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Message: 3400 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:43:17 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Saturday, January 19, 2002 7:52 AM >> Subject: Re: [tuning-math] A top 20 11-limit superparticularly generated> linear temperament list >> > >> >> Hi Gene, >>>>> From: genewardsmith <genewardsmith@j...> >>> To: <tuning-math@y...> >>> Sent: Friday, January 18, 2002 8:22 PM >>> Subject: [tuning-math] A top 20 11-limit superparticularly generated>> linear temperament list >>> >>> >> But ... it would be really nice if you could explain, as >> only tw examples, exactly what all this means. Since I've >> already played around with these particular unison-vectors, >> explaining what you did here would help me a lot to >> understand the rest of your work. >>>>>> Number 46 Monzo >>> >>> >>> [64/63, 81/80, 100/99, 176/175] >>> >>> ets 7, 12 >>> >>> [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]] >>> >>> [.4190088422, 1] >>> >>> a = 5.0281/12 = 502.8106107 cents >>> >>> badness 312.5112733 >>> rms 28.87226550 >>> g 4.174754057 >> >> >> >> >> -monz > >> OK, I gave this a whirl thru my spreadsheet and this is > what I got: > > > kernel > > 2 3 5 7 11 unison vectors ~cents > > [ 1 0 0 0 0 ] = 2:1 0 > [ 4 0 -2 -1 1 ] = 176:175 9.864608166 > [ 2 -2 2 0 -1 ] = 100:99 17.39948363 > [ 6 -2 0 -1 0 ] = 64:63 27.2640918 > [-4 4 -1 0 0 ] = 81:80 21.5062896 > > adjoint > > [ 0 0 0 -0 0 ] > [ 0 1 1 -1 0 ] > [ 0 4 4 -4 0 ] > [ 0 -2 -2 2 0 ] > [ 0 6 6 -6 0 ] > > determinant = | 0 | > > > mapping of Ets (top row above) to Uvs Which ETs? > > [ 1 1/3 2/3 -2/3 0 ] > [ 4 2&2/3 2&2/3 -2&2/3 0 ] > [ 2 1&1/3 1&1/3 -1&1/3 0 ] > [ 6 4 4 -4 0 ] > [-4 -2&2/3 -2&2/3 2&2/3 0 ] > > > I don't really understand what this is saying either. > > (Many of the "0"s were actually given by Excel as > tiny numbers such as "2.22045 * 10^-16", which is > what it actually gave as the determinant.)This is a linear temperament. If you want a periodicity block, you have to add one "chromatic" unison vector to the list of "commatic" unison vectors Gene gave.

Message: 3401 - Contents - Hide Contents Date: Mon, 21 Jan 2002 12:20 +0 Subject: Heuristics (Was: Hi Dave K.) From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a2g36a+7sl3@xxxxxxx.xxx> Gene:>> 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? Paul:> 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.My experience of generating and sorting linear temperaments from the 5- to the 21-limit is that the "right" error metric for one can be wildly inappropriate for others. One assumption behind the heuristic is that the error is proportional to the size/complexity of the unison vector. If you measure complexity as the number of consonant intervals, that's the best case of tempering it out. 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. 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. 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. 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. 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. It still has a complexity of 4, whereas 13**3 is already 2197 and 23**2 is 529. Yes, 12168:12167 is a 23-limit comma with a complexity of 3. (8*9*13*13):(13**3). I'd prefer to see a heuristic for how complex a temperament produced for a set of unison vectors or pair of ETs will be. Or one for how small the error will be when it's generated by ETs. Graham

Message: 3402 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:57:09 Subject: homotetrachordal (was Re: The 'Arabic' temperament From: paulerlich --- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:> Definition needed for homotetrachordal please!Homotetrachordal: (of an octave species) Having two identical ~4/3 spans, separated by either a ~4/3 or a ~3/2. Example: the octave species C Db E F G Ab B C is homotetrachordal, since the interval pattern from C to F is identical to that from G to C.> Feel free to point out > differences with omnitetrachordalOmnitetrachordal: (of a scale) All octave species are homotetrachordal.

