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Message: 3425 - Contents - Hide Contents Date: Mon, 21 Jan 2002 14:13:13 Subject: Re: the Lattice Theory Homepage From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, January 21, 2002 12:39 PM > Subject: [tuning-math] Re: the Lattice Theory Homepage > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Wow! ... check out the diagram and text at the top >> of this page : >> >> Lattice Theory * [with cont.] (Wayb.) >> I'm afraid that's the wrong kind of lattice--as came up > before, there are two different things called "lattice" > in English-language mathematics. This kind is a kind of > partial ordering, which is important in universal algebra > among other things, which is why the univeral algebraists > in Hawaii care about it.Are there any other types of lattices or just these two? (not counting the kind which hold up rose-bushes, etc., of course!) While I hardly understand it, I'm surprised to see that Minkowski reduction applies to "our" lattices as well as the regular mathematical kind, since I knew they are different. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3426 - Contents - Hide Contents Date: Mon, 21 Jan 2002 03:20:46 Subject: Re: lattices of Schoenberg's rational implications From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: However, I think the only reality for Schoenberg's> system is a tuning where there is ambiguity, as defined by the kernel > <33/32, 64/63, 81/80, 225/224>. BTW, is this Minkowski-reduced?Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.

Message: 3427 - Contents - Hide Contents Date: Mon, 21 Jan 2002 22:35:04 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "clumma" <carl@l...> wrote: >>> Actually, Partch considered having far more than 43, stopping at >> 43 only for pragmatic reasons (according to him), and often using >> far less. I can think of at least 4 separate diatribes given by >> Partch at different times on the association of the 43-tone scale >> with his music. He thought of his working area as the infinite >> space of JI. >> Don't forget Ben Johnston, who often composed with 81 tones or more!OK. I stand corrected. And I apologise for the "religion" remark. One might just as easily say "This subset of hemiennealimmal is so close to RI we might as well try to tune the instruments to the RI scale since that's easier to calculate and tunable by ear (assuming harmonic timbres).

Message: 3428 - Contents - Hide Contents Date: Mon, 21 Jan 2002 03:22:48 Subject: Re: Hi Dave K. From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> I'm not getting this last part. Will it help make my heuristic work?I don't know. What I'd like to know is what a version of your heuristic would be which applies to sets of commas--is this what you are aiming at?

