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Message: 3475 - Contents - Hide Contents Date: Tue, 22 Jan 2002 21:51:50 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> The point of my question is: even tho the shape of this particular > PB is different from the 4 identical ones from which it was coalesced, > all the same UVs are in effect, so it's still identical to those 4, yes?Right . . .
Message: 3476 - Contents - Hide Contents Date: Tue, 22 Jan 2002 21:52:24 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Well, Monz, it would be good to know if Partch exploited _only_ the >> hemiennealimmal equivalencies, or _only_ the MIRACLE equivalencies, >> or what. Wilson seems to have felt that he exploited enough >> equivalencies that a closed 41-tone system (as in 41-tET) was >> actually implied. But Wilson never seems to have thought much about >> MIRACLE, let along hemiennealimmal. > >> All good points, Paul. I was careful to add "without examining > the actual mathematics", because I don't even know what > "hemiennealimmal" is!!!!! (... but I'm still studying!) > > Why don't you guys take a look at that? Sounds interesting.We would need Partch's scores in a readable form . . .
Message: 3477 - Contents - Hide Contents Date: Tue, 22 Jan 2002 22:06:45 Subject: Re: deeper analysis of Schoenberg unison-vectors From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>> Have you ever considered the [34, 54, 79, 96] version of twintone? >> You mentioned playing with someone who had a 34-et guitar, I think.> The 34-tET version of twintone? It's better than 12, but much worse > than 22.Depends on what you are using the 7-limit stuff to do, I would think--it is sweeter so far as the 5-limit goes.
Message: 3478 - Contents - Hide Contents Date: Tue, 22 Jan 2002 22:10:13 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> I've always been careful to emphasize that our tuning-theory use >> of "lattice" is different from the mathematician's strictly define >> uses of the term.> This is not correct, Monz. There are several mathematical definitions > of "lattice" -- the one we use is most certainly one of these, as > we've discussed numerous times on the tuning list, and applied for > example in crystallographic theory.When I use lattice on these lists it always means a discrete subgroup of R^n.
Message: 3479 - Contents - Hide Contents Date: Tue, 22 Jan 2002 22:11:19 Subject: Re: deeper analysis of Schoenberg unison-vectors From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>>>> Have you ever considered the [34, 54, 79, 96] version of twintone? >>> You mentioned playing with someone who had a 34-et guitar, I think. >>> The 34-tET version of twintone? It's better than 12, but much worse >> than 22. >> Depends on what you are using the 7-limit stuff to do, I wouldthink--it is sweeter so far as the 5-limit goes. Then we're talking diaschismic, not twintone. Or if they're the same thing, we need another word for "paultone".
Message: 3480 - Contents - Hide Contents Date: Wed, 23 Jan 2002 10:32:54 Subject: Re: hemiennealimmal / MIRACLE Partch? (was: A top 20 11-limit ...)) From: monz> From: jonszanto <jonszanto@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 22, 2002 4:25 PM > Subject: [tuning-math] Re: hemiennealimmal / MIRACLE Partch? (was: A top20 11-limit ...)> > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Well ... there are two available: >> Nope, there's more, and you probably will find the last very useable: > Richard Kassel has a big book, MUSA 9, that is a complete > transcription, analysis, history, and background on Barstow (the > piece, not the town). He uses the 1968 version, but I believe the > entire thing is translated into Johnston notation complete with > harmonic analysis.Thanks, Jon! I know Richard, but didn't know about this! Unfortunately, as with Hackbarth's transcription of _Daphne_, anyone wanting to analyze Partch's music must first learn how to decode Johnston's notation ... and that ain't the easiest thing to do. My suggestion is to start with this: An examination of * [with cont.] (Wayb.) And, BTW, I had speculated on the MIRACLE-Partch connection last summer, and drew some inferences from an examination of what Partch said in _Genesis_ about various ETs and other tunings: An examination of * [with cont.] (Wayb.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3481 - Contents - Hide Contents Date: Wed, 23 Jan 2002 20:21:08 Subject: Re: new Dictionary entry: "torsion" From: paulerlich> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> new Dictionary entry: "torsion" >> >> Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) >> >> >> Feedback appreciated. >> >> (and thanks for the helpful criticisms, Paul)Monz, this might be confusing because even the 12-tone Duodene can be split in this way (into two augmented scale PBs a 9:8 apart) and yet it does not possess torsion. The key point is that some unison vector (or combination thereof, depending on how you define UV) is an integer power (or multiple, in cents) of some interval within the block. In the particular example, the "unison vector" in question goes from one corner of the block to the opposite corner, and is the syntonic comma squared.