Message: 3403 - Contents - Hide Contents Date: Mon, 21 Jan 2002 08:23:42 Subject: Minkowski reduction (was: ...Schoenberg's rational implications) From: monz Hey Paul,> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, January 20, 2002 7:48 PM > Subject: [tuning-math] Re: lattices of Schoenberg's rational implications > > > --- 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): > > <tables of ratios snipped>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. 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? Please set me straight on this. Here's an article that you (et al) might find useful: "Finding a shortest vector in a two-dimensional lattice modulo m" Finding a shortest vector in a two-dimensional... * [with cont.] (Wayb.) Please, let me know what it means after you've read it. :) 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? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3404 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:58:45 Subject: Re: deeper analysis of Schoenberg unison-vectors From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> OK, the 5th column is like the one you already explained to > me before, where 11 is mapped to a note 1 generator more than > the 12-tET value, like on a second keyboard tuned a quarter-tone > higher.Hmm . . . quarter-tones should _not_ figure into an analysis of a 12- tone periodicity block. Gene, am I missing something?

Message: 3405 - Contents - Hide Contents Date: Mon, 21 Jan 2002 08:50:08 Subject: the Lattice Theory Homepage From: monz Wow! ... check out the diagram and text at the top of this page : Lattice Theory * [with cont.] (Wayb.) Note what it says about cylindrical wrapping! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3406 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:01:42 Subject: Re: deeper analysis of Schoenberg unison-vectors From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Hmm . . . quarter-tones should _not_ figure into an analysis of a 12- > tone periodicity block. Gene, am I missing something?Quarter-tones have nothing to do with it--it's telling us that 11 is being mapped inconsistently.

Message: 3407 - Contents - Hide Contents Date: Mon, 21 Jan 2002 09:29:16 Subject: "I didn't bring up the term religion here..." From: monz Hopefully this will be seen as a little levity ... ;-)> 2823 From: dkeenanuqnetau <d.keenan@u...> > Date: Sun Jan 20, 2002 5:14pm > Subject: Re: A comparison of Partch's scale in RI and Hemiennealimmal > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>> ... 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. > 2827 From: clumma <carl@l...> > Date: Sun Jan 20, 2002 7:56pm > Subject: Re: A top 20 11-limit superparticularly generated linear temperament list > > > 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. ... > 2828 From: genewardsmith <genewardsmith@j...> > Date: Sun Jan 20, 2002 8:02pm > Subject: Re: A top 20 11-limit superparticularly generated linear temperament list > > > > Hmmm? What's "religious" about looking at the mathematics of > someone's scale? > 2841 From: clumma <carl@l...> > Date: Sun Jan 20, 2002 4:07pm > Subject: Re: A top 20 11-limit superparticularly generated linear temperament list > > > I didn't bring up the term religion here, and bringing up religion > is a very religious thing to do. Yes, numerology has many traits > in common with some religions, and numerology has seeped in to RI. > But there are other traits of religion, including the re-casting of > history into one's own perspective.Hmmm ... guess it's time for me to stir up a little trouble ... ;-) Erv Wilson, 1974, "Bosanquet -- A Bridge -- A Doorway to Dialog", Xenharmonikôn 2 <http://www.anaphoria.com/xen2.PDF - Type Ok * [with cont.] (Wayb.)> :>> " ... We've >> always had the Fifth or the Third on some kind of borrowing system that >> takes from Peter to payu Paul. In the positive systems -- and FOR THE FIRST >> TIME IN _WESTERN_ HISTORY we have both the Fifth and the Third, both >> Pythagorean and Just. But instead of borrowing from 3 to pay 5, in linear >> temperaments (especially 41 approximations) we now borrow from 5 to pay >> 7 and 11, _far lesser apostles_.(all emphases Wilson's) :-P -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3408 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:05:41 Subject: Re: deeper analysis of Schoenberg unison-vectors From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>>> 2 3 5 7 11 unison vectors ~cents >>>> >>>> [ 1 0 0 0 0 ] = 2:1 0 >>>> [-5 2 2 -1 0 ] = 225:224 7.711522991 >>>> [-4 4 -1 0 0 ] = 81:80 21.5062896 >>>> [ 6 -2 0 -1 0 ] = 64:63 27.2640918 >>>> [-5 1 0 0 1 ] = 33:32 53.27294323Here is the contents of the Fokker parallelepiped defined by these UVs, at one (arbitrary) position in the lattice: cents numerator denominator 84.467 21 20 203.91 9 8 315.64 6 5 386.31 5 4 498.04 4 3 590.22 45 32 701.96 3 2 813.69 8 5 905.87 27 16 996.09 16 9 1088.3 15 8 1200 2 1 It's pretty clear that most of the consonances will straddle across different instances of the PB, rather than being contained mostly within this set of JI pitches.

