it more obscure.
>
> > > It could be called "fifth of eleventh".
> >
> > Sounds like a borg.
>
> Tee hee. Yeah. One with a lisp. A cute minor.

ok, so now we're pedophiles, are we? :)

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

> I'm happy with both of those. The only worry: there are lots of
things
> called chromas in Manuel's intnam.par.

this is even worse than 'diesic'.

> Would "chromatic" be better, or
> would that tend to suggest something else?

chromatic unison vectors.

> > > generators 126.2382718 1200

This is about 2/3 of a whole tone, in Latin "bes toni".
So how about bestonic?

Manuel

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I'm happy with both of those. The only worry: there are lots of
> things
> > called chromas in Manuel's intnam.par.
>
> this is even worse than 'diesic'.

When I came up with "chromic", I thought of certain compounds of chromium, such as chromic acid. I suppose we could just call the thing "chrome".

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > > > generators 126.2382718 1200
>
> This is about 2/3 of a whole tone, in Latin "bes toni".
> So how about bestonic?

16875/16384, right? "best tonic" -- 16384 is a power of 2 . . .
hmm . . .

sounds like 'bastoni' (italian bread). fits in well with 'injera'
(ethiopian bread). actually this works well, since 225/224^49/48
(bastoni) is in the same 'aisle' as 81/80^50/49 (injera).

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> so 7625597484987:7629394531250 would be the first major one we're
> cutting off? it's of course important as the 5-limit aspect of
> ennealimmal. you might argue that ennealimmal is a higher-limit
> consideration.

Absolutely.

> but will the current criteria dave is using preclude
> its inclusion when we get to higher limits? i certainly hope not. we
> don't want to go through this same subjective process over again for
> every 'limit'.

I doubt it. I'll probably lose interest by then. ;-)

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

> sounds like 'bastoni' (italian bread). fits in well with 'injera'
> (ethiopian bread). actually this works well, since 225/224^49/48
> (bastoni) is in the same 'aisle' as 81/80^50/49 (injera).

Next time I'm stuck for a name, I'll think bread.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > sounds like 'bastoni' (italian bread). fits in well with 'injera'
> > (ethiopian bread). actually this works well, since 225/224^49/48
> > (bastoni) is in the same 'aisle' as 81/80^50/49 (injera).
>
> Next time I'm stuck for a name, I'll think bread.

or spiny mammal :)

>> It's an acronym for "acute minor third", from its generator.
>
>Lets call it that then, since AMT isn't pronouncable and why save one
>syllable just to make it more obscure.

Amt sure is pronouncable. I must say I'm quite amt about the keeping
the name. :)

Actually, I don't care what you call it.

Except "acute minor thirds". That's a terrible name:

1. Three words. Temperaments should have cool, single-word
names.

2. I find it perverse to name temperaments by their relation to
diatonic intervals. I guess this counts against Amt too.

-Carl

>chromatic unison vectors.

You're suggesting this as a name for a temperament???

-Carl

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

> >chromatic unison vectors.
>
> You're suggesting this as a name for a temperament???

no!

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Actually, I don't care what you call it.
>
> Except "acute minor thirds". That's a terrible name:
>
> 1. Three words. Temperaments should have cool, single-word
> names.

Well yeah, but if possible, they should give you some clue as to what
the temperament is. This can be

(a) The size of the generator (and possibly how many periods to the
octave when there is more than one):

neutral thirds
acute minor thirds
hemithirds
quadrafourths/bestonic
chrome
augmented (indirectly)
diminished (indirectly)
pelogic (indirectly)
orwell (obscurely)

and possibly semisuper if "semi" is meant to indicate the half-octave
period and "super" is short for super fourth. However I don't think
most people distinguish between semi and hemi, so I think "twin" (or
"double") is a better indicator of a half-octave period, as in "twin
chains", which generalises to triple, quadruple, quintuple, sextuple,
septuple, octuple, ?...

