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Message: 6503 Date: Wed, 23 Jan 2002 04:31:17 Subject: Re: new Dictionary entry: "torsion" From: monz new Dictionary entry: "torsion" Definitions of tuning terms: torsion, (c) 2002 by Joe Monzo * > I would suggest adding something to explain why it is called "torsion" Done. Thanks, Gene! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 6504 Date: Wed, 23 Jan 2002 12:46 +0 Subject: Re: twintone, paultone (was something about schoenberg From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <200201231200.OAA67852@xxxxxx.xxx.xxxxx.xxx> Robert C Valentine wrote: > More quick definitions needed : twintone, paultone and quick > comparison with diaschismic. Any additional commentary regarding > 34 welcome too. I'm trying to keep track of these at <Catalogue of linear temperaments *>. I do need some updates, especially for this Pelog thing. Diaschismic is any temperament with a period of a half octave, and a mapping of [1, -2]. That is, you have a semitone generator. A perfect fifth is a tritone plus a semitone. A major third is a tritone minus two semitones. Paultone, which now seems to be being called twintone, is a particular 7-limit diaschismic consistent with 22-equal. The mapping is [1, -2, -2] so a 7:4 is an octave minus two semitones. 34-equal is an accurate, 5-limit diaschismic. But it isn't consistent in the 7-limit. There are two 7-limit mappings which converge at 34, and give good results for different tunings. Neither of them is paultone. The typical diaschismic mapping, for me at least, is that consistent with 46- and 58-equal. That's [1, -2, -8] and it's accurate to about 6 cents. A script for finding the linear temperament consistent with (the prime mappings of) a pair of equal temperaments can be found at <temperament finding scripts *> Graham
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Message: 6505 Date: Thu, 24 Jan 2002 13:57 +0 Subject: Re: twintone, paultone From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a2n9fh+hqeh@xxxxxxx.xxx> Me: > > 34-equal is an accurate, 5-limit diaschismic. But it isn't > > consistent in the 7-limit. There are two 7-limit mappings which > > converge at 34, and give good results for different tunings. Neither > > of them is paultone. Gene: > This is getting ridiculous. Why so? You haven't addressed anything I said there, so I'll have to do it myself. > > 34-equal is an accurate, 5-limit diaschismic. The worst-tuned 5-limit interval in 34-equal is 5:4, which is 4 cents sharp, or about 1/9 steps of 34. That's pretty good. > > But it isn't consistent in the 7-limit. The worst-tuned 7-limit interval in h34 (34-equal using the best approximations to ratios of prime numbers) is 7:4, which is 0.56 steps flat. Hence 34-equal is not 7-limit consistent. > > There are two 7-limit mappings which > > converge at 34, and give good results for different tunings. h34&h46 (the temperament consistent with 46-equal and the prime mapping of 34-equal) is accurate to 5.9 cents in the 7-limit with a generator of 103.9 cents. h34&h22 is accurate to 6.9 cents in the 7-limit with a generator of 107.5 cents. These are the most accurate 7-limit diaschismics I know of. (You could probably improve on them with much more complex mappings, and there's also Shrutar and the like.) They both involve the prime mapping of 34-equal, hence converge at 34-equal. > > Neither > > of them is paultone. h34&h22 has a period-generator mapping of [(2, 0), (3, 1), (5, -2), (4, 9)] h34&h46 has a period-generator mapping of [(2, 0), (3, 1), (5, -2), (7, -8)] paultone has a period-generator mapping of [(2, 0), (3, 1), (5, -2), (6, -2)] These are clearly three different temperaments. So what's the problem? > Here is a comparison of twintone as tuned in h22 and g34, which I think > shows g34 is perfectly practical and arguably preferable: Preferable to what? All this seems to show is that 34-equal is inconsistent in the 7-limit, as I stated above. > 3: 7.13 3.93 > 5: -4.50 1.92 > 7: 12.99 19.41 > 5/3: -11.63 -2.01 > 7/3: 5.86 15.48 > 7/5: 17.49 17.49 > > The 22-et version has better 7/4s and 7/6s, and the 34-et version has > better 3/2s, 5/4s, and 5/3s; they share the twintone 7/5 of sqrt(2). Of course g34 has good 5-limit intervals, they're the same as those from h34 ! The 7-limit as a whole is much worse, and most errors are in the same direction, so a compromise temperament won't improve matters. As it happens, my optimum for paultone/twintone is a 109.4 cent generator, with a worst 7-limit error of 17 cents. 3/34 octaves are 105.9 cents and 2/22 octaves are 109.1 cents. So the twintone optimum is not only closer to 22 than 34, but falls the other side of 22 than 34. That makes 34-equal far less characteristic of paultone/twintone than the other two diaschismics I mention above. So again, what's the problem? Graham
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Message: 6506 Date: Thu, 24 Jan 2002 13:57 +0 Subject: Re: Thank you Graham! From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a2ntkf+353u@xxxxxxx.xxx> paulerlich wrote: > I don't see any precedent for the kind of stuff we're dealing with in > our paper, though . . . It derives a linear temperament by tempering out a unison vector. I thought Fokker only did equal temperaments. Graham
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Message: 6508 Date: Thu, 24 Jan 2002 21:30:06 Subject: Re: twintone, paultone From: clumma >You can't hear consistency, so why is this relevant? You can hear consistency, when neighboring chords involve different approximations to the same interval. -Carl
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Message: 6509 Date: Thu, 24 Jan 2002 22:08:47 Subject: Re: twintone, paultone From: clumma I wrote... >>You can't hear consistency, so why is this relevant? > >You can hear consistency, when neighboring chords involve >different approximations to the same interval. Which is not to say that this is in any way "bad". -Carl
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Message: 6510 Date: Thu, 24 Jan 2002 19:58:22 Subject: Re: OUR PAPER From: monz Just wanted to mention that I'll offer space on my website for anything anyone wants to publish there regarding this stuff. I plan to explain as much as I can understand of the tuning-math proceedings, with *LOTS AND LOTS* of lattice diagrams! -monz ----- Original Message ----- From: jpehrson2 <jpehrson@xxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Thursday, January 24, 2002 6:06 PM Subject: [tuning-math] Re: OUR PAPER > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > Yahoo groups: /tuning-math/message/2958 * > > > Hello? > > > > Let's push forward, shall we? > > > > ... > > I would propose (if I may humbly do that for a micromini second, or a > mathmicromini second) that there are actually *two* papers... > > One the "intense" "real" one, and the other a kind of "synopsis" > along the lines of Paul Erlich's *very* fine... in fact *very, very* > fine "The Forms of Tonality" which was a very readable and *broadly- > based* effort, directed to the larger microtonal community. And it > had nice *pictures* too. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 6511 Date: Thu, 24 Jan 2002 00:04:10 Subject: Re: Proposed dictionary entry: torsion From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > > torsion > > > > Torsion describes a condition wherein an independent set of n unison vectors (<uvector.htm>) fails to define a periodicity block of dimension n... > > No agreement has been reached on what peridicity block means, so this could also read "defines an anomalous periodicity block". That would be preferable.
