Tuning-Math Digests messages 4225 - 4249

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Message: 4225

Date: Tue, 12 Mar 2002 01:00:54

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> ok. gene, once again, this means that in the 'gens' calculation, the 
> number of generators in the 3:1 should be multiplied by log(3), the 
> number of generators in the 5:3 should be multiplied by log(5), the 
> number of generators in the 5:1 should be multiplied by log(5).

And to make the result meaningful (i.e. comparable to the unweighted 
values) then after you take the RMS of these weighted values you 
should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).


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Message: 4226

Date: Tue, 12 Mar 2002 09:59:07

Subject: Re: amt

From: dkeenanuqnetau

--- 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.


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Message: 4227

Date: Tue, 12 Mar 2002 01:27:29

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Dave's getting 7.3 too, with this:
> 
> SQRT((E13^2+F13^2+(E13-F13)^2)/3)*1200/L13
> 
> Anybody care to explain why this isn't total rubbish?  Putting
> both the individual gens per idenitity (E13 and F13) and the
> total width of the chain (E13-F13) together into the rms calc???

That's unweighted rms complexity which is

SQRT((gens(1:3)^2+gens(1:5)^2+gens(3:5)^2)/3)*1200/period

where period is in cents and 

gens(3:5) = gens(1:5)-gens(1:3), where the gens are signed quantities, 
not absolute values. So it's not necessarily the total width.

Max-absolute complexity (your total width) is
MAX(ABS(gens(1:3),ABS(gens(1:5),ABS(gens(3:5))*1200/period
which is equivalent to
ABS(MAX(gens(1:3),gens(1:5))-MIN(gens(1:3),gens(1:5)))*1200/period

By expanding gen(3:5)^2, rms complexity could be calculated as
SQRT((gens(1:3)^2+gens(1:5)^2-gens(1:3)*gens(1:5))*2/3)*1200/period
but who cares.


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Message: 4228

Date: Tue, 12 Mar 2002 10:12:13

Subject: Re: Dave's 23 best 5-limit temperaments

From: dkeenanuqnetau

--- 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?


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Message: 4229

Date: Tue, 12 Mar 2002 02:37:24

Subject: Re: Dave's 18 best 5-limit temperaments

From: genewardsmith

--- 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.


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Message: 4230

Date: Tue, 12 Mar 2002 10:14:25

Subject: Re: Dave's 23 best 5-limit temperaments

From: paulerlich

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'.


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Message: 4231

Date: Tue, 12 Mar 2002 04:30:25

Subject: Re: Dave's 18 best 5-limit temperaments

From: dkeenanuqnetau

--- 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.


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Message: 4232

Date: Tue, 12 Mar 2002 10:15:19

Subject: Re: amt

From: paulerlich

--- 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? :)


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Message: 4233

Date: Tue, 12 Mar 2002 22:49:02

Subject: Re: Systematic naming of new temperaments (was: amt)

From: Carl Lumma

>> 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


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Message: 4234

Date: Tue, 12 Mar 2002 04:51:08

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >carl, if 'identity' is defined as 'consonant interval', then the 
> >*only* thing going in here is the individual gens per identity. 
> >that's all. E13-F13 is the major sixth or minor third.
> 
> Oh.  Well, I'm not sure how that's significant, since in regular
> temperaments it will always be the difference of the 3 and 5
> mappings.

right -- so what's your objection?


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Message: 4235

Date: Tue, 12 Mar 2002 10:16:05

Subject: Re: Dave's 23 best 5-limit temperaments

From: paulerlich

--- 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.


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Message: 4236

Date: Tue, 12 Mar 2002 05:04:19

Subject: Re: Dave's 18 best 5-limit temperaments

From: paulerlich

--- 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 . . .


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Message: 4237

Date: Tue, 12 Mar 2002 11:19:46

Subject: Re: A common notation for JI and ETs

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

[Secor]
>> That's something that I don't like about the Sims notation -- down
>> arrows used in conjunction with sharps, and up arrows with flats.

[Keenan]
>I think Manuel exempts sharps and flats from this criticism.

Yes indeed, for example, Eb/ is always the nearest tone to 6/5
as E\ is always nearest to 5/4.

Manuel


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Message: 4239

Date: Tue, 12 Mar 2002 10:30:20

Subject: Re: A proposed list of 5-limit not-quite-Just-thingies

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> What happened to chromic?

I don't need it on the list, and by your badness it is _much_ worse 
than any of those on the list. You'd have to include lots of others if 
you included chromic. I should never have tried to name it. Sorry.