Also note that in lowest terms the generator for "semisuper" is in
fact a better example of a chromatic semitone than the one for
"chrome".

or
(b)The comma that vanishes:
schismic
kleismic
diaschismic
wuerschmidt
tiny diesic
minimal diesic
limmal
meantone (indirectly)

or is "meantone" indicating the generator size indirectly?

Unfortunately this method doesn't generalise well to higher limits
since more than one comma vanishes. We could stick to using the single
comma that vanishes in its 5-limit subset, but that won't work for
temperaments that don't _have_ a 5-limit subset.

A problem with having these two different systems is that when the
comma is bigger than a chromatic semitone (71c) or the generator is
smaller than a large limma (133 c) it isn't clear whether the name
refers to the generator or the comma. limmal and chrome being
examples.

(c) No clue whatsoever:
porcupine
starling
pajara
magic
miracle

There's nothing particularly magical or miraculuos about Magic and
Miracle at the 5-limit.

> 1. Three words. Temperaments should have cool, single-word
> names.

I can't see any reason to stop calling neutral thirds by a two word
three syllable name. I think number of syllable is more relevant than
number of words. I think 2 words is ok if no more than 4 syllables.
di-a-schis-mic, par-a-kleis-mic, sem-i-su-per. And a-cute mi-nor
thirds isn't as bad as min-i-mal di-es-ic.

> 2. I find it perverse to name temperaments by their relation to
> diatonic intervals. I guess this counts against Amt too.

I know what you mean, but it's not too perverse for 5-limit since the
JI diatonic is 5-limit. But hey, that's simply how we name intervals,
of any limit. I think we're stuck with it.

I favour naming temperaments based on the size of the generator and
the number of periods to the octave. That immediately tells someone
how to make a scale from it with only basic math.

I favour describing the size of the generator as the appropriate
fraction of the interval (within the specified limit) that needs the
fewest generators and only whole octaves, provided it is accurate
enough and provided it can be done in four syllables or less. This is
more-or-less the approach Gene used with:

hemithirds
quadrafourths (except this should be trithirds, my mistake)

Using this system, AMT becomes

pentelevenths

I'm unsure whether we should put an "s" on the end or not. But I think
we should, because "neutral third temperament" sounds odd to me.

Porcupine would become hemiminorthirds. Too many syllables. We could
shorten it to haemorrhoids. No? OK, I guess it stays porcupine. :-)

Chrome would be quadraminorthirds. Too many syllables again. Is there
a shorter way to say "a quarter of"? Is there a shorter way to say
"minor thirds"? There _is_ a shorter way to say "half of", "bi" as in
bicarbonate.

Tiny diesic becomes hemisixths.
Minimal diesic becomes quadrafifths.
4294967296/4271484375 becomes septathirds.

Any valid generator could be used, not just the smallest one.
So semisuper becomes twin tritenths or double tritenths.

But it's no use calling pajara/twintone/paultone "twin fourths" or
"twin fifths", since that could apply to twin meantone as well.

And Magic and Wuerschmidt would both be "major thirds" so they had
better stay as they are.

>Well yeah, but if possible, they should give you some clue as to what
>the temperament is. This can be

Well, crazy names won't tell you as much, but are more memorable,
and might make up for it. See also my complaint about using diatonic
intervals for temperaments where the diatonic scale may not even
be supported.

>hemithirds
>quadrafourths/bestonic
>chrome

I still don't know the generators for these.

>augmented (indirectly)
>diminished (indirectly)
>pelogic (indirectly)

Where historical names exist, I think they should
be used.

>orwell (obscurely)

But you'll never forget it once you've heard it.

>and possibly semisuper if "semi" is meant to indicate the half-octave
>period and "super" is short for super fourth. However I don't think
>most people distinguish between semi and hemi,

I'm not sure I do. I'd guess semi is partial, while hemi is
exactly half.

Super is short for super fourth?

>(b)The comma that vanishes:
>schismic
>kleismic
>diaschismic
>wuerschmidt
>tiny diesic
>minimal diesic
>limmal
>meantone (indirectly)

These are my favs -- I consider unison vectors far more informative
than generator size. Except for stuff like "tiny" diesic vs. "minimal"
diesic, which I still don't know as I write this.

