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Message: 2625 - Contents - Hide Contents Date: Sat, 22 Dec 2001 03:52:09 Subject: Re: My top 5--for Paul From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> I suggest the "Woolhouse" as a name for this temperament, >> Tricky -- "Woolhouse temperament" clearly means 7/26-comma meantone > to me. So this one falls inside the cutoff but the 612-tET-related > one doesn't?That's what happens with cutoffs--this one has an even lower badness, not that it makes a difference at this point. I don't know if it deserves a name; I agree Woolhouse won't do, and it's really more of a 1783 system anyway. I tried to give it a name because of itsvery low badness, but it's kind of absurd. I'd favor reeling in the cutoff . . . schismic is pretty> complex already . . .I could continue analyzing them until we are clearly running off the rails into idiocy, which is why I suggested the other cutoff. Finally, I'd like to> reinstate my strong belief that the "g" measure should be _weighted_.You think 3 should counts for more than 5, etc?
Message: 2626 - Contents - Hide Contents Date: Sat, 22 Dec 2001 15:29:14 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 22, 2001 12:21 PM > Subject: [tuning-math] coordinates from unison-vectors (was: 55-tET) > > > Paul, can you please explain the procedure you use to find > coordinates from a given set of unison-vectors, as you did > here? Thanks.I've figured out how to use Excel to calculate the coordinates within the unit square of the inverse of a 2-dimensional matrix, and even how to have it centered on 0,0... I think. Here are my results for the periodicity-block [3^x * 5^y] with unison-vectors (x y) of (4 -1) and (19 9): 0/55 0/55 9/55 1/55 18/55 2/55 27/55 3/55 -19/55 4/55 -10/55 5/55 -1/55 6/55 8/55 7/55 17/55 8/55 26/55 9/55 -20/55 10/55 -11/55 11/55 -2/55 12/55 7/55 13/55 16/55 14/55 25/55 15/55 -21/55 16/55 -12/55 17/55 -3/55 18/55 6/55 19/55 15/55 20/55 24/55 21/55 -22/55 22/55 -13/55 23/55 -4/55 24/55 5/55 25/55 14/55 26/55 23/55 27/55 -23/55 -27/55 -14/55 -26/55 -5/55 -25/55 4/55 -24/55 13/55 -23/55 22/55 -22/55 -24/55 -21/55 -15/55 -20/55 -6/55 -19/55 3/55 -18/55 12/55 -17/55 21/55 -16/55 -25/55 -15/55 -16/55 -14/55 -7/55 -13/55 2/55 -12/55 11/55 -11/55 20/55 -10/55 -26/55 -9/55 -17/55 -8/55 -8/55 -7/55 1/55 -6/55 10/55 -5/55 19/55 -4/55 -27/55 -3/55 -18/55 -2/55 -9/55 -1/55 Right? The graph of this is at Yahoo groups: /tuning-math/files/monz/inv-matr... * [with cont.] (I think my axis labels may be wrong, but the graph appears to be showing the correct periodicity-block shape.) But now how do I go about "transforming them back to the lattice (using the original Fokker matrix)" as described at <Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>, to get the actual lattice coordinates? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2627 - Contents - Hide Contents Date: Sat, 22 Dec 2001 03:53:57 Subject: Re: Four funky ones From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >> badness: 46.1 >> Is this a typo? Should this be 461? I might revise my badness cutoff > now . . .I was cross-eyed again, but so where you--you missed my correction. :) If you want to keep this, 500 is working for you. Do you think I've missed anything, by the way?
