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Message: 3375 - Contents - Hide Contents Date: Sun, 20 Jan 2002 11:28 +0 Subject: Re: more questions about adjoints and mappings From: graham@xxxxxxxxxx.xx.xx monz wrote:> Here's a simple example: Ellis's Duodene > > > kernel > > 2 3 5 unison vectors ~cents > > [ 1 0 0 ] = 2:1 1200 > [ -4 4 -1 ] = 81:80 21.5062896 > [ 7 0 -3 ] = 128:125 41.05885841 > > > adjoint > > [ -12 0 0 ] > [ -19 -3 1 ] > [ -28 0 4 ] > > determinant = | -12 | > > > "mapping" of UVs > > [ 1 0 0 ] > [ 0 1 0 ] > [ 0 0 1 ] > > > So here, I can see that the h12 mapping does not > temper out the 2:1 ... and I still don't understand > what that means. Is it simply because any "8ve"-based > ET must include 2:1 by definition?Tempering out the 2:1 is the same as enforcing octave equivalence. ETs aren't octave-equivalent. In 12-equal, a unison is 0 steps, an octave is 12 steps and two octaves are 24 steps. These certainly aren't the same.> I can also see that the third column of the adjoint > specifies some kind of meantone, which tempers out > the 2:1 and the 81:80, but not the diesis 128:125. > Is there a way to tell what flavor of meantone it is?No, because all you asked is for an octave-equivalent temperament that tempers out the 81:80. All meantones do that. You could set the enharmonic diesis to be just, following the method I outline in <Linear temperaments from matrix formalism * [with cont.] (Wayb.)> and that happens to give quarter-comma meantone.> And the middle column of the adjoint specifies some > temperament which does not temper out the 81:80. > But can someone explain what kind of tuning this is?It's an octave-equivalent tuning that tempers out the enharmonic diesis. That's the difference between three 5:4 major thirds and an octave. So, if you temper it out, the octave must divide into three equal parts, each of which can be called a 5:4. The column tells you that by having a GCD of 3, and a zeros in the 2:1 and 5:1 rows. The 3:1 column doesn't have a zero in it, so you can choose whatever size for a fifth you like. It's independent of the major third and octave. Graham