Message: 3429 - Contents - Hide Contents Date: Mon, 21 Jan 2002 14:37:36 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: monz Hi Graham,> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, January 21, 2002 9:44 AM > Subject: [tuning-math] Re: Minkowski reduction (was: ...Schoenberg's rational implications) > >>> What's the purpose of wanting to find the Minkowski-reduced >> version of the PB instead of the actual one defined by >> Schoenberg's ratios? How much of a difference is there? >> It means you get the simplest set that define the same temperament. Also > that you have a canonical set to compare with others, although we use > wedgies for that now. If you follow the link I gave before for my unison > vector CGI, you'll see its attempts at Minkowski reduction among the > results. They should all be correct, except for the ones that are wildly > incorrect. That's something I'm still working on. It's something to do > with short vectors.You're talking about Temperament result * [with cont.] (Wayb.) I put in the four unison-vectors which I had determined from p 1-184 of _Harmonielehre_, that is, <225/224, 81/80, 63/64, 33/32>, and got these results:>> [output from Graham's temperament calculator] >> >> unison vectors >> >> 225:224 >> 81:80 >> 64:63 >> >> lower limit, got an ET (12, 19, 28, 34) >> >> --------------- >> >> unison vectors >> >> 225:224 >> 81:80 >> 33:32 >> >> calculated >> unison vectors >> >> 33:32 >> 55:54 >> 77:75 >> >> 0/1, 1892.5 cent generator >> >> basis: >> (1.0, 1.57708393667) >> >> mapping by period and generator: >> [(1, 0), (0, 1), (-4, 4), (-13, 10), (5, -1)] >> >> mapping by steps: >> [(1, 0), (-1, 1), (-8, 4), (-23, 10), (6, -1)] >> >> highest interval width: 11 >> complexity measure: 11 (12 for smallest MOS) >> highest error: 0.036516 (43.819 cents) >> >> ------------- >> >> unison vectors >> >> 225:224 >> 64:63 >> 33:32 >> >> calculated >> unison vectors >> >> 33:32 >> 64:63 >> 242:225 >> >> 0/1, 1908.8 cent generator >> >> basis: >> (0.5, 1.59064251985) >> >> mapping by period and generator: >> [(2, 0), (0, 1), (11, -2), (12, -2), (10, -1)] >> >> mapping by steps: >> [(2, 0), (-1, 1), (13, -2), (14, -2), (11, -1)] >> >> highest interval width: 4 >> complexity measure: 8 (10 for smallest MOS) >> highest error: 0.061434 (73.721 cents) >> >> ------------ >> >> unison vectors >> >> 81:80 >> 64:63 >> 33:32 >> >> calculated >> unison vectors >> >> 22:21 >> 33:32 >> 36:35 >> >> 0/1, 1902.9 cent generator >> >> basis: >> (1.0, 1.58576219547) >> >> mapping by period and generator: >> [(1, 0), (0, 1), (-4, 4), (6, -2), (5, -1)] >> >> mapping by steps: >> [(1, 0), (-1, 1), (-8, 4), (8, -2), (6, -1)] >> >> highest interval width: 6 >> complexity measure: 6 (7 for smallest MOS) >> highest error: 0.066315 (79.577 cents)So yes, I can see that the last one found both the 36/35 and the 22/21 which Gene found for the Minkowski-reduced version, to replace two of the UVs I determined. The second one replaces (225/224 and 81/80) with (77/75 and 55/54), and the third one replaces 225/224 with 242/225. So the only difference between your results and Gene's is that your calculator seems to be fishing around for the 50/49 UV, but it got two of the others. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3430 - Contents - Hide Contents Date: Mon, 21 Jan 2002 03:48:28 Subject: Re: lattices of Schoenberg's rational implications From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > However, I think the only reality for Schoenberg's>> system is a tuning where there is ambiguity, as defined by the kernel >> <33/32, 64/63, 81/80, 225/224>. BTW, is this Minkowski-reduced? >> Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.Awesome. So this suggests a more compact Fokker parallelepiped as "Schoenberg PB" -- here are the results of placing it in different positions in the lattice (you should treat the inversions of these as implied): 0 1 1 84.467 21 20 203.91 9 8 315.64 6 5 386.31 5 4 470.78 21 16 617.49 10 7 701.96 3 2 786.42 63 40 933.13 12 7 968.83 7 4 1088.3 15 8 0 1 1 119.44 15 14 203.91 9 8 315.64 6 5 386.31 5 4 470.78 21 16 617.49 10 7 701.96 3 2 786.42 63 40 933.13 12 7 968.83 7 4 1088.3 15 8 0 1 1 119.44 15 14 155.14 35 32 301.85 25 21 386.31 5 4 470.78 21 16 617.49 10 7 701.96 3 2 772.63 25 16 884.36 5 3 968.83 7 4 1088.3 15 8 0 1 1 84.467 21 20 155.14 35 32 266.87 7 6 386.31 5 4 470.78 21 16 582.51 7 5 701.96 3 2 737.65 49 32 884.36 5 3 968.83 7 4 1053.3 147 80

Message: 3431 - Contents - Hide Contents Date: Mon, 21 Jan 2002 14:42:11 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: monz Hi Dave,> From: dkeenanuqnetau <d.keenan@xx.xxx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, January 21, 2002 2:35 PM > Subject: Re: A top 20 11-limit superparticularly generated linear temperament list > > > ... One might just as easily say "This subset of > hemiennealimmal is so close to RI we might as well try > to tune the instruments to the RI scale since that's > easier to calculate and tunable by ear (assuming > harmonic timbres).Hmmm ... given remarks on the tuning list in the past by Daniel Wolf about the multiple senses (other than the two obvious ones) or other ratios implied by Partch's use of his 43-tone scale, now *that* sounds like something pretty close to the mark! Partch apparently wove harmonic structures into his compositions which sometimes require the listener to infer different rational implications from his scale than the obvious ones. Without examining the actual mathematics of it, your revised statement here seems to me to be a good way to model that aspect of Partch's compositional practice. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3432 - Contents - Hide Contents Date: Mon, 21 Jan 2002 03:50:34 Subject: Re: Hi Dave K. From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> I'm not getting this last part. Will it help make my heuristic work? >> I don't know. What I'd like to know is what a version of your >heuristic would be which applies to sets of commas--is this what you >are aiming at?Eventually. It would probably involve some definition of the dot product of the commas in a tri-taxicab metric. But I like to start simple, and perhaps if we can formulate the right error measure in 5- limit, we can generalize it and use it for 7-limit even without knowing how one would apply the heuristic.