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Message: 6512 Date: Thu, 24 Jan 2002 00:05:44 Subject: Re: Proposed dictionary entry: torsion From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > > Well then your definition doesn't seem to work, because if the basis > > is the diesis and the schisma, the syntonic comma squared is in the > > kernel, but not in the basis. > > That's why I said "not products of the proposed set of unison >vectors" instead of "not members of the proposed set of unision >vectors". Oops! I read it wrong, somehow. But "products" might still fail to capture a case like a^2/b where a and b are in the basis. Can't we think of a better terminology?
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Message: 6514 Date: Thu, 24 Jan 2002 03:04:47 Subject: Thank you Graham! From: paulerlich Thanks for the papers you sent me, Graham! I don't see any precedent for the kind of stuff we're dealing with in our paper, though . . .
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Message: 6515 Date: Fri, 25 Jan 2002 18:11:23 Subject: Re: twintone, paultone From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > What about both > 0-10-18-25 and 0-10-18-26 in the 31-et? The first is a 7-limit tetrad or German augmented sixth, the second is a classical dominant seventh (I know, we argued about this a while ago, you do not believe this, but I sure do).
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Message: 6516 Date: Fri, 25 Jan 2002 10:29:43 Subject: Re: Proposed dictionary entry: torsion From: monz Hi Gene, Two things. 1) I love your new definition of "torsion". What exactly should I replace in my old definition? Everything? Please be as specific as possible. 2) The last post on this subject was this one from Paul, with a question to which you have not yet responded: > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 23, 2002 4:05 PM > Subject: [tuning-math] Re: Proposed dictionary entry: torsion > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > > > > Well then your definition doesn't seem to work, because if the > basis > > > is the diesis and the schisma, the syntonic comma squared is in > the > > > kernel, but not in the basis. > > > > That's why I said "not products of the proposed set of unison > >vectors" instead of "not members of the proposed set of unision > >vectors". > > Oops! I read it wrong, somehow. But "products" might still fail to > capture a case like a^2/b where a and b are in the basis. Can't we > think of a better terminology? I'd appreciate your help in getting the definiton webpage of "torsion" finished. Thanks. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 6517 Date: Fri, 25 Jan 2002 18:16:02 Subject: Re: Our Paper From: paulerlich --- In tuning-math@y..., graham@m... wrote: > Or perhaps an introduction to some of > the new temperaments for the imminent Xenharmonikon, It's not as imminent as some have made it out to be. Despite reports to the contrary, no papers have yet been accepted for the next issue of XH. Maybe we should all write separate papers that fit together and reference one another.
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Message: 6519 Date: Fri, 25 Jan 2002 18:12:38 Subject: Re: OUR PAPER From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > Hello? > > > > Let's push forward, shall we? > > > > Graham, do you agree with the way Gene's doing things? > > > > If so, you guys have a plurality, against Dave and myself, who both > > seem to be resisting in different areas. > > One possibility would be for me to write up my own approach to the >theory part, and have that as a separate paper. Would they publish >it? I have a feeling John Chalmers would probably publish it in Xenharmonikon -- he's a chemist, so he might be a little more favorably disposed to something so heavily mathematical.
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Message: 6520 Date: Fri, 25 Jan 2002 00:30:18 Subject: Re: twintone, paultone From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote: > >You can't hear consistency, so why is this relevant? > > You can hear consistency, when neighboring chords involve > different approximations to the same interval. That wouldn't happen in the case Gene is talking about.
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Message: 6522 Date: Fri, 25 Jan 2002 00:50:40 Subject: Re: twintone, paultone From: clumma >>>You can't hear consistency, so why is this relevant? >> >>You can hear consistency, when neighboring chords involve >>different approximations to the same interval. > >That wouldn't happen in the case Gene is talking about. I didn't say it would. Gene's was a general dismissal of consistency. -Carl
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Message: 6524 Date: Fri, 25 Jan 2002 00:54:59 Subject: Re: twintone, paultone From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote: > >>>You can't hear consistency, so why is this relevant? > >> > >>You can hear consistency, when neighboring chords involve > >>different approximations to the same interval. > > > >That wouldn't happen in the case Gene is talking about. > > I didn't say it would. Gene's was a general dismissal of > consistency. You can have neighboring chords involve different approximations to the same interval even in a consistent tuning. I see this happening in 76-tET, where one could modulate between twintone, meantone, double-diatonic, as well as other systems.
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