Message: 3409 - Contents - Hide Contents Date: Mon, 21 Jan 2002 17:44 +0 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <001301c1a297$fe1d9ec0$af48620c@xxx.xxx.xxx> monz wrote:> Here's an article that you (et al) might find useful: > > "Finding a shortest vector in a two-dimensional lattice modulo m" > Finding a shortest vector in a two-dimensional... * [with cont.] (Wayb.) > > Please, let me know what it means after you've read it. :)Web searches are, unfortunately, more likely to turn up research articles than beginners' guides. That may be why my matrix tutorial is so popular. I do now have a book that covers short vectors. They look very important, but I still don't understand them (haven't even got to that chapter).> 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. Graham

Message: 3410 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:07:48 Subject: Re: deeper analysis of Schoenberg unison-vectors From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > Hi Graham, > > >> From: <graham@m...> >> To: <tuning-math@y...>>> Sent: Sunday, January 20, 2002 12:27 PM >> Subject: [tuning-math] Re: deeper analysis of Schoenberg unison- vectors >> >> >> monz wrote: >>>>> The most I can do with the 3rd column is this: the GCD is 2, >>> so that's equivalent to dividing the 8ve in half, right? >>> Which makes the tritone the interval of equivalence? So if >>> I divide the whole column by 2, I get [0 1 -2 -2 -1]. So >>> does this tell me how many generators away from 12-tET this >>> tuning maps 3, 5, 7, and 11? And exactly what *is* the generator? >>>> The house terminology is that you have a period of tritone, but the >> interval of equivalence is still an octave. > >> OK, sorry ... I realize that I should have made that distinction > myself. But ... what *is* that distinction? Does "period of tritone" > mean that some form of tritone is the generator?The 1/2-octave can be thought of as a generator in the same way that 1 octave can be thought of as a generator in the usual cases, say meantone for example. Normally we refer to the _other_ generator as _the_ generator, and 1/2-octave or 1 octave or 1/n octave as the period. It's the interval of repetition.

Message: 3411 - Contents - Hide Contents Date: Mon, 21 Jan 2002 12:02:29 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: monz> Message 2850 > From: paulerlich <paul@s...> > Date: Sun Jan 20, 2002 10:48pm > Subject: Re: lattices of Schoenberg's rational implications > > > --- 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>.Ah ... so then, Paul, you agreed with me that this PB is a valid one for p 1-184 of _Harmonielehre_?>>> 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 80With variant alternate pitches written on the same line -- and thus with invariant ones on a line by themselves -- these scales are combined into: 1/1 21/20 15/14 35/32 9/8 7/6 25/21 6/5 5/4 21/16 7/5 10/7 3/2 49/32 25/16 63/40 5/3 12/7 7/4 147/80 15/8 My first question is: this is a 7-limit periodicity-block, so can you explain how the two 11-limit unison-vectors disappeared? I've been trying to figure it out but don't see it. One thing I did notice in connection with this, is that 147/80 is only a little less than 4 cents wider than 11/6, which is one of the pitches implied in Schoenberg's overtone diagram (p 23 of _Harmonielehre_) : vector ratio ~cents [ -4 1 -1 2 0 ] = 147/80 1053.2931 - [ -1 -1 0 0 1 ] = 11/6 1049.362941 -------------------- [ -3 2 -1 2 -1 ] = 441/440 3.930158439 So I know that 441/440 is tempered out. But I don't see how to get this as a combination of two of the other unison-vectors. Anyway, regarding the 7-limit PB itself: I could see that all of those pairs and triplets of alternate pitches are separated by either or both of the two 7-limit unison-vectors. I made a lattice of this combination of PBs: Monzo lattice of 4 variant 12-tone 7-limit periodicity-blocks calculated by Paul Erlich from Minkowski-reduced form of my PB for p 1-184 of Schoenberg's _Harmonielehre_ : Yahoo groups: /tuning-math/files/monz/mink-red... * [with cont.] Dotted lines connect the alternate pitches: -- the long, somewhat horizontal dotted line represents the 50:49 = [1 0 2 -2] between the pairs of notes: 21/20 : 15/14 , 7/6 : 25/21 , 7/5 : 10/7 , 49/32 : 25/16 , 147/80 : 15/8 -- and the short, nearly vertical dotted line represents the 36:35 [-2 -2 1 1] between the pairs of notes: 35/32 : 9/8 , 7/6 : 6/5 , 49/32 : 63/40 , 5/3 : 12/7 . These intervals are portrayed graphically in my list of the scale above. I was startled by the unusual number of different symmetries I saw on this lattice.> Message 2861 > From: graham@m... > Date: Mon Jan 21, 2002 0:44pm > Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) > > > > In-Reply-To: <001301c1a297$fe1d9ec0$af48620c@d...> > monz wrote: >>> 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.Then, I reasoned that since all of these pitches are separated by one or two of the unison vectors which define this set of PBs, the lattice could be further reduced to a 12-tone set, one that can still "define the same temperament": Monzo lattice of Monzo's ultimate reduction of Paul Erlich's 4 variant Minkowski-reduced 7-limit PBs for p 1-184 of Schoenberg's _Harmonielehre_, to one 12-tone PB : Yahoo groups: /tuning-math/files/monz/ult-red.gif * [with cont.] Correct? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3412 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:07:51 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: clumma>> > 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?I didn't bring up the term religion here, and bringing up religion is a very religious thing to do. Yes, numerology has many traits in common with some religions, and numerology has seeped in to RI. But there are other traits of religion, including the re-casting of history into one's own perspective. There's no doubt in my mind that Partch's music can be said to ignore all sorts of commas -- who cares? Will we then pronounce that Partch would have been better off using temperament x? By what criteria will we say a composer's work was not tempered? Temperament gets a lot of attention, because RI is too simple to occupy theorists. But it is not too simple to occupy composers, and theorists do a disservice when they do not actively point that out. -Carl