Message: 2628 - Contents - Hide Contents Date: Sat, 22 Dec 2001 23:45:09 Subject: Kleismic & co From: genewardsmith 2109375/2097152 = 2^-21 3^3 5^7 Orwell Map: [ 0 1] [ 7 0] [-3 3] Generators: a = 19.01127197/84; b = 1 badness: 305.93 rms: .8004 g: 7.257 errors: [-.828, -1.082, -.255] ets: 22,31,53,84 15625/15552 = 2^-6 36-5 5^6 Kleismic Map: [ 0 1] [ 6 0] [ 5 1] Generators: a = 14.00435233/53 (~6/5); b = 1 badness: 97 rms: 1.030 g: 4.546 errors: [.523, -.915, -1.438] ets: 19,34,53,68,72,87,140 1600000/1594323 = 2^9 3^-13 5^-2 Acute Minor Third system Map: [ 0 1] [-5 3] [-13 6] Generators: a = 28.00947813/99 (~243/200); b = 1 badness: 305.53 rms: .3831 g: 9.273 error: [-.5009, .0716, -.4293] All three of the above systems are done quite well by the 53-et, with different minor third generators; Orwell is 12/53, Kleismic is 14/53, and AMT is 15/53 2^8 3^14 5^-13 Parakleismic Map: [ 0 1] [-13 5] [-14 6] Generators: a = 30.99967314/118 (~6/5); b = 1 badness: 373 rms: .2766 g: 11.045 errors: [-.2169, .1735, .3904]
Message: 2629 - Contents - Hide Contents Date: Sat, 22 Dec 2001 08:04:13 Subject: For Pierre, from tuning From: genewardsmith> Given two unison vectors U and V determining a block having periodicity= > in octave, how to determine the minimal polytope centered around the > unison in which the transposition space of a possible simplest mode m > would be contained?If I start with the basis <16/15,81/80,25/24> I get a corresponding inverse= matrix [h7,h3,h5]: [ 7 3 5] [11 5 8] [16 7 12] If I apply your transformation to the two commatic unisons, I get <16/15,128/135,81/80>^(-1) = [h7,-h5,h3-h5] = [h7,-h5,-h2] <16/15,25/24,128/135>^(-1) = [h7,h5-h3,-h3] = [h7,h2,-h3] I can produce corresponding blocks, starting from 1, for each of these by m= eans of (16/15)^n (81/80)^round(3n/7) (25/24)^round(5n/7) (16/15)^n (128/135)^round(-5n/7) (81/80)^round(-2n/7) (16/15)^n (25/24)^round(2n/7) (128/135)^round(-3n/7) where n runs from 0 to 6 and "round" means round to the nearest integer. The scales we get in this way are scale1: [1, 10/9, 6/5, 4/3, 3/2, 5/3, 9/5] scale2: [1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9] scale3: [1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8] There is overlap among these; their union is [1, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 27/16, 16/9, 9/5= , 15/8]; Does this have anything to do with what you are saying?> By the way, I would ask if the present state of your maths permits to sho=w that these  systems are the simplest in <2 3 5>Z3 ? I need a definition before I get a statement or a proof—what do you mean by= "simplest"?
Message: 2630 - Contents - Hide Contents Date: Sat, 22 Dec 2001 19:35:26 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 22, 2001 3:29 PM > Subject: Re: [tuning-math] coordinates from unison-vectors (was: 55-tET) > > >>> From: monz <joemonz@xxxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Saturday, December 22, 2001 12:21 PM >> Subject: [tuning-math] coordinates from unison-vectors (was: 55-tET) >> >> >> Paul, can you please explain the procedure you use to find >> coordinates from a given set of unison-vectors, as you did >> here? Thanks.Never mind! Trial and error wins again! By brute force, a bit of research into matrix transformations, and a whole lot of luck, I figured out how to do it. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2631 - Contents - Hide Contents Date: Sat, 22 Dec 2001 20:08:50 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, December 22, 2001 7:35 PM > Subject: Re: [tuning-math] coordinates from unison-vectors (was: 55-tET) > > > By brute force, a bit of research into matrix transformations, > and a whole lot of luck, I figured out how to do it.I've figured out how to do what I was trying to do, but I'm still puzzled over the way changing the sign of the exponents in the unison-vector changes the shape of the periodicity-block. I know that if the sign of the 3-exponent is changed, the sign for the 5-exponent must be reversed accordingly. But I find sometimes that using, for example, (4 -1) for the syntonic comma doesn't always give me the PB I expected, whereas making it (-4 1) does. Now that I have an Excel spreadsheet set up to graph the periodicity-blocks, I can simply play with the signs until I get the block I'm looking for. But I'm curious as to why the shape changes as it does. Can anyone explain this? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2632 - Contents - Hide Contents Date: Sun, 23 Dec 2001 20:59:10 Subject: Re: My top 5--for Paul From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Finally, I'd like to>> reinstate my strong belief that the "g" measure should be _weighted_. >> You think 3 should counts for more than 5, etc?That's right. Harmonic progression by 3 is more comprehensible than progression by 5 or 5/3. I suggest weights of 1/log(3), 1/log(5), and 1/log(5), to conform with the geometry of the lattice and with what seems to me to be a properly octave-reduced Tenney metric.