>or is "meantone" indicating the generator size indirectly?

Meantone is just historical. It doesn't have to make sense.

>Unfortunately this method doesn't generalise well to higher limits
>since more than one comma vanishes.

I was just going to mention that.

>We could stick to using the single comma that vanishes in its
>5-limit subset, but that won't work for temperaments that don't
>_have_ a 5-limit subset.

Indeed. That's when we resort to shamelessly locking in all of our
surnames! :)

>starling

When was this ratified?

>There's nothing particularly magical or miraculuos about Magic and
>Miracle at the 5-limit.

That's okay. These names are historical now, too.

>>1. Three words. Temperaments should have cool, single-word
>>names.
>
>I can't see any reason to stop calling neutral thirds by a two word
>three syllable name.

I don't think any established (as in, more than a month or two)
names should be changed at all.

>I think number of syllable is more relevant than number of words.

I don't care if it's hard to say, I just want people to want to say
it. Compare "Acute minor thirds" to "kleismic" here. Characters
from novels, breads, etc., are also good.

>> 2. I find it perverse to name temperaments by their relation to
>> diatonic intervals. I guess this counts against Amt too.
>
>I know what you mean, but it's not too perverse for 5-limit since the
>JI diatonic is 5-limit. But hey, that's simply how we name intervals,
>of any limit. I think we're stuck with it.

I'm willing to accept this, but I don't have to like it. I try
to avoid it, at any rate.

>Tiny diesic becomes hemisixths.
>Minimal diesic becomes quadrafifths.
>4294967296/4271484375 becomes septathirds.
>
>Any valid generator could be used, not just the smallest one.
>So semisuper becomes twin tritenths or double tritenths.
>
>But it's no use calling pajara/twintone/paultone "twin fourths" or
>"twin fifths", since that could apply to twin meantone as well.
>
>And Magic and Wuerschmidt would both be "major thirds" so they had
>better stay as they are.

There's not enough variety in this naming scheme for my taste. In
effect, I'm going to have to think about the name each time I hear
the temperament, whereas "orwell" lives in my mind as its own entity.

-Carl

Does anyone have a list of the proper latin-derived prefixes for the
various fractions.

And shouldn't we use "semi" for 1/2 since we already have semitone.
And hemi is greek, not latin, as are tetra, penta, hexa, hepta, (octa
is both L and Gr). And please ignore my suggestion of "bi", it
cleraly means 2, not 1/2.

Similarly I realise we can't use "tri" to mean 1/3 since it already
clearly means 3 in tritone. Is "tertia" clearly 1/3 and not 3?

Do any of them beyond 2 clearly mean 1/n and not also n?

I think "quarta" is more clearly 1/4 than "quadra" which sounds to me
like it could equally mean 4.

Is "quinta" clearly 1/5 and not 5?

Or is it the case that you change them from a multiple to a fraction
by changing the final vowel e.g.

semi
terti
quarti
quinti
sexti
septi
octi

Even if there's no such rule, we can make one, it's pretty suggestive,
following on from semi.

Certainly a quartile and a quintile are 1/4 and 1/5. I could only find
that a tertian is 1/3 of a tun (liquid measure). And a sextary is 1/6
of a congius or modius. And there's sextant meaning 1/6 part.

So we have the following:
semiminorthirds (porcupine)
tertiminorsixths (orwell)
semisixths (tiny diesic)
quintelevenths (AMT)
quartififths (minimal diesic)
tertithirds (quadrafourths)
twin tertitenths (semisuper)
septithirds (4294967296/4271484375)
quartiminorthirds (chrome)
semithirds (hemithirds) [not on my list for the paper]

The only prefix I'm not real keen on there is "terti", maybe 'cause it
sounds like "dirty". Other suggestions?

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >Well yeah, but if possible, they should give you some clue as to
what
> >the temperament is. This can be
>
> Well, crazy names won't tell you as much, but are more memorable,
> and might make up for it.