Message: 3413 - Contents - Hide Contents Date: Mon, 21 Jan 2002 20:39:48 Subject: Re: the Lattice Theory Homepage From: genewardsmith --- 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 algebraamong other things, which is why the univeral algebraists in Hawaii care about it.

Message: 3414 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:12:36 Subject: Re: lattices of Schoenberg's rational implications From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> If you want a notation, yes. One which makes the matrix >> unimodular, ie with determinant +-1. > >> So what's the secret to finding that?Forget it. I don't know why you want to bother with Gene's "notation" here. The "notation" would allow you to specify just ratios unambiguously. 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?

Message: 3415 - Contents - Hide Contents Date: Mon, 21 Jan 2002 20:44:33 Subject: Re: "I didn't bring up the term religion here..." From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Hopefully this will be seen as a little levity ... ;-)Pretty good. :) Why is music, even here, so rife with arch-conservativism? In other fields you seem to be able to express a thought without people jumping you, but here even the radicals are conservatives.

Message: 3416 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:16:45 Subject: Re: deeper analysis of Schoenberg unison-vectors From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> 1) Where is it written that 12-tET is consistent with both >> meantone and twintone? This kind of stuff needs to be >> in my Dictionary. >> 12-et is the only thing consistent with both twintone and meantone.I noticed that quite a long time ago. Practicing my 22-tET guitar is a great lesson in twintone that I can apply back on my 12-tET guitar. Same for 31-tET and meantone. Both guitars make me better at 12-tET. Of course, 12-tET always sounds out-of-tune afterwards.

Message: 3417 - Contents - Hide Contents Date: Mon, 21 Jan 2002 12:47:38 Subject: Re: deeper analysis of Schoenberg unison-vectors From: monz Hi Paul, Here is the earlier PB you calculated, before Minkowski reduction:> Message 2839 > From: paulerlich <paul@s...> > Date: Sun Jan 20, 2002 9:05pm > Subject: Re: deeper analysis of Schoenberg unison-vectors > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> 2 3 5 7 11 unison vectors ~cents >> >> [ 1 0 0 0 0 ] = 2:1 0 >> [-5 2 2 -1 0 ] = 225:224 7.711522991 >> [-4 4 -1 0 0 ] = 81:80 21.5062896 >> [ 6 -2 0 -1 0 ] = 64:63 27.2640918 >> [-5 1 0 0 1 ] = 33:32 53.27294323 >> Here is the contents of the Fokker parallelepiped defined by these > UVs, at one (arbitrary) position in the lattice: > > cents numerator denominator > 84.467 21 20 > 203.91 9 8 > 315.64 6 5 > 386.31 5 4 > 498.04 4 3 > 590.22 45 32 > 701.96 3 2 > 813.69 8 5 > 905.87 27 16 > 996.09 16 9 > 1088.3 15 8 > 1200 2 1I later wrote, about the Minkowski-reduced PB:> Message 2862 > From: monz <joemonz@y...> > Date: Mon Jan 21, 2002 3:02pm > Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) > > ... > > My first question is: this is a 7-limit periodicity-block, > so can you explain how the two 11-limit unison-vectors disappeared? > I've been trying to figure it out but don't see it.And I pose the same question here. I've added and subtracted all pairs of UVs in this kernel, and I don't see anything that should eliminate 11 from the PB. *Please* explain! (I understand intuitively how it works, but I don't see it happening in these numbers.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3418 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:18:06 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: paulerlich --- 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!