Message: 2633 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:26:46 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: paulerlich OK, there's something in the way you're calculating things that forces you to use the determinant again at the end. So, for now, I would recommend dividing by abs(n).
Message: 2634 - Contents - Hide Contents Date: Sun, 23 Dec 2001 20:59:51 Subject: Re: Four funky ones 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: > >>> badness: 46.1 >>>> Is this a typo? Should this be 461? I might revise my badness cutoff >> now . . . >> I was cross-eyed again, but so where you--you missed my correction. :) > If you want to keep this, 500 is working for you. Do you think I'vemissed anything, by the way? Why, are you done with the list?
Message: 2635 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:28:26 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> All you need for the transformation in one direction is the matrix >> itself; and in the other direction, its inverse. You don't >> _additionally_ apply the determinant. > >> But when the inverse is described in integer terms, the > determinant is part of it!Yes, it's part of the usual calculation of the inverse. But it's certainly not part of the usual application of the matrix itself at the end of the process.
Message: 2636 - Contents - Hide Contents Date: Sun, 23 Dec 2001 21:03:39 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Wednesday, December 19, 2001 1:36 PM >> Subject: [tuning-math] Re: 55-tET >> >> >> ... So in your [monz's] view, the 55 tones would be much >> better understood as the Fokker periodicity block defined >> by the two unison vectors (-4 4 -1) and (-51 19 9). Since >> I'm sure you're interested, here are the coordinates of >> these 55 tones in the (3,5) lattice: >> >> 3 5 >> --- ---- >> >> -11 -4 >> -10 -4 >> -9 -4 >> -8 -4 >> -7 -4 <etc. -- snip> > > >> Paul, can you please explain the procedure you use to find > coordinates from a given set of unison-vectors, as you did > here? Thanks. > > > > -monzThis is explained in the _Gentle Introduction, part 3. Though there I'm dealing with a 3-d case, here it's only a 2-d case. Gene has a better way, though, where he can give a single formula to produce _all_ the points and _only_ those points in one fell swoop.
Message: 2637 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:29:26 Subject: Re: 55-tET & 1/6-comma meantone From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Your Excel lattices, though, are currently referring to JI ratios, >> including some rather complex ones -- we have to get rid of this >> feature first. Gene, any clever ideas? >> I thought you were the visual aids wizard; but if his lattices havenumbers all over them why not take them off and see if you can get that to work for starters? Ok, this point obviously didn't get across well :)
Message: 2638 - Contents - Hide Contents Date: Sun, 23 Dec 2001 21:09:03 Subject: Re: I don't understand (was: inverse of matrix --> for what?) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: genewardsmith <genewardsmith@j...> >> To: <tuning-math@y...