I think some of these systematic names sound pretty crazy and if any
of them had had any history I thing you'd remember them just fine. I
been remembering several temperaments just fine up till now, simply by
the generator size in cents. I just say to myself "oh that's 126 cent
temperament" or whatever. Orwell was "272 cent temperament" or
"subminor thirds temperament" to me for a long time.

> See also my complaint about using
> diatonic
> intervals for temperaments where the diatonic scale may not even
> be supported.

That's your problem. I long ago ceased to think of them as in any way
related to the diatonic scale, they are just particular sized
intervals near particular ratios.

> >hemithirds
> >quadrafourths/bestonic
> >chrome
>
> I still don't know the generators for these.

I thought you said you'd rather know the comma. The generators are, as
my most recent name suggestions approximately tell you,
semithirds, half a major third, 193 c, (rms is 193 c)
tertithirds, a third of a third, 129 c, (rms is 126 c)
quartiminorthirds, a quarter of a minor third, 79c, (rms is 78 c)

> >augmented (indirectly)
> >diminished (indirectly)
> >pelogic (indirectly)
>
> Where historical names exist, I think they should
> be used.

Absolutely!

> >orwell (obscurely)
>
> But you'll never forget it once you've heard it.

Oh yeah. If folks were told once that it was a 19/84 oct
generator, I'll bet some would come back to it months later wondering
"was that a temperament consistent with both 19-tET and 84-tET".

> >and possibly semisuper if "semi" is meant to indicate the
half-octave
> >period and "super" is short for super fourth. However I don't
think
> >most people distinguish between semi and hemi,
>
> I'm not sure I do. I'd guess semi is partial, while hemi is
> exactly half.

Nah they're both half. Hemi is greek semi is latin. We need latin,
since semi is already used for exactly the purpose I want to use it,
in semi-tone.

> Super is short for super fourth?

Beats me. Gene?

> >(b)The comma that vanishes:
> >schismic
> >kleismic
> >diaschismic
> >wuerschmidt
> >tiny diesic
> >minimal diesic
> >limmal
> >meantone (indirectly)
>
> These are my favs -- I consider unison vectors far more informative
> than generator size. Except for stuff like "tiny" diesic vs.
"minimal"
> diesic, which I still don't know as I write this.

Indeed. I think gene made up the "tiny diesis" to start with. But if I
say semisixths and quartififths anyone who knows the system (and
even the system is not too hard to figure out for yourself) can figure
out the rest if they want to.

Very few musicians and composers even know the difference between a
pythagorean and a syntonic comma, let alone what a diaschisma or a
kleisma is. The historical comma-based names can stay but I say we
don't base any more on obscurely named commas.

In contrast, everyone knows what thirds fourths etc. are, even if they
will assume the 12-tET version instead of the just.

> >or is "meantone" indicating the generator size indirectly?
>
> Meantone is just historical. It doesn't have to make sense.

Good grief! I never meant to suggest we rename meantone.

> >Unfortunately this method doesn't generalise well to higher limits
> >since more than one comma vanishes.
>
> I was just going to mention that.

A serious problem, yes?

> >We could stick to using the single comma that vanishes in its
> >5-limit subset, but that won't work for temperaments that don't
> >_have_ a 5-limit subset.
>
> Indeed. That's when we resort to shamelessly locking in all of our
> surnames! :)

Tee hee.

> >starling
>
> When was this ratified?

Sorry I mentioned it. It's a 7-limit planar temperament by Herman
Miller where the 125:126 vanishes.

> >There's nothing particularly magical or miraculuos about Magic and
> >Miracle at the 5-limit.
>
> That's okay. These names are historical now, too.

Good grief! Only months old and only on this list. That's not
historical. But anyway I don't propose to change these.

> >>1. Three words. Temperaments should have cool, single-word
> >>names.
> >
> >I can't see any reason to stop calling neutral thirds by a two word
> >three syllable name.
>
> I don't think any established (as in, more than a month or two)
> names should be changed at all.
>
> >I think number of syllable is more relevant than number of words.
>
> I don't care if it's hard to say, I just want people to want to say
> it. Compare "Acute minor thirds" to "kleismic" here.