Message: 3482 - Contents - Hide Contents Date: Wed, 23 Jan 2002 20:41:09 Subject: Re: new Dictionary entry: "torsion" From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Done. Thanks, Gene!Here are more useful mathworld definitions: http://mathworld.wolfram.com/Group.html * [with cont.] Abelian Group -- from MathWorld * [with cont.] http://mathworld.wolfram.com/FreeAbelianGroup.... * [with cont.] * [with cont.] http://mathworld.wolfram.com/CyclicGroup.html * [with cont.] http://mathworld.wolfram.com/GroupHomomorphism... * [with cont.] http://mathworld.wolfram.com/IsomorphicGroups.... * [with cont.] http://mathworld.wolfram.com/Nullspace.html * [with cont.] * [with cont.] http://mathworld.wolfram.com/PointLattice.html * [with cont.] http://mathworld.wolfram.com/QuadraticForm.html * [with cont.] http://mathworld.wolfram.com/PositiveDefiniteQ... * [with cont.]
Message: 3483 - Contents - Hide Contents Date: Wed, 23 Jan 2002 20:42:11 Subject: Re: twintone, paultone (was something about schoenberg From: clumma I'd seen this before, but only now taken the time to read it thoroughly. All I can say is, "Thank you, thank you, thank you, thank you, thank you, Graham!". -Carl
Message: 3484 - Contents - Hide Contents Date: Wed, 23 Jan 2002 12:47:52 Subject: Re: new Dictionary entry: "torsion" From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 23, 2002 12:21 PM > Subject: [tuning-math] Re: new Dictionary entry: "torsion" > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> new Dictionary entry: "torsion" >> >> Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) >> > >> Monz, this might be confusing because even the 12-tone Duodene can be > split in this way (into two augmented scale PBs a 9:8 apart) and yet > it does not possess torsion. The key point is that some unison vector > (or combination thereof, depending on how you define UV) is an > integer power (or multiple, in cents) of some interval within the > block. In the particular example, the "unison vector" in question > goes from one corner of the block to the opposite corner, and is the > syntonic comma squared.Try it now. I included Paul's post verbatim under my description of the "classic example", but also changed a bit in the first paragraph (i.e., the actual definition part). Is that first paragraph good enough now? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3485 - Contents - Hide Contents Date: Wed, 23 Jan 2002 20:50:50 Subject: Re: new Dictionary entry: "torsion" From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, January 23, 2002 12:21 PM >> Subject: [tuning-math] Re: new Dictionary entry: "torsion" >> >> >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>> new Dictionary entry: "torsion" >>> >>> Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) >>> >> >>>> Monz, this might be confusing because even the 12-tone Duodene can be >> split in this way (into two augmented scale PBs a 9:8 apart) and yet >> it does not possess torsion. The key point is that some unison vector >> (or combination thereof, depending on how you define UV) is an >> integer power (or multiple, in cents) of some interval within the >> block. In the particular example, the "unison vector" in question >> goes from one corner of the block to the opposite corner, and is the >> syntonic comma squared. > >> Try it now. > > I included Paul's post verbatim under my description of the > "classic example", but also changed a bit in the first paragraph > (i.e., the actual definition part). Is that first paragraph > good enough now?I don't see the change as addressing my objection above in any way. Do you?
Message: 3486 - Contents - Hide Contents Date: Wed, 23 Jan 2002 20:54:18 Subject: Re: new Dictionary entry: "torsion" From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> From: paulerlich <paul@s...> >>> To: <tuning-math@y...> >>> Sent: Wednesday, January 23, 2002 12:21 PM >>> Subject: [tuning-math] Re: new Dictionary entry: "torsion" >>> >>> >>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>>>> new Dictionary entry: "torsion" >>>> >>>> Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) >>>> >>> >>>>>> Monz, this might be confusing because even the 12-tone Duodene > can be>>> split in this way (into two augmented scale PBs a 9:8 apart) and > yet>>> it does not possess torsion. The key point is that some unison > vector>>> (or combination thereof, depending on how you define UV) is an >>> integer power (or multiple, in cents) of some interval within the >>> block. In the particular example, the "unison vector" in question >>> goes from one corner of the block to the opposite corner, and is > the>>> syntonic comma squared. >> >>>> Try it now. >> >> I included Paul's post verbatim under my description of the >> "classic example", but also changed a bit in the first paragraph >> (i.e., the actual definition part). Is that first paragraph >> good enough now? >> I don't see the change as addressing my objection above in any way. > Do you?Sorry, Monz, I didn't hit refresh. Well, the definition now is quite incorrect even as a description (let alone a definition), as you can see by comparing what you say with the example you give. "Torsion describes a condition where a set of pitch-classes appears to create a periodicity-block, when in reality that apparent periodicity-block is actually composed of two or more (call it n) identical smaller periodicity-blocks, each exactly 1/n as large as the apparent one, whose pitch-classes are separated by some interval which acts as a unison-vector and which is an integer power of an interval included within the actual smaller periodicity-block itself." You won't find sets of pitch classes separated by such an interval.