Message: 3378 - Contents - Hide Contents Date: Sun, 20 Jan 2002 11:43:46 Subject: Re: more questions about adjoints and mappings From: monz Thanks, Graham, this helps a lot! -monz> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, January 20, 2002 3:28 AM > Subject: [tuning-math] Re: more questions about adjoints and mappings > > > monz wrote: >>> Here's a simple example: Ellis's Duodene >> >> >> kernel >> >> 2 3 5 unison vectors ~cents >> >> [ 1 0 0 ] = 2:1 1200 >> [ -4 4 -1 ] = 81:80 21.5062896 >> [ 7 0 -3 ] = 128:125 41.05885841 > > ... >> Tempering out the 2:1 is the same as enforcing octave equivalence > <etc.> _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3379 - Contents - Hide Contents Date: Sun, 20 Jan 2002 12:05:27 Subject: Re: deeper analysis of Schoenberg unison-vectors From: monz Hi Graham and Gene,> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, January 20, 2002 2:39 AM > Subject: Re: [tuning-math] deeper analysis of Schoenberg unison-vectors > > >> >>> Message 2798>>> From: "monz" <joemonz@y...> >>> Date: Sat Jan 19, 2002 2:40 pm >>> Subject: deeper analysis of Schoenberg unison-vectors >> Yahoo groups: /tuning-math/message/2798 * [with cont.] >>> >>> ... >>> >>> So as of p 184 in _Harmonielehre_, we can construct as system >>> valid for Schoenberg's theories, as follows: >>> >>> kernel >>> >>> 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 >>> >>> adjoint >>> >>> [ 12 0 0 0 0 ] >>> [ 19 1 2 -1 0 ] >>> [ 28 4 -4 -4 0 ] >>> [ 34 -2 -4 -10 0 ] >>> [ 41 -1 -2 1 12 ] >>> >>> determinant = | 12 | > >>> From: monz <joemonz@xxxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Sunday, January 20, 2002 12:08 AM >> Subject: Re: [tuning-math] deeper analysis of Schoenberg unison-vectors > > UV map >>> [ 1 0 0 0 0 ] >> [ 0 1 0 0 0 ] >> [ 0 0 1 0 0 ] >> [ 0 0 0 1 0 ] >> [ 0 0 0 0 1 ] > >> So in other words, the way Gene would write it: > > h12(225/224) = h12(81/80) = h12(64/63) = h12(33/32) = 0 > h12(2/1) = 1 > > > But how do you label those other four columns? Well, for > the time being, I'll call them h0, g0, f0, and e0, respectively > from left to right, so that: > > h0(2/1) = h0(81/80) = h0(63/64) = h0(33/32) = 0 , h0(225/224) = 1 > > g0(2/1) = g0(225/224) = g0(63/64) = g0(33/32) = 0 , g0(81/80) = 1 > > f0(2/1) = f0(225/224) = f0(81/80) = f0(33/32) = 0 , f0(64/63) = 1 > > e0(2/1) = e0(225/224) = e0(81/80) = e0(64/63) = 0 , e0(33/32) = 1 > > > So, the 2nd and 4th column-vectors in the adjoint (h0 and f0, > respectively) define two versions of meantone: > > - one (h0) in which 7 maps to the "minor 7th" = -2 generators, > and which tempers out all the UVs except 225/224; > > - one (f0) in which 7 maps to the "augmented 6th" = +10 generators, > and which tempers out all the UVs except 64/63; > > and both of which map 11 to the "perfect 4th" = -1 generator. > > > But what about the 3rd and 5th column-vectors in the adjoint > (g0 and e0, respectively)? What tunings are they? I don't get it. > > And what relevance to these other mappings have to Schoenberg's > theory?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. So I understand that. 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? Thanks. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3380 - Contents - Hide Contents Date: Sun, 20 Jan 2002 20:27 +0 Subject: Re: deeper analysis of Schoenberg unison-vectors From: graham@xxxxxxxxxx.xx.xx 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. As for the generator, well, either entry of 1 will map to it. Graham