> >> Sent: Tuesday, December 18, 2001 3:17 PM >> Subject: [tuning-math] Re: inverse of matrix --> for what? >> >> >> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>>> For 5-limit, we will only need two unison vectors to define >>> an ET, in this case 55-tET. One of these unison vectors should >>> of course 81:80, the unison vector that defines meantone. >>>> I got two of the commas on my list--one, of course, 81/80, and >> the other 6442450944/6103515625 = 2^31*3*5^(-14). >> Thanks for responding to this, but I'm afraid it's all too cryptic > for me, and I don't understand any of it. I'm sure that you've > discussed much of this in tuning-math posts which went over my > head... if you have links to relevant posts, I'd appreciate it.This is not hard. These are the two simplest ratios for defining 55- tET in the 5-limit. You can't find a simpler pair.> > Now for the specific questions: > >>> My badness score for the associated temperament is 6590, but some >> of the other commas do better--in particular, 2^47 3^(-15) 5^(-10) >> scores 1378; which hardly compares with the score of 108 for >> meantone and would not make my best list, where I have a cutoff >> of 500, but it isn't garbage. > > > What's "badness"?It's some function that penalizes boththe complexity and the tuning error of a linear temperament. Not too relevant for you here.>> The period matrix is >> >> [ 0 5] >> [ -2 11] >> [ 3 7] > >> ?? -- what does this mean?This means that the 2:1 is obtained by combining 0 of generator a and 5 of generator b. the 3:1 is obtained by combining -2 of generator a and 11 of generator b. the 5:1 is obtained by combining 3 of generator a and 7 of generator b.>> and the generators are a = 19.98/65 and b = 1/5; >> ?? -- Why is the generator not 2^(38/65), which is the closest > thing in 65-EDO to a 3:2? What do these numbers mean?This is a linear temperament with a generator of 2^(19.98/65), and an interval of repetition of 2^(1/5) (instead of the usual 2).>> it really is more of a 65-et system than a 55-et system, and >> scores as well as it does since it is in much better tune than >> the 55-et itself, with errors: >> >> 3: .317 >> 5: .228 >> 5/3: -.040 > >> By "much better in tune", you mean that 65-EDO is a better > approximation to the JI ratios than 55-EDO? Yes. > What is the unit > of measurement for these "errors"? Cents.
Message: 2639 - Contents - Hide Contents Date: Sun, 23 Dec 2001 22:58:35 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Yes, it's part of the usual calculation of the inverse. But it's > certainly not part of the usual application of the matrix itself at > the end of the process.Maybe he would be better off calculating the adjoint instead.
Message: 2640 - Contents - Hide Contents Date: Sun, 23 Dec 2001 13:11:27 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: monz ----- Original Message ----- From: paulerlich <paul@xxxxxxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Sunday, December 23, 2001 1:03 PM Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET)> This is explained in the _Gentle Introduction, part 3. Though there > I'm dealing with a 3-d case, here it's only a 2-d case.Hmmm... I studied all the pages in your _Gentle Introduction_ series *except* the 3-d (7-limit) one!> Gene has a better way, though, where he can give a single formula to > produce _all_ the points and _only_ those points in one fell swoop.I'm interested in seeing the differences between his formula and the method I jury-rigged. :) ... always searching for greater elegance ... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2641 - Contents - Hide Contents Date: Sun, 23 Dec 2001 23:10:19 Subject: Note to Monz From: paulerlich I suppose the cylinder could change shape _slightly_ as the meantone type changes. For me, the lengths of the connections are based on _consonance_, so one could alter these lengths to correspond with the altered level of consonance these simple ratios acheive in the different meanontes.