Yes. I don't want to say "Acute minor thirds" because it is hard to
say.

> Characters from novels, breads, etc., are also good.

They are fun when you're in the in-crowd who knows, but they are
totally mystifying to newcomers. I dunno about you, but I get tired of
explaining terms to newbies or telling them where to find the list or
dictionary. I'd rather newbies had at laest some chance of figuring it
out for themselves.

> >I know what you mean, but it's not too perverse for 5-limit since
the
> >JI diatonic is 5-limit. But hey, that's simply how we name
intervals,
> >of any limit. I think we're stuck with it.
>
> I'm willing to accept this, but I don't have to like it. I try
> to avoid it, at any rate.

Understood.

> >Tiny diesic becomes hemisixths.
> >Minimal diesic becomes quadrafifths.
> >4294967296/4271484375 becomes septathirds.
...
> There's not enough variety in this naming scheme for my taste. In
> effect, I'm going to have to think about the name each time I hear
> the temperament, whereas "orwell" lives in my mind as its own
entity.
>

But Carl, that's only because it has a history with you.

There is at least one temperament name that essentially uses this
system that has been around for a long time. I'm sure you don't have
any problem recognising it. neutral thirds.

Sure the terti quarti quinti ones are a little more difficult but they
will usually be the more complex and therefore less popular ones
anyway. When we go to higher limits, the best ones often have a
generator which is a whole consonant interval. Like "subminor thirds
temperament", the name that was used before "Orwell", for that 7-limit
temperament.

>> Characters from novels, breads, etc., are also good.
>
>They are fun when you're in the in-crowd who knows, but they are
>totally mystifying to newcomers. I dunno about you, but I get tired of
>explaining terms to newbies or telling them where to find the list or
>dictionary. I'd rather newbies had at laest some chance of figuring it
>out for themselves.

The names will be the least of their worries.

>> >Tiny diesic becomes hemisixths.
>> >Minimal diesic becomes quadrafifths.
>> >4294967296/4271484375 becomes septathirds.
>...
>> There's not enough variety in this naming scheme for my taste. In
>> effect, I'm going to have to think about the name each time I hear
>> the temperament, whereas "orwell" lives in my mind as its own
>> entity.
>
>But Carl, that's only because it has a history with you.

Yes, but my more general point was that I like off-the-wall
associations. Maybe just me, too.

>There is at least one temperament name that essentially uses this
>system that has been around for a long time. I'm sure you don't have
>any problem recognising it. neutral thirds.

True, but it's the only one like that.

>Sure the terti quarti quinti ones are a little more difficult but they
>will usually be the more complex and therefore less popular ones
>anyway. When we go to higher limits, the best ones often have a
>generator which is a whole consonant interval. Like "subminor thirds
>temperament", the name that was used before "Orwell", for that 7-limit
>temperament.

And we used to say "chain-of-minor-thirds" for kleismic!

How do you differentiate two different mappings with the same
generator?

-Carl

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

I
> been remembering several temperaments just fine up till now, simply by
> the generator size in cents. I just say to myself "oh that's 126 cent
> temperament" or whatever. Orwell was "272 cent temperament" or
> "subminor thirds temperament" to me for a long time.

That doesn't really work in any exact way, and it incorrectly leaves the impression that temperaments are octave-based; if you want to really be scientific about it I suggest thinking of them in terms of the wedgies. To me wedgies do the precise naming thing, and verbal labels do the "remember this" thing.
> > Super is short for super fourth?
>
> Beats me. Gene?

It refers to 3125/2304, which I claimed was a semisuper fourth for the purpose of coming up with a name for the temperament. This, by the way, shows why I prefer "hemi" to mean "half"--"semi" is much more vague.
> > These are my favs -- I consider unison vectors far more informative
> > than generator size.