Message: 3381 - Contents - Hide Contents Date: Sun, 20 Jan 2002 13:03:19 Subject: Re: deeper analysis of Schoenberg unison-vectors From: monz Hi Graham,> From: <graham@xxxxxxxxxx.xx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > 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?> As for the generator, well, either entry of 1 will map to it.Hmmm ... but the signs are opposite, which I think is why I'm confused. If the mapping of both 3 and 11 showed "1", then it would be more understandable: 3 and 11 both map to the generator, which is somewhere in the vicinity of a tritone ... that makes sense to me. But here we have 1 and -1, respectively. I called this column g0, so from the adjoint, we have g0(3)=2 and g0(11)=-4. Does the sign not matter because the tritone splits the interval of equivalence exactly in half? And can you also explain how 5 and 7 both map to the same number of generators in this case? And yet again I ask:> And what relevance to these other mappings have to Schoenberg's > theory? Still confused, -monz _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3382 - Contents - Hide Contents Date: Sun, 20 Jan 2002 13:09:46 Subject: Re: lattices of Schoenberg's rational implications From: monz Help! I set up an Excel spreadsheet to calculate the notes of a periodicity-block according to Gene's formula as expressed here:> Message 2164 > From: "genewardsmith" <genewardsmith@j...> > Date: Wed Dec 26, 2001 3:27 am > Subject: Re: lattices of Schoenberg's rational implications Yahoo groups: /tuning-math/message/2164 * [with cont.] > > > We can extend the set {33/32,64/63,81/80,45/44} to an > 11-limit notation in various ways, for instance > > <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5] > > where g7 differs from h7 by g7(7)=19. Using this, > we find the corresponding block is > > (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12) > (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9; > the Pythagorean scale. We don't need anything new to find > a 12-note scale; we get > > 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--16/9--15/8 > > or variants, the variants coming from the fact that 12 is even, > by using 12 rather than 7 in the denominator. > Message 2185 > From: "genewardsmith" <genewardsmith@j...> > Date: Wed Dec 26, 2001 6:25 pm > Subject: Re: Gene's notation & Schoenberg lattices Yahoo groups: /tuning-math/message/2185 * [with cont.] > > ... > > For any non-zero I can define a scale by calculating for 0<=n<d > > step[n] = (56/55)^round(7n/d) (33/32)^round(12n/d) > (64/63)^round(7n/d) (81/80)^round(-2n/d) (45/44)^round(5n/d)It worked just fine for both of these examples, the 7-tone and 12-tone versions. But for the kernel I recently posted for Schoenberg ...> kernel > > 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 > > adjoint > > [ 12 0 0 0 0 ] > [ 19 1 2 -1 0 ] > [ 28 4 -4 -4 0 ] > [ 34 -2 -4 -10 0 ] > [ 41 -1 -2 1 12 ] > > determinant = | 12 |... it doesn't work. All I get are powers of 2. Why? How can it be fixed? Do I need yet another independent unison-vector instead of 2:1? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3383 - Contents - Hide Contents Date: Sun, 20 Jan 2002 21:49:08 Subject: Re: deeper analysis of Schoenberg unison-vectors From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> So in other words, the way Gene would write it: > > h12(225/224) = h12(81/80) = h12(64/63) = h12(33/32) = 0 > h12(2/1) = 1Actually, I'd write it h12(2) = 12; by definition, in fact, hn(2) = n.> But what about the 3rd and 5th column-vectors in the adjoint > (g0 and e0, respectively)? What tunings are they? I don't get it.g0 is Twintone, aka Paultone. e0 sends everything to 0 mod 12, and is not a temperament.> And what relevance to these other mappings have to Schoenberg's > theory?(1) It is consistent with twintone as well as meantone, and so is a 12-et theory (2) It sends all your commas to 0 mod 12, so again it is a 12-et theory.

Message: 3384 - Contents - Hide Contents Date: Sun, 20 Jan 2002 14:01:05 Subject: Re: deeper analysis of Schoenberg unison-vectors From: monz Hi Gene, Regarding:> So as of p 184 in _Harmonielehre_, we can construct as system > valid for Schoenberg's theories, as follows: > > kernel > > 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 > > adjoint > > [ 12 0 0 0 0 ] > [ 19 1 2 -1 0 ] > [ 28 4 -4 -4 0 ] > [ 34 -2 -4 -10 0 ] > [ 41 -1 -2 1 12 ] > > determinant = | 12 | > > > UV map > > [ 1 0 0 0 0 ] > [ 0 1 0 0 0 ] > [ 0 0 1 0 0 ] > [ 0 0 0 1 0 ] > [ 0 0 0 0 1 ] >Gene replied to my questions:> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, January 20, 2002 1:49 PM > Subject: [tuning-math] Re: deeper analysis of Schoenberg unison-vectors > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> So in other words, the way Gene would write it: >> >> h12(225/224) = h12(81/80) = h12(64/63) = h12(33/32) = 0 >> h12(2/1) = 1 >> Actually, I'd write it h12(2) = 12; by definition, in fact, > hn(2) = n.Right, I understand that, but it's for the values in the *adjoint*. This is for the values in the "UV map" matrix, which simply shows which UVs are tempered out and which are not.>> But what about the 3rd and 5th column-vectors in the >> adjoint (g0 and e0, respectively)? What tunings are they? >> I don't get it. >> g0 is Twintone, aka Paultone.I still don't know what that is, and need to do some studying.> e0 sends everything to 0 mod 12, and is not a temperament.So then exactly what *is* that column-vector telling us? Simply that this is 12-tET?>> And what relevance to these other mappings have to >> Schoenberg's theory? >> (1) It is consistent with twintone as well as meantone, > and so is a 12-et theory > > (2) It sends all your commas to 0 mod 12, so again it > is a 12-et theory.Ah ... a few flickers illuminate the dark cave! Two new questions, then: 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. 2) How can we see that "it sends all your commas to 0 mod 12"? Is that what the adjoint's 5th column-vector ("e0") is saying? Thanks! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3385 - Contents - Hide Contents Date: Sun, 20 Jan 2002 22:08:55 Subject: Re: lattices of Schoenberg's rational implications From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> determinant = | 12 |> ... it doesn't work.This determinant is why. In my example, the determinant had an absolute value of 1, and so we get what I call a "notation", meaning every 11-limit interval can be expressed in terms of integral powers of the basis elements. You have a determinant of 12, and therefore torsion. In fact, you map to the cyclic group C12 of order 12, and the twelveth power (or additively, twelve times) anything is the identity.> Why? How can it be fixed? Do I need yet another > independent unison-vector instead of 2:1?If you want a notation, yes. One which makes the matrix unimodular, ie with determinant +-1.