Message: 2642 - Contents - Hide Contents Date: Sun, 23 Dec 2001 15:31:23 Subject: a different example (was: coordinates from unison-vectors) From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 2:16 PM > Subject: [tuning-math] Re: coordinates from unison-vectors (was: 55-tET) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> I'm interested in seeing the differences between his formula and >> the method I jury-rigged. :) >> >> ... always searching for greater elegance ... >> Why don't you give an example and I'll work it out in a different way? > From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 1:41 PM > Subject: Re: [tuning-math] Re: coordinates from unison-vectors >> >>>> I've posted the Excel spreadsheet to the Files section. >> Yahoo groups: /tuning-math/files/monz/matrix m... * [with cont.]OK, here's a different example I've been playing around with. Also note that it brings up another problem with my calculations: my intention was to always have the periodicity-block centered on 1/1, but it doesn't always work that way, even way I try using reversed signs for the unison-vector exponent integers. So anyway, I put in the matrix: ( 6 -14) (-4 1) and could see that the resulting periodicity-block had a strong correlation (in the sense of my meantone-JI implied lattices) with the neighborhood of 2/7- to 3/11-comma meantone. The determinant is 50, so this agrees with my observation. So then I made lattices comparing the JI periodicity-block derived from that matrix with various fraction-of-a-comma meantones, and can see by eye that the 7/25-comma meantone axis goes right down the middle of this periodicity-block. Therefore, my conclusion is that 7/25-comma meantone, 50-EDO, and the (6 -14),(-4 1) periodicity-block are all intimately related, and essentially identical. Other related meantones are (in order of decreasing relatedness): 5/18-, 3/11-, and 2/7-comma. This is getting closer to describing what I was asking about a couple of weeks ago, for elegant mathematics which would describe these kinds of relationships. Looking forward to your reply. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2643 - Contents - Hide Contents Date: Sun, 23 Dec 2001 21:13:47 Subject: Re: 55-tET & 1/6-comma meantone From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> My idea was simply this: since 67-EDO approximates 1/6-comma >> meantone better than 55-EDO, there should be a unison-vector >> derived from 67-EDO which (along with 81:80) better defines >> a periodicity-block for my "acoustically implied ratios" lattice >> for 1/6-comma, than the one I got from 55-EDO, which was >> (2^-51 * 3^19 * 5^9). >> If I LLL reduce the above pair I get 2^34 * 3^5 * 5^-18 for the >second comma; TM reducing this then gives 2^38 * 3 * 5^-17. These >are certainly better in the sense of simpler, though as commas the >badness of the resulting temperaments is worse, since they are also >quite a bit larger.Gene, is this one of those mysterious magical facts? Or are you forgetting that we are defining ETs here and not linear temperaments?>> I'm having a hard time following Gene's comments because >> I don't understand why (2^62 * 3^-23 * 5^-11) "really doesn't >> work very well for anything *but* 65-et" when in fact it >> *is* also a 67-EDO comma. >> I thought when you asked me to run it through my program you meant > to analyze the linear temperament it defines;That's not what Monz meant. Plus Monz has an additional conceptual idiosyncracy -- he sees the JI scale implied _in toto_, not just via consonant intervals, and thus the smaller (in cents) unison vectors have a special meaning to him which to us, they do not.
Message: 2644 - Contents - Hide Contents Date: Sun, 23 Dec 2001 15:47:12 Subject: Re: a different example (was: coordinates from unison-vectors) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 3:31 PM > Subject: [tuning-math] a different example (was: coordinates from unison-vectors) > > > So anyway, I put in the matrix: > > ( 6 -14) > (-4 1) > > ... <snip> > > Therefore, my conclusion is that 7/25-comma meantone, 50-EDO, > and the (6 -14),(-4 1) periodicity-block are all intimately > related, and essentially identical. Other related meantones > are (in order of decreasing relatedness): 5/18-, 3/11-, and > 2/7-comma.Oops... my bad. That should be the matrix: ( 6 -14) ( 4 -1) and "the (6 -14),(4 -1) periodicity-block". -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2645 - Contents - Hide Contents Date: Sun, 23 Dec 2001 21:16:03 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Saturday, December 22, 2001 12:21 PM >> Subject: [tuning-math] coordinates from unison-vectors (was: 55- tET) >> >> >> Paul, can you please explain the procedure you use to find >> coordinates from a given set of unison-vectors, as you did >> here? Thanks. > > >> I've figured out how to use Excel to calculate the coordinates > within the unit square of the inverse of a 2-dimensional matrix, > and even how to have it centered on 0,0... I think.Good for you! So you read part 3 already?> But now how do I go about "transforming them back to the lattice > (using the original Fokker matrix)" as described at > <Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.)>, > to get the actual lattice coordinates?Just multiply each pair of coordinates _inside the box_ by the original, non-inverted, Fokker matrix.
Message: 2646 - Contents - Hide Contents Date: Sun, 23 Dec 2001 21:18:26 Subject: Re: coordinates from unison-vectors (was: 55-tET) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> I know that if the sign of the 3-exponent is changed, the sign > for the 5-exponent must be reversed accordingly. But I find > sometimes that using, for example, (4 -1) for the syntonic comma > doesn't always give me the PB I expected, whereas making it > (-4 1) does.As long as it's IN THE SAME FORM when you apply the inverse of the matrix as well as when you apply the matrix itself, it won't matter -- if you're centering around (0,0).