There you go--stick with wedgies. :)
> > >Unfortunately this method doesn't generalise well to higher limits
> > >since more than one comma vanishes.
> >
> > I was just going to mention that.
>
> A serious problem, yes?

Write the wedgies as wedge products of an MT reduced basis?

> > >starling
> >
> > When was this ratified?
>
> Sorry I mentioned it. It's a 7-limit planar temperament by Herman
> Miller where the 125:126 vanishes.

When was that discussed? I was going over 7-limit planars on tuning, and I recall someone saying that 126/125 had been looked at, but I thought starling was a scale!
>
> > >Tiny diesic becomes hemisixths.
> > >Minimal diesic becomes quadrafifths.
> > >4294967296/4271484375 becomes septathirds.

Oog.

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> And we used to say "chain-of-minor-thirds" for kleismic!

Oh yeah. How embarrassing. Not.

> How do you differentiate two different mappings with the same
> generator?

I just say 5-limit whatever versus 7-limit whatever, and so on.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> I
> > been remembering several temperaments just fine up till now,
simply by
> > the generator size in cents. I just say to myself "oh that's 126
cent
> > temperament" or whatever. Orwell was "272 cent temperament" or
> > "subminor thirds temperament" to me for a long time.
>
> That doesn't really work in any exact way,

It doesn't need to, most of the time, and when it might lead to
confusion between temperaments, _then_ we can use other names.

>
and it incorrectly leaves
the impression that temperaments are octave-based;
>

Huh? All the one's we've been talking about for this paper _are_
octave based.

> if you want to
really be scientific about it I suggest thinking of them in terms of
the wedgies. To me wedgies do the precise naming thing, and verbal
labels do the "remember this" thing.
>

I'm sorry. I've never managed to understand what a wedgie is, and
frankly I don't see any need to, in order to understand temperaments.

> Write the wedgies as wedge products of an MT reduced basis?

Doesn't really sound like something you can base a snappy name on.

> When was that discussed? I was going over 7-limit planars on tuning,
and I recall someone saying that 126/125 had been looked at, but I
thought starling was a scale!
> >

It is. Maybe I was the first to apply the term to the temperament the
scale is in. In referring to Genes recent 8-noter as Starling-8.

> > > >Tiny diesic becomes hemisixths.
> > > >Minimal diesic becomes quadrafifths.
> > > >4294967296/4271484375 becomes septathirds.
>
> Oog.

Thanks for your helpful comments.

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

> Huh? All the one's we've been talking about for this paper _are_
> octave based.

Not in my head; maybe in yours. :)

> I'm sorry. I've never managed to understand what a wedgie is, and
> frankly I don't see any need to, in order to understand temperaments.

One thing they might help with is with this idea that things are octave-based. If 81/80 defines meantone, where's the octave?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> > And we used to say "chain-of-minor-thirds" for kleismic!
>
> Oh yeah. How embarrassing. Not.
>
> > How do you differentiate two different mappings with the same
> > generator?
>
> I just say 5-limit whatever versus 7-limit whatever, and so on.

that won't necessarily do the trick.

look at the fourth and fifth entries here:

http://x31eq.com/limit7.txt

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Huh? All the one's we've been talking about for this paper _are_
> > octave based.
>
> Not in my head; maybe in yours. :)

gene means that the period is not always one octave, even in your
list, dave.

I've added proposed systematic names to my spreadsheet now, for all
the non-historicals. I've included the previous names as well, even if
they were only invented yesterday.

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

There would be no harm in giving more than one name for each
temperament in THE PAPER.

This has been full-on addictive for me the past few days. You'll all
be relieved to hear I'm gonna have to go cold turkey for a while, or I
won't have a job or a family.

Carl, I believe I have provided the "crucial BS control" that you were
hoping for.

I hope that by the time I come back you will all have long-ago agreed
on the final list for 5-limit and moved on to 7-limit. I suggest doing
that before coming back to the incomplete ones, for the same reason we
did 5-limit first; we're familiar with it.

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

> gene means that the period is not always one octave, even in your
> list, dave.