Message: 3386 - Contents - Hide Contents Date: Sun, 20 Jan 2002 22:14:11 Subject: Re: A comparison of Partch's scale in RI and Hemiennealimmal From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> RI: > > edges: 18, 32, 64, 88 > > connectivity: 0, 0, 0, 2 > > > Hemiennealimmal: > > edges: 64, 106, 159, 219 > > connectivity: 0, 0, 0, 4 > > > The numbers are edges/connectivity in the 5, 7, 9 and 11-limits. Iconclude 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.

Message: 3387 - Contents - Hide Contents Date: Sun, 20 Jan 2002 14:23:24 Subject: Re: lattices of Schoenberg's rational implications From: monz ----- Original Message ----- From: genewardsmith <genewardsmith@xxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Sunday, January 20, 2002 2:08 PM Subject: [tuning-math] Re: lattices of Schoenberg's rational implications> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> determinant = | 12 | >>> ... it doesn't work. >> This determinant is why. In my example, the determinant > had an absolute value of 1, and so we get what I call a > "notation", meaning every 11-limit interval can be expressed > in terms of integral powers of the basis elements. You have > a determinant of 12, and therefore torsion.OK ... I still have lots to learn about torsion.> In fact, you map to the cyclic group C12 of order 12, Huh? > and the twelveth [_sic_: twelfth] power (or additively, > twelve times) anything is the identity.OK, I can follow that.>> Why? How can it be fixed? Do I need yet another >> independent unison-vector instead of 2:1? >> If you want a notation, yes. One which makes the matrix > unimodular, ie with determinant +-1.So what's the secret to finding that? And so then is there or is there not any value in calculating the kernel which has 2:1 in it? If yes, then what? If no, then why not? Don't we need determinants <-1 and >1 in order to have a denominator with which to find the JI scale? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3388 - Contents - Hide Contents Date: Sun, 20 Jan 2002 22:37:05 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> But the question seems fairly irrelevant >> of any temperament, since no-one I know wants to have as many as 72 >> notes per octave on a keyboard or fretboard and no composer I know >> wants to have to deal with that many notes (choosing always some more >> manageable subset). >> This is assuming that you must be using a keyboard or fretboard.Not in the case of the composers. But I was wrong to say they always choose some more manageable subset. Sometimes they just use continuously gliding tones etc.> Partch, after all, *did* have 43 actual tones per octave in play, so Idon'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.> Also I don't find it likely that anyone would want>> to play Partch's scale in more than one "key" per piece. >> Even if they did not, tuning Partch's scale in this way would giveyou some equivalences for free (deriving from 2401/2400, 3025/3024,> 4375/4374 and 9801/9800) which would make tempering Partch's 43tones in this way a perfectly reasonable option. Yes. Certainly. But those equivalences would also be quite acceptable _without_ tempering. Perhaps one day we'll have all of Partch's works in some machine-readable form and can check to see if he ever used any of them.