Message: 3434 - Contents - Hide Contents Date: Mon, 21 Jan 2002 00:56:06 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: clumma>But I was wrong to say they always choose some more manageable >subset. Sometimes they just use continuously gliding tones etc.What about Maneri and his students (72 tones)?>> Partch, after all, *did* have 43 actual tones per octave in play, >> so I don't see how this theory holds up. >>Huh? 43 is considerably less than 72, being only about 60% of it. >So it _supports_ this theory.Actually, Partch considered having far more than 43, stopping at 43 only for pragmatic reasons (according to him), and often using far less. I can think of at least 4 separate diatribes given by Partch at different times on the association of the 43-tone scale with his music. He thought of his working area as the infinite space of JI.>> The numbers are edges/connectivity in the 5, 7, 9 and 11-limits. >> I conclude that a great deal is gained by tempering in this way, >> and nothing significant is conceded in terms of quality of >> intonation. Of course, 72-et would do much better yet, but then >> some concessions will have been made. >>I totally agree. With the discovery of microtemperaments like >this, an insistence on strict RI starts to look more like a >religion than an informed decision.I agree. But there's something else that's starting to look like a religion -- the insistence on re-casting everyone else's scale choices in terms of temperament. If you've ever sat down to use one of these scales, you know that they already have so many more resources than a system which has been the life's-work of three centuries of our best minds, that adding resources by tempering may not be the first thing on the composer's mind. That's not to say it isn't. But if it isn't, it is no sign of ignorance. -Carl

Message: 3435 - Contents - Hide Contents Date: Mon, 21 Jan 2002 22:52:54 Subject: Re: "I didn't bring up the term religion here..." From: clumma>What I meant by arch-conservative is the tendency to mug anyone >who does not maintain an attitude of worshipful awe towards Those >Who Have Gone Before.Gene, I certainly didn't mean to jump on you, Dave, or anyone else. And I don't see anything in my post that worships Those Who Have Gone Before. Did I miss something?>My fields are math and philosophy, and it doesn't work like that >there. Why is music different, even among the reformers? Why are >the reformers so very few, come to that?Monz posted an excellent answer over on metatuning (I think), which should cover the general case. More specifically, many of us, notably David Doty, and to some extent Paul and myself, feel that music was confused in the last century. It had seen the end of its resources, but did not know how to expand them. It tried anyway, but wound up pushing out the wrong stuff, and got gibberish. In a sense, these composer/theorists want to return to 1900 and try again.>I would think that they would be *less* conservative than the >mainstream, since they are willing, even eager, to try something >different. As for letting it go, I am not going to let go of my >right to think my own thoughts, even if the Spanish Inquistion >breaks in with their dreaded Comfy Chair and Dish Rack.Gene, who has tried to take away your free thought? My point was simply that though every tuning may be considered a temperament, that it _must be_ requires an added assumption -- namely, that chords/notes matters to everyone in every situation. That assumption may or may not be founded, but until there's a rigorous proof, it deserves to be on the table. There have been many people who have come to the list, full of the enthusiasm of sharing their ideas, only to be told (or so they thought), in a language that requires considerable expertise outside of music to even enter discussion, that their tuning was worthless. In one case, a person was so discouraged that he stopped work on his instrument, and on alternate tunings, until another fellow who had been in the same situation talked him out of it. I personally can't imagine how that would happen from what I've seen on the list, but I was raised in an academic environment, etc., etc. And please don't think it was the mention of Partch that set this off -- I've been harping on this point for years. Unlike the self-appointed Partch experts around here, I take Harry's artistic vision seriously -- which is to say, I refuse to make him into a diety (even though I think he is every bit as worthy of this as Mozart or others who have been deified). -Carl

Message: 3436 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:02:57 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> I totally agree. With the discovery of microtemperaments like >> this, an insistence on strict RI starts to look more like a >> religion than an informed decision.> I agree. But there's something else that's starting to look like > a religion -- the insistence on re-casting everyone else's scale > choices in terms of temperament.Hmmm? What's "religious" about looking at the mathematics of someone's scale? This really isn't anything radical--the same point could have been made at any time with ets, by looking at the the versions of Partch's scale which results from using the likes of the 224, 270, 342, 494 or 612 ets.

Message: 3438 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:30:13 Subject: Re: Hi Dave K. From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> Thanks, but I don't think RMS will work. That implies a Euclidean >> metric, but a "taxicab" metric seems to be what we want here. >> Yes of course. Sorry. Just replace every ocurrence of "rms" with > "taxicab" in what I wrote.How do you calculate "taxicab" error?