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Message: 4240

Date: Tue, 12 Mar 2002 05:30:45

Subject: Dave's 23 best 5-limit temperaments

From: genewardsmith

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


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Message: 4241

Date: Tue, 12 Mar 2002 11:33:47

Subject: Re: Dave's 23 best 5-limit temperaments

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

> > > generators   126.2382718   1200

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

Manuel


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Message: 4242

Date: Tue, 12 Mar 2002 10:39:41

Subject: Re: Dave's 23 best 5-limit temperaments

From: genewardsmith

--- 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".


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Message: 4243

Date: Tue, 12 Mar 2002 06:16:12

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > ok. gene, once again, this means that in the 'gens' calculation, 
the 
> > number of generators in the 3:1 should be multiplied by log(3), 
the 
> > number of generators in the 5:3 should be multiplied by log(5), 
the 
> > number of generators in the 5:1 should be multiplied by log(5).
> 
> And to make the result meaningful (i.e. comparable to the 
unweighted 
> values) then after you take the RMS of these weighted values you 
> should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).

why do you want them to be comparable to the unweighted values?


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Message: 4244

Date: Tue, 12 Mar 2002 10:40:02

Subject: Re: A proposed list of 5-limit not-quite-Just-thingies

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: 
> What happened to chromic?

Oh I think I realise why you ask, since you wouldn't want it by your 
own badness measure. You think I'm violating my own badness measure. 
The thing is, I've changed my parameters from 7.4 cents and 0.5 power, 
to 5.5 cents and 0.43 power to make it agree better with your list.

See my latest version spreadsheet 
http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Ok *

I've fixed the names.


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Message: 4245

Date: Tue, 12 Mar 2002 06:47:56

Subject: amt

From: genewardsmith

--- 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.


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Message: 4246

Date: Tue, 12 Mar 2002 10:43:13

Subject: Re: Dave's 23 best 5-limit temperaments

From: paulerlich

--- 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).


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Message: 4247

Date: Wed, 13 Mar 2002 21:39:18

Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments)

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > > ok. gene, once again, this means that in the 'gens' 
calculation, 
> > the 
> > > > number of generators in the 3:1 should be multiplied by log
(3), 
> > the 
> > > > number of generators in the 5:3 should be multiplied by log
(5), 
> > the 
> > > > number of generators in the 5:1 should be multiplied by log
(5). 
> > this 
> > > > will cause temperaments generated by the fifth to look better 
> > than 
> > > > they currently do, relative to those that aren't.
> 
> Paul, don't you mean "divided by". log(3) is smaller than log(5) so 
if 
> you want to favour fifths ...

oh yeah, i meant divided by . . . i said it the right way last time, 
about two months ago . . .

and i don't buy graham's objections one bit. yes, it favors those 
systems which are *aspects* of lower-limit systems -- but very often 
you get more than one such aspect per lower-limit system! this is a 
very positive feature, not a bug -- it sits nicely with a view of 
higher-limit systems often evolving out of lower-limit ones. and of 
course it reflects musical reality much better.


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Message: 4248

Date: Wed, 13 Mar 2002 21:43:18

Subject: Re: Degenerate temperaments (was: A proposed list of 5-limit not-quite-Just-things)

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > > twin meantone = garbage
> > > half meantone-fourth = ditto
> > > half meantone-fifth = ditto
> 
> Paul, you say they are not temperaments because they do not map to 
a 
> _the_ JI lattice, and are therefore merely "tuning systems". But of 
> course they map to two (or more) disconnected JI lattices which is 
a 
> hell of a lot better than a mere "tuning system". How about we call 
> them degenerate temperaments?

decent, though torsion, rather than contorsion, would seem closer to 
the usage of 'degeneracy' that i'm familiar with from physics and 
math, when applied to temperaments.


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Message: 4249

Date: Wed, 13 Mar 2002 13:49:43

Subject: Re: Degenerate temperaments (was: A proposed list of 5-limit not-quite-Just-things)

From: Carl Lumma

>> hell of a lot better than a mere "tuning system". How about we call 
>> them degenerate temperaments?
>
>decent, though torsion, rather than contorsion, would seem closer to 
>the usage of 'degeneracy' that i'm familiar with from physics and 
>math, when applied to temperaments.

Agree.  They're not instances of degenerate temperament.  They're
instances of superposed, perfectly legitimate temperament_s_.

-Carl


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