Message: 3389 - Contents - Hide Contents Date: Sun, 20 Jan 2002 23:23:51 Subject: Re: deeper analysis of Schoenberg unison-vectors From: genewardsmith --- 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. The twintone wedgie is tt = [-2,4,4,-2,-12,11] and the meantone wedgie is mt = [-1,-4,-10,-12,13,-4]. We can take wedge products of the wedgie with either vals (leading to intervals) or intervals (leading to vals); in particular we have mt ^ 64/63 = h12 tt ^ 81/80 = -h12 (according to the basis I used in my program). Hence h12 is the only thing which works with both of them. Taking wedge products of wedgies with vals gives us stuff like mt ^ h22 = 225/224 tt ^ h15 = 63/64 and so forth.> 2) How can we see that "it sends all your commas to 0 mod 12"? > Is that what the adjoint's 5th column-vector ("e0") is saying?Right--it sends everything to 0 but 33/32, and it sends that to 12.

Message: 3390 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:33:48 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: paulerlich> 1. Hemiennealimmal [...]They all had b = 1 octave?

Message: 3391 - Contents - Hide Contents Date: Mon, 21 Jan 2002 04:44:05 Subject: Re: deeper analysis of Schoenberg unison-vectors From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> But what about the 3rd and 5th column-vectors in the >>> adjoint (g0 and e0, respectively)? What tunings are they? >>> I don't get it. >>>> g0 is Twintone, aka Paultone. > >> I still don't know what that is http://www-math.cudenver.edu/~jstarret/22ALL.pdf - Type Ok * [with cont.] (Wayb.)Especially the section, "Tuning the Decatonic Scale", page 22.

Message: 3392 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:37:18 Subject: Re: Hi Dave K. From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> How do you calculate "taxicab" error?Taxicab is L1, so it should be a sum (perhaps weighted) of the absolute values of the errors, should it not?

Message: 3393 - Contents - Hide Contents Date: Mon, 21 Jan 2002 07:15:16 Subject: Re: deeper analysis of Schoenberg unison-vectors From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> 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.The one problem with microtonality is that it does this. 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. It came up whenI was playing with the twintone wedgie, since tt ^ 4375/4374 = [34, 54, 79, 96]. I looked for more things I might have overlooked in this way, but didn't find much: ennealimmal ^ 2048/2025 = h126, and [-16,-2,-5,-6,-37,34] ^ 64/63 = h37.

Message: 3394 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:39:12 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> 1. Hemiennealimmal > [...] >> They all had b = 1 octave?This one has b = 1/18; it's in the [a, b] thing which gives the generators.

Message: 3395 - Contents - Hide Contents Date: Mon, 21 Jan 2002 11:02 +0 Subject: Re: deeper analysis of Schoenberg unison-vectors From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <005201c1a1f5$e389c180$af48620c@xxx.xxx.xxx> monz wrote:> Hmmm ... but the signs are opposite, which I think is why I'm confused. > If the mapping of both 3 and 11 showed "1", then it would be more > understandable: 3 and 11 both map to the generator, which is somewhere > in the vicinity of a tritone ... that makes sense to me. But here > we have 1 and -1, respectively. I called this column g0, so from > the adjoint, we have g0(3)=2 and g0(11)=-4. Does the sign not > matter because the tritone splits the interval of equivalence > exactly in half?You can either have 3:1 or 11:1, tritone reduced, as the generator. Or you could have either (3:1 the same as 1:11) or (1:3 the same as 11:1). The latter makes sense, because 4:3 and 11:8 are already within the tritone.> And can you also explain how 5 and 7 both map to the same > number of generators in this case?5:4 is 386 cents, and 7:4 is 969 cents. Tritone-reduced, 7:4 is 369 cents. 386 and 369 cents are close enough to be approximated equal. Note you can also go to <Linear Temperament Finding Home * [with cont.] (Wayb.)>, choose the "temperaments from unison vectors" option (without the 2:1) and plug in your unison vectors. It happens to give generators larger than an octave currently, but that's not important.> And yet again I ask: >>> And what relevance to these other mappings have to Schoenberg's >> theory?That's for to you to work out. Graham

Message: 3396 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:41:14 Subject: Re: Hi Dave K. From: genewardsmith --- 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.