Message: 3440 - Contents - Hide Contents Date: Tue, 22 Jan 2002 00:12:17 Subject: Re: "I didn't bring up the term religion here..." From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Gene, I certainly didn't mean to jump on you, Dave, or anyone > else. And I don't see anything in my post that worships Those > Who Have Gone Before. Did I miss something?I had the impression you were saying that someone's choice of tuning was so sanctified by tradition that pointing out that using the apparant standard of what constitutes "JI" in that particular context allows for a great many more "JI" consonances than strict RI would allow was a "relgious" remark. This struck me as a continuation of a trend of thought I am finding increasingly tiresome.> More specifically, many of us, notably David Doty, and to some > extent Paul and myself, feel that music was confused in the > last century. It had seen the end of its resources, but did > not know how to expand them. It tried anyway, but wound up > pushing out the wrong stuff, and got gibberish. In a sense, > these composer/theorists want to return to 1900 and try again.That's interesting, because I've never said anything nearly so harsh, and yet I'm the one characterized as arrogant.> Gene, who has tried to take away your free thought? My point > was simply that though every tuning may be considered a > temperament, that it _must be_ requires an added assumption -- > namely, that chords/notes matters to everyone in every situation.I was applying Partch's standards, to the best of my recollection, to his scale. What's the problem?> That assumption may or may not be founded, but until there's a > rigorous proof, it deserves to be on the table."Rigorous proof" in this context makes no sense. I hope *that* isn't arrogant or out of bounds.> There have been many people who have come to the list, full of > the enthusiasm of sharing their ideas, only to be told (or so > they thought), in a language that requires considerable expertise > outside of music to even enter discussion, that their tuning was > worthless.I've seen some things which suggested that to me, I admit. On the other hand, it can be discouraging even to hear that your tuning has been thought of before, or that here is another tuning which seems to do what you want to do, only better.

Message: 3441 - Contents - Hide Contents Date: Tue, 22 Jan 2002 12:11:48 Subject: Re: deeper analysis of Schoenberg unison-vectors From: paulerlich --- 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.

Message: 3442 - Contents - Hide Contents Date: Tue, 22 Jan 2002 14:28:30 Subject: Re: the Lattice Theory Homepage From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 22, 2002 9:34 AM > Subject: [tuning-math] Re: the Lattice Theory Homepage > > > Monz, > > If you're looking at MathWorld, the definition of lattice that we > care about is found under "Point Lattice". Thanks, Paul!What about all the stuff listed under "see also"? : "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 Inequality, Lattice Path, Lattice Sum, Leech Lattice, Minkowski Convex Body Theorem, Modular Lattice, N-Cluster, Nosarzewska's Inequality, Pick's Theorem, Random Walk, Schinzel's Theorem, Schröder Number, Torus, Unit Lattice, Visible Point, Voronoi Polygon" Any relevance of those to what we do here? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3443 - Contents - Hide Contents Date: Tue, 22 Jan 2002 00:22:46 Subject: Re: "I didn't bring up the term religion here..." From: clumma Jon (and all); I've replied on metatuning. Thanks, -C.

Message: 3444 - Contents - Hide Contents Date: Tue, 22 Jan 2002 12:22:50 Subject: Re: Heuristics (Was: Hi Dave K.) From: paulerlich --- In tuning-math@y..., graham@m... wrote:> My experience of generating and sorting linear temperaments fromthe 5- to> the 21-limit is that the "right" error metric for one can be wildly > inappropriate for others.Can you give an example?> One assumption behind the heuristic is that the error is proportional to > the size/complexity of the unison vector.You can call it an assumption, if you wish -- I've verified its approximate correctness for all 10 (wildly different) temperaments I've tried, against Gene's rms measures.> If you measure complexity as > the number of consonant intervals, that's the best case of tempering it > out.What does that mean?> Higher-limit linear temperaments tend not to be best cases, but the > proportionality might still work. At least if you can magically produce > orthogonal unison vectors. I'll have to look at lattice theory more.Well, so far I've only considered the case where one unison vector is tempered out.> The other assumption is that the octave-specific Tenney metric > approximates the number of consonant intervals a comma's composed of. I'm > not sure how closely this holds.This is based on the Kees van Prooijen lattice metric, and again its good approximation was verified relative to Gene's rms measure.> The Tenney metric is a good match for > the first-order odd limit of small intervals. But extended limits can > behave differently. > > For example, 2401:2400 works well in the 7-limit because the numerator > only involves 7, so it has a complexity of 4 despite being fairly complex > and superparticular.This is only one possible complexity measure, not the one Gene's currently using, which already showed a good match with the heuristic. A better one awaits . . .> Whereas a comma involving 11**4, or 14641, still > only has a complexity of 4 in the 11-limit. So if you could get a > superparticular like that, it'd lead to a much smaller error.You're missing the lattice justification for the heuristic. No wonder you're skeptical!> It should follow that 5**4:(13*3*2**4) or 625:624 will be particularly > inefficient between the 13- and 23-limits relative to what the heuristic > would predict.Why? Try a 2D system based on 5 and 13. The heuristics should work fine, especially if you weight ratios of 13 as less important than ratios of 5.