Message: 3419 - Contents - Hide Contents Date: Mon, 21 Jan 2002 21:27:49 Subject: Re: Minkowski reduction (was: ...Schoenberg's rational implications) From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> With variant alternate pitches written on the same line > -- and thus with invariant ones on a line by themselves -- > these scales are combined into: > > 1/1 > 21/20 15/14 > 35/32 9/8 > 7/6 25/21 6/5 > 5/4 > 21/16 > 7/5 10/7 > 3/2 > 49/32 25/16 63/40 > 5/3 12/7 > 7/4 > 147/80 15/8 1--21/20--9/8--6/5--5/4--21/16--7/5--3/2--25/16--5/3--7/4--15/8is one version of this, and has scale steps (25/24)^2 * (21/20)^3 * (16/15)^4 * (15/14)^3 = 2 Looks awfully familiar...a slight permutation of the order of the steps, and we have 1--21/20--9/8--6/5--5/4--4/3--7/5--3/2--8/5--5/3--7/4--15/8 The very first scale I ever constructed.> My first question is: this is a 7-limit periodicity-block, > so can you explain how the two 11-limit unison-vectors disappeared?The block wasn't that big, and perhaps was a little flat in the 11-direction.> So I know that 441/440 is tempered out. But I don't see > how to get this as a combination of two of the other > unison-vectors.Maybe it isn't. You should set up the linear equation and solve for it.

Message: 3420 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:21:49 Subject: Re: Hi Dave K. From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> How do you calculate "taxicab" error? >> What corresponds to rms in L1 (unweighted taxicab) is the median in >place of the mean, and the mean of the sum of the absolute values of >the deviations from the median in place of rms.THANK YOU GENE! Can this be applied to a _triangular_, instead of quadrangular, city-block graph? If so, can you tell be how to apply that to this lattice metric: Searching Small Intervals * [with cont.] (Wayb.) Note that the only distances I'm concerned with "accurately capturing" are those of intervals m:n where m/n is approximately 1.

Message: 3422 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:48:40 Subject: Re: Hi Dave K. From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> THANK YOU GENE! Can this be applied to a _triangular_, instead of > quadrangular, city-block graph? If so, can you tell be how to apply > that to this lattice metric: > > Searching Small Intervals * [with cont.] (Wayb.)The hexagonal region H consisting of all (m,n) with measure less than or equal to 1 is convex, so this defines a norm for a normed vector space: if v = (m, v) then ||v|| = r is the maximum nonnegative r such that r*v is in H. Given a set of vectors {v1, ... , vk} you could then seek to find a vector t which minimizes ||v1 - t|| + ||v2-t|| + ... + ||vk-t|| which could be used as a central point, and define error as the minimum value thus achieved.

Message: 3423 - Contents - Hide Contents Date: Mon, 21 Jan 2002 21:48:46 Subject: Re: "I didn't bring up the term religion here..." From: genewardsmith --- In tuning-math@y..., "jonszanto" <jonszanto@y...> wrote:> That's pretty good. When I posted something about artists pushing the > envelope, you replied back that things were better in the 18th > century! That's pretty "radical conservative" in my book!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. 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?> But this is one area that I can well agree on: the tuning community, > those that stick around at least, are some of the most musically > conservative I've come across. And it has *always* befuddled me, but > since music is personal choice, I just let it go.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.

Message: 3424 - Contents - Hide Contents Date: Mon, 21 Jan 2002 02:52:57 Subject: Re: Hi Dave K. From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> THANK YOU GENE! Can this be applied to a _triangular_, instead of >> quadrangular, city-block graph? If so, can you tell be how to apply >> that to this lattice metric: >> >> Searching Small Intervals * [with cont.] (Wayb.) >> The hexagonal region H consisting of all (m,n) with measure lessthan or equal to 1 is convex, so this defines a norm for a normed vector space: if v = (m, v) then ||v|| = r is the maximum nonnegative r such that r*v is in H.> > Given a set of vectors {v1, ... , vk} you could then seek to find avector t which minimizes> > ||v1 - t|| + ||v2-t|| + ... + ||vk-t|| > > which could be used as a central point, and define error as theminimum value thus achieved. I'm not getting this last part. Will it help make my heuristic work?

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