Message: 3487 - Contents - Hide Contents Date: Wed, 23 Jan 2002 21:20:49 Subject: Re: twintone, paultone From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> 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.This is getting ridiculous. If h22 is the map [22, 35, 51, 62] and g34 the map [34, 54l 79, 96] then: Proof 1: h22 ^ g34 = [-2, 4, 4, -2, -12, 11] = h10 ^ h12 = twintone wedgie. Proof 2: h22(50/49) = g34(50/49) = 0; h22(64/63) = g34(64/63) = 0. In other words, yes it *is* twintone and *no*, no new word is required. Here is a comparison of twintone as tuned in h22 and g34, which I think shows g34 is perfectly practical and arguably preferable: 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).
Message: 3488 - Contents - Hide Contents Date: Wed, 23 Jan 2002 00:22:13 Subject: Re: Heuristic? From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Why? This is not the heuristic for complexity we're looking at here -- > it's the heuristic for error!So small values of your heuristuc applied to 5-limit commas should correspond to good 5-limit temperaments, and larger values to not-so-good?
Message: 3489 - Contents - Hide Contents Date: Wed, 23 Jan 2002 13:37:06 Subject: Re: new Dictionary entry: "torsion" From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 23, 2002 12:54 PM > Subject: [tuning-math] Re: new Dictionary entry: "torsion" > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> new Dictionary entry: "torsion" >> >> Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) > >> Well, the definition now is quite incorrect even as a description > (let alone a definition), as you can see by comparing what you say > with the example you give. > > "Torsion describes a condition where a set of pitch-classes appears > to create a periodicity-block, when in reality that apparent > periodicity-block is actually composed of two or more (call it n) > identical smaller periodicity-blocks, each exactly 1/n as large as > the apparent one, whose pitch-classes are separated by some interval > which acts as a unison-vector and which is an integer power of an > interval included within the actual smaller periodicity-block itself." > > You won't find sets of pitch classes separated by such an interval.ACK!! OK, I don't have time to do anything else now. You guys hash out how this definition should be corrected, and I'll change it when I read your follow-up posts later tonight. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3490 - Contents - Hide Contents Date: Wed, 23 Jan 2002 00:17:09 Subject: Re: the Lattice Theory Homepage From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> "Barnes-Wall Lattice, Blichfeldt's Theorem, Browkin's Theorem, Circle > Lattice Points, Coxeter-Todd Lattice, Ehrhart Polynomial, Elliptic Curve,= > Gauss's Circle Problem, Golygon, Integration Lattice, Jarnick's Inequalit= y, > Lattice Path, Lattice Sum, Leech Lattice, Minkowski Convex Body Theorem, > Modular Lattice, N-Cluster, Nosarzewska's Inequality, Pick's Theorem, Ran= dom > Walk, Schinzel's Theorem, Schröder Number, Torus, Unit Lattice, Visible > Point, Voronoi Polygon" > Any relevance of those to what we do here?Blichfeldt's theorem, Minkowski's theorem, and the Jarnick-Nosarzewska ineq= uality are quite relevant, and the Voronoi cell is a bit of terminology I've= been considering introducing. I keep thinking I'll apply Pick's theorem, bu= t never have, however I have had occasion to mention random lattice walks. If you want specific lattices to look at, the root lattice An and its dual = An* are the key ones, I think.
Message: 3491 - Contents - Hide Contents Date: Wed, 23 Jan 2002 22:11:38 Subject: Proposed dictionary entry: torsion From: genewardsmith torsion Torsion describes a condition wherein an independent set of n unison vectors (<uvector.htm>) fails to define a periodicity block of dimension n, because of the existence of torsion elements, meaning intervals which are not products of the proposed set of unison vectors, but some power of which are. Torsion may be tested by forming the n by n+1 matrix whose rows correspond to the unison vectors, and calculating the gcd(<http://mathworld.wolfram.com/GreatestCommonDiv... * [with cont.] >) of the minors (<http://mathworld.wolfram.com/Minor.html * [with cont.] >) of the matrix. If the rows are linearly independent but the gcd is not one, we have torsion. The term comes from mathematical usage, see < * [with cont.] >.