Message: 3445 - Contents - Hide Contents Date: Tue, 22 Jan 2002 22:33:39 Subject: Re: the Lattice Theory Homepage From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Tuesday, January 22, 2002 9:34 AM >> Subject: [tuning-math] Re: the Lattice Theory Homepage >> >> >> Monz, >> >> If you're looking at MathWorld, the definition of lattice that we >> care about is found under "Point Lattice". > > > Thanks, Paul! >> What about all the stuff listed under "see also"? :At least "Torus" is relevant.

Message: 3446 - Contents - Hide Contents Date: Tue, 22 Jan 2002 00:35:07 Subject: Re: "I didn't bring up the term religion here..." From: clumma>> >ore specifically, many of us, notably David Doty, and to some >> extent Paul and myself, feel that music was confused in the >> last century. It had seen the end of its resources, but did >> not know how to expand them. It tried anyway, but wound up >> pushing out the wrong stuff, and got gibberish. In a sense, >> these composer/theorists want to return to 1900 and try again. >>That's interesting, because I've never said anything nearly so >harsh, and yet I'm the one characterized as arrogant.I don't think you're arrogant at all, Gene. Man sakes alive, what is it coming to? To paraphrase Doty, 'this resulted in music which the vast majority of people find unlistenable' ... 'because it ignores the design of the hearing system'. You can read it for yourself over on the Just Intonation Network website (it's the first chapter of the Primer). Paul and I certainly have a far weaker view.>> Gene, who has tried to take away your free thought? My point >> was simply that though every tuning may be considered a >> temperament, that it _must be_ requires an added assumption -- >> namely, that chords/notes matters to everyone in every situation. >>I was applying Partch's standards, to the best of my recollection, >to his scale. What's the problem?I don't know.>> That assumption may or may not be founded, but until there's a >> rigorous proof, it deserves to be on the table. >>"Rigorous proof" in this context makes no sense. I hope *that* >isn't arrogant or out of bounds.Of course. I was speaking figuratively. Do you disagree?>I've seen some things which suggested that to me, I admit.It used to be much, much worse.>On the other hand, it can be discouraging even to hear that your >tuning has been thought of before, or that here is another tuning >which seems to do what you want to do, only better.As I say, I have a hard time seeing where they're coming from. With the exception of my point, above, which I check in to make every once in a while. -Carl

Message: 3447 - Contents - Hide Contents Date: Tue, 22 Jan 2002 12:25:52 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: paulerlich --- 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.> I've been searching the web to learn about > Minkowksi reduction, and so now it appears to me that we are > talking about the strict mathematical definition after all, yes?We have been all along.> What's the purpose of wanting to find the Minkowski-reduced > version of the PB instead of the actual one defined by > Schoenberg's ratios?There's no purpose, as Schoenberg clearly meant for all the unison vectors to be tempered out, and thus for 12-tET rather than JI to be used. Tempering out the original set of unison vectors you posted is exactly equivalent to tempering out the Minkowski-reduced set.

Message: 3448 - Contents - Hide Contents Date: Tue, 22 Jan 2002 14:37:56 Subject: hemiennealimmal / MIRACLE Partch? (was: A top 20 11-limit ...)) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 22, 2002 1:52 PM > Subject: [tuning-math] Re: A top 20 11-limit ... temperament list > >>> Why don't you guys take a look at that? Sounds interesting. >> We would need Partch's scores in a readable form . . .Well ... there are two available: - Glenn Hackbarth's dissertation has a transcription of _Daphne of the Dunes_ into Johnston notation, and - I've transcribed _The Intruder_ into HEWM in my book -- and _Perspectives of New Music_ has published an analysis of that piece by Bob Gilmore, so the hardest part of the work has already been done for that piece. I'll scan my transcription of _The Intruder_ later on today, and upload it to my website. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3449 - Contents - Hide Contents Date: Tue, 22 Jan 2002 02:30:14 Subject: Re: "I didn't bring up the term religion here..." From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:>>> That assumption may or may not be founded, but until there's a >>> rigorous proof, it deserves to be on the table. >>>> "Rigorous proof" in this context makes no sense. I hope *that* >> isn't arrogant or out of bounds. >> Of course. I was speaking figuratively. Do you disagree?Considering I've used methods which only work in RI, why would I?

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