I mean more than that. My point of departure is that for example 5-limit meantone is defined by 81/80. I now decide to look at this in connection with octaves, and take 2/\81/80 = [0,1,4], giving me my map to generators. Then [0,1,4]/\[a,b,c] = 2^(c-4b) 3^4a 5^-a so if a=1, b=1, and c=0 I get [0,1,4]/\[1,1,0] = 81/80. This may therefore be used as a map, giving me generators of an octave and meantone fifth.

What if I look at 720 instead of 2? I now have 720/\81/80 = [-6,0,24].
This has a gcd of 6, so I divide out the 6, getting [-1,0,4]. I now wedge [-1,0,4]/\[a,b,c] = 2^(-b) 3^(4a+c) 5^(-b), so a=b=1, c=0 will work to give me 81/80: [-1,0,4]/\[1,1,0] = 81/80. I now use this as my map, setting the condition that 720 must be pure if I want. I get generators of about 3/2 and 3, and if 720 is pure, then the 3 is really 720^(1/6), and is my period, and the meantone fifth is my generator. Now meantone consists of six parallel lines of generators an approximate twelvth apart!

My point is, meantone is not really defined by the choice to priviledge octaves, which is a separate consideration.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> > > How do you differentiate two different mappings with the same
> > > generator?
> >
> > I just say 5-limit whatever versus 7-limit whatever, and so on.
>
> that won't necessarily do the trick.
>
> look at the fourth and fifth entries here:
>
> http://x31eq.com/limit7.txt

You'll have to do better than that.

270.8 cent generator, [-2, -3, -1] mapping, 45 cent minimum MA error

The generator here must function as 7:8, 6:7 and 5:6. Look at the
minimum error. As a 7-limit temperament it is garbage. But if forced
to find a meaningful name for it I would call it "7:8 = 5:6" or
"supermajor second minor thirds" or "diminished third minor thirds",
after Gene's naming of "fourth thirds".

271.3 cent generator, [7, -3, 8] mapping, 4.5 cent minimum MA error

This is of course 7-limit subminor thirds or 7-limit orwell.

I must point out that Gene invented the "generator as fraction of
consonant interval" nomenclature, with "quadrafourths". I'm just
arguing that we should
(a) extend its use,
(b) refine the prefixes to more clearly indicate fractions as opposed
to multiples,
(c) use latin consistently (not mix it with greek), and
(d) apply it always to the interval that is made up of the fewest
generators (and doesn't contain any periods except as whole octaves).

Regardsing (c), I wouldn't mind if they were consistently greek except
I feel the latin ones work better regarding criterion (b) (fractions
versus multiples).

Regarding (d): In the case where there is more than one consonant
interval which is made up of the minimum number of generators, then
the largest and smallest in size should both be mentioned, as above.
This was Gene's invention too, with fourth thirds.

I also think that when a temperament _does_ extend to being good at a
higher limit, fewer generators map to a consonant interval at the
higher limit, then it fine to use the higher limit name prefixed by
the limit. So 5-limit Orwell can also be called 5-limit subminor
thirds and we can forget about calling it tertiminorsixths.

Kleismic and parakleismic is a better example of the problem you were
setting up for me. I approve of these names, and para (presumably
meaning "near to") might generally be used for the badder of the two
since para can also mean "improper".

So the horrible 270.8 cent temperament above could also be para-orwell
or para-subminor-thirds.

However I remember Gene using cata- meaning "down". Its opposite is
ana- (up). I quite like these two, and they (or para) would remove the
temptation to use semi to mean "almost", and leave it to mean "half",
as in semitone or semicircle.

So the horrible 270.8 cent temperament above could also be cata-orwell
or cata-subminor-thirds. And parakleismic could be catakleismic.

Gene, What was it that you used cata- for?

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

> Gene, What was it that you used cata- for?