Message: 3397 - Contents - Hide Contents Date: Mon, 21 Jan 2002 11:29:45 Subject: 'Arabic' example From: genewardsmith I've uploaded an example in the 31-et version of Arabic Diatonic, which is in effect in the 'Arabic' temperament. It's supposed to show its omnichordality or whatever you would like to call it--the property of being an 11-limit chord. Yahoo groups: /tuning-math/files/Gene/tamwind.mid * [with cont.]

Message: 3398 - Contents - Hide Contents Date: Mon, 21 Jan 2002 01:41:17 Subject: ERROR IN CARTER'S SCHOENBERG (Re: badly tuned remote overtones) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Friday, January 18, 2002 1:04 PM >> Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned > remote overtones) >> >>>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>> I think you misunderstand me, Paul. I just mean that there's >>> probably a good chance that at least some of the time, Schoenberg >>> thought of the "Circle of 5ths" in a meantone rather than a >>> Pythagorean sense. >>>> I doubt it. For Schoenberg, the circle of 5ths closes after 12 >> fifths -- which is closer to being true in Pythagorean than in most >> meantones. > >> I understand that, Paul ... but if one is trying to ascertain > the potential rational basis behind Schoenberg's work, how does > one decide which unison-vectors are valid and which are not?Only ones he specifically pointed to are valid. If it's not enough to give you a PB, it'll still give you a linear temperament, which should be interesting enough!> Schoenberg was very clear about what he felt were the "overtone" > implications of the diatonic scale (and later, the chromatic > as well), but as I showed in my posts, the only "obvious" > 5-limit unison-vector is the syntonic comma, and it seemed > to me that there always needed to be *two* 5-limit unison-vectors > in order to have a matrix of the proper size (so that it's square).In 5-limit, yes.> > (I realize that by transposition it need not be a 5-limit UV,By transposition? Not sure what you're getting at, but none of the unison vectors need to necessarily be 5-limit when constructing an 11- limit PB.>>> This reference to you is only meant to credit you for opening >>> my eyes to the strong meantone basis behind a good portion of >>> the "common-practice" European musical tradition. >>>> OK -- but you're confusing two completely unrelated facts -- that >> 128:125 is just in 1/4-comma meantone, and that 128:125 is one of the >> simplest unison vectors for defining a 12-tone periodicity block. > >> OK, I'm willing to take note of your point, but ... *why* are > these two facts "completely unrelated"? Isn't it possible that > there *is* some relation between them that no-one has noticed > before?128:125 is just in 1/4-comma meantone. If 128:125 is tempered out, the meantone is transformed into 12-tET. Then again, you could just as well temper 2048:2025 out of meantone and get 12-tET, or temper 32768:32805 out of meantone and get 12-tET. These intervals are not just in 1/4-comma meantone. 128:125 is a UV of choice for this purpose _only_ because it's the simplest, _not_ because it's just in 1/4-comma meantone.

Message: 3399 - Contents - Hide Contents Date: Mon, 21 Jan 2002 03:31:44 Subject: Re: lattices of Schoenberg's rational implications From: monz Gene, Paul, Graham,> Message 2850 > From: "paulerlich" <paul@s...> > Date: Sun Jan 20, 2002 10:48 pm > Subject: Re: lattices of Schoenberg's rational implications Yahoo groups: /tuning-math/message/2850 * [with cont.] > > > --- 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 scales snipped>Paul, thanks!!! This *is* awesome! Just what I've been waiting and hoping for since that posting on Christmas day. Actually, it's something I've been trying to achieve since about 1988 or so. Fantastic. Thanks for your help too, Gene, and for your explanations, Graham. Lattices and webpages to follow soon!! And then, some serious work on retuning Schoenberg MIDIs! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

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