Message: 3493 - Contents - Hide Contents Date: Wed, 23 Jan 2002 22:33:07 Subject: Re: Proposed dictionary entry: torsion From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> torsion > > Torsion describes a condition wherein an independent set of nunison vectors (<uvector.htm>) fails to define a periodicity block of dimension n, because of the existence of torsion elements, meaning intervals which are not products of the proposed set of unison vectors, but some power of which are.> > Torsion may be tested by forming the n by n+1 matrix whose rowscorrespond to the unison vectors, and calculating the gcd (<http://mathworld.wolfram.com/GreatestCommonDiv... * [with cont.] >) of the minors (<http://mathworld.wolfram.com/Minor.html * [with cont.] >)> of the matrix. If the rows are linearly independent but the gcd isnot one, we have torsion. This is awesome!! Should you say "field of unison vectors" or "ring of unison vectors" or some such as opposed to "set of unison vectors" above? The idea is to eliminate the ambiguity that arises from the two usages of "unison vector" that you brought up -- a member of the basis for the kernel, or a member of the kernel. P.S. An early example of torsion: Yahoo groups: /tuning/message/9694 * [with cont.] Kees' corrections: Yahoo groups: /tuning/message/9757 * [with cont.]
Message: 3494 - Contents - Hide Contents Date: Wed, 23 Jan 2002 00:32:50 Subject: Re: deeper analysis of Schoenberg unison-vectors From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>>>> Have you ever considered the [34, 54, 79, 96] version of > twintone?>>>> You mentioned playing with someone who had a 34-et guitar, I > think.>> Depends on what you are using the 7-limit stuff to do, I would> think--it is sweeter so far as the 5-limit goes. > > Then we're talking diaschismic, not twintone. Or if they're the same > thing, we need another word for "paultone".Why? I mean the linear temperament [-2,4,4,-2,-12,11]. Add 245/243 to the mix and you have 22-et, add 4375/4374 instead and you have 34-et in theversion where 7/4 is 19.4 cents sharp, rather than being flat. But they are both twintone.
Message: 3495 - Contents - Hide Contents Date: Wed, 23 Jan 2002 22:41:20 Subject: Proposed definition: regular temperament From: genewardsmith A temperament (Definitions of tuning terms: temperament, (c) ... * [with cont.] (Wayb.)) which maps rational intonation of a given rank in a consistent way to an intonation of smaller rank. A regular temperament is equal, linear, or planar depending on whether the rank, meaning the number of generators, is one, two, or three. In general a regular temperament is described as being of dimension d when it has d+1 generators, and so is of rank d+1. Mathematically, a regular temperament is an epimorphism (a homomprhism (<http://mathworld.wolfram.com/GroupHomomorphism... * [with cont.] > which is onto http://mathworld.wolfram.com/Onto.html * [with cont.] ) of a rational note group onto a note group of smaller rank. Regular temperaments may be classified according to their wedge invariant, or wedgie.
Message: 3496 - Contents - Hide Contents Date: Wed, 23 Jan 2002 00:59:43 Subject: Re: Heuristic? From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Why? This is not the heuristic for complexity we're looking at here -- >> it's the heuristic for error! >> So small values of your heuristuc applied to 5-limit commas should >correspond to good 5-limit temperaments, and larger values to not-so- >good?One heuristic, |n-d|/(d*log(d)), gives you the the "error" component of goodness, while another, log(d), gives you the "gens"/"complexity" component. Please look again at Yahoo groups: /tuning-math/message/2491 * [with cont.] You'll need to use "Expand Messages".
Message: 3497 - Contents - Hide Contents Date: Wed, 23 Jan 2002 22:50:58 Subject: Re: Proposed dictionary entry: torsion From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Should you say "field of unison vectors" or "ring of unison vectors" > or some such as opposed to "set of unison vectors" above?You could say "group generated by the unison vectors", but I thought I made it clear with "set" that I was talking about a basis for the kernel, not the kernel itself. It seems to me that makes fewer demands on the reader by way of knowledge of mathematics.
Message: 3498 - Contents - Hide Contents Date: Wed, 23 Jan 2002 23:00:47 Subject: Re: Proposed dictionary entry: torsion From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Should you say "field of unison vectors" or "ring of unison vectors" >> or some such as opposed to "set of unison vectors" above? >> You could say "group generated by the unison vectors", but I >thought I made it clear with "set" that I was talking about a basis >for the kernel, not the kernel itself.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.
Message: 3499 - Contents - Hide Contents Date: Wed, 23 Jan 2002 01:25:30 Subject: Re: new Dictionary entry: "torsion" From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > new Dictionary entry: "torsion" > > Definitions of tuning terms: torsion, (c) 2002... * [with cont.] (Wayb.) > > > Feedback appreciated. > > (and thanks for the helpful criticisms, Paul) > > > > -monzWhy hyphenate periodicity-block? You don't hyphenate vacuum-cleaner, do you?
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