It had a generator in the 7 and 11 limit a whisker below the
kleismic 5-limit generator, but I really shouldn't have called it that, since it is a whisker *above* the h19^h15 = [6,5,3,-7,12,-6]
7-limit generator, which I was taking to be the 7-limit kleismic.
So it's a misnomer. The other reason is that it sounds like
"cataclysmic"--ha ha. Get it? I don't either. :)

On Wed, 13 Mar 2002 08:39:23 -0000, "dkeenanuqnetau" <d.keenan@uq.net.au>
wrote:

>> When was that discussed? I was going over 7-limit planars on tuning,
>and I recall someone saying that 126/125 had been looked at, but I
>thought starling was a scale!
>> >
>
>It is. Maybe I was the first to apply the term to the temperament the
>scale is in. In referring to Genes recent 8-noter as Starling-8.

It's referred to as "Starling temperament" on the Warped Canon page,
although of course the canon doesn't take advantage of the 7-limit commas
in this case. I believe I also referred to it as a temperament in some old
posts to the tuning list. I described it as essentially what we'd now call
a planar temperament with (approximate) major third and minor third
generators, having 126/125 as a unison vector. This would have been
sometime in 1999, but I don't have the exact reference.

The prototypical Starling scale had a 312 cent minor third and a 388 cent
major third. A later variation had slightly tempered octaves. But I'm
starting to think that the exact size of the thirds isn't the most
definitive feature of the tuning. Those were just the sizes of the
geenrators that optimized a particular set of 7-limit intervals I was
interested in at the time.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

On Tue, 12 Mar 2002 05:30:45 -0000, "genewardsmith"
<genewardsmith@juno.com> wrote:

>1990656/1953125 Extends to the 1029/1024^126/125 =
>[9,5,-3,-21,30,-13] system, and needs a name.)
>
>map [[0, 9, 5], [1, 1, 2]]
>
>generators 77.96498962 1200
>
>keenan 12.03289099 rms 2.983295872 g 6.377042156

This looks like a good candidate for the "Starling" name. Besides having
126/125 as a unison vector, its generator is a prominent melodic interval
in Starling temperament.

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> On Tue, 12 Mar 2002 05:30:45 -0000, "genewardsmith"
> <genewardsmith@j...> wrote:
>
> >1990656/1953125 Extends to the 1029/1024^126/125 =
> >[9,5,-3,-21,30,-13] system, and needs a name.)
> >
> >map [[0, 9, 5], [1, 1, 2]]
> >
> >generators 77.96498962 1200
> >
> >keenan 12.03289099 rms 2.983295872 g 6.377042156
>
> This looks like a good candidate for the "Starling" name. Besides
having
> 126/125 as a unison vector, its generator is a prominent melodic
interval
> in Starling temperament.

wouldn't this be a *tempering of* starling? i thought that starling
was the *planar* temperament defined by *only* 126/125 vanishing;
this here is one *linear* temperament of that (1029/1024 vanishing as
well), with a 'fifth' of 702 cents, and a 'major third' of 390 cents.
this generator would imply that the most important scales have 15,
16, 31, 46, 77 . . . notes per octave. is this really starling?

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> On Tue, 12 Mar 2002 05:30:45 -0000, "genewardsmith"
> <genewardsmith@j...> wrote:
>
> >1990656/1953125 Extends to the 1029/1024^126/125 =
> >[9,5,-3,-21,30,-13] system, and needs a name.)
> >
> >map [[0, 9, 5], [1, 1, 2]]
> >
> >generators 77.96498962 1200
> >
> >keenan 12.03289099 rms 2.983295872 g 6.377042156
>
> This looks like a good candidate for the "Starling" name. Besides
having
> 126/125 as a unison vector, its generator is a prominent melodic
interval
> in Starling temperament.

This is the one you suggested calling chrome. Starling? No. As Paul
said, its only one possible further tempering of the real Starling
planar temperament. But also it's a very uninteresting temperament,
having so many generators to the fifth, with errors still as large as
3 cents. With a smallest MOS of 15 notes (having one triad) it's also
fairly uninteresting melodically. It's marginally more interesting at
7-limit, but beyond that forget it.

Quarter minor thirds is a good enough name for it for me.