Message: 2647 - Contents - Hide Contents Date: Sun, 23 Dec 2001 23:47:47 Subject: Re: a different example (was: coordinates from unison-vectors) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Therefore, my conclusion is that 7/25-comma meantone, 50-EDO, > and the (6 -14),(-4 1) periodicity-block are all intimately > related, and essentially identical.It would seem that you're totally ignoring the issue of how the intervals are _tuned_, whether they're wolves or not, and focusing on a rather superficial kind of similarity that results from sticking to a 2-d JI lattice, and not using a cylinder to view 7/25-comma meantone, and a torus to view 50-tET. In the latter cases, your association of fixed JI pitches to each note in the tuning system is at work, and Dave Keenan and I have been fighting to explain for years now, it's the _intervals_ that matter in these lattices, not the _pitches_. Think of the simple example of "where do you put the 1/1". C? D? The key of the piece? The key of the section? The current chord? What if you have a chord like C-A-G-E-D? As to the meantones in this range, I don't think your lattices illuminate their similarity to any greater degree than is already obvious from, say, the cents values of the fifths. The numeric data on ratios you're pointing to in these studies has, I argue, no musical or acoustical relevance to the perception of these meantones.
Message: 2648 - Contents - Hide Contents Date: Sun, 23 Dec 2001 15:55:26 Subject: Re: a different example From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, December 23, 2001 3:47 PM > Subject: [tuning-math] Re: a different example (was: coordinates from unison-vectors) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> Therefore, my conclusion is that 7/25-comma meantone, 50-EDO, >> and the (6 -14),(-4 1) periodicity-block are all intimately >> related, and essentially identical. >> It would seem that you're totally ignoring the issue of how the > intervals are _tuned_, whether they're wolves or not, and focusing on > a rather superficial kind of similarity that results from sticking to > a 2-d JI lattice, and not using a cylinder to view 7/25-comma > meantone, and a torus to view 50-tET. In the latter cases, your > association of fixed JI pitches to each note in the tuning system is > at work, and Dave Keenan and I have been fighting to explain for > years now, it's the _intervals_ that matter in these lattices, not > the _pitches_. Think of the simple example of "where do you put the > 1/1". C? D? The key of the piece? The key of the section? The current > chord? What if you have a chord like C-A-G-E-D? > > As to the meantones in this range, I don't think your lattices > illuminate their similarity to any greater degree than is already > obvious from, say, the cents values of the fifths. The numeric data > on ratios you're pointing to in these studies has, I argue, no > musical or acoustical relevance to the perception of these meantones.Paul, I recognize that the *ambiguity* of the rational implications of temperaments has become an integral part of Eurocentric musical theory and practice. It's hard for me to visualize what happens on a cylinder or torus since I'm dealing with planar graphs. But if I'm sufficiently interested in pursuing this line of reasoning, then why not just go ahead and make the calculations, graphs, spreadsheets, etc., and then eventually if others who are interested in my approach do some experiments to test their validity, perhaps some may be found. :) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2649 - Contents - Hide Contents Date: Sun, 23 Dec 2001 21:20:27 Subject: Re: Temperament names From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>> 2109375/2097152 = 2^-21 3^3 5^7 Orwell >> I like it, but the comma is called the semicomma on Manuel's list. > >> 27/25 >> "Large limmic"? Margo seemed dubious about calling it any kind of a limma. > >> 16/15 > > Semitonic? No. > >> 135/128 >> Major Limmic? Major Chromic? > > Map: > > [ 0 1] > [-1 2] > [ 3 1] > > Generators: a = 10.0215 / 23; b = 1 > > badness: 46.1 > rms: 18.1 > g: 2.94 > errors: [-24.8, -17.7, 7.1] > > 25/24 > > I called it "Neutral Thirds". "Minor chromic" is another possibility. > > >> 648/625 >> Too many things called a diesis, I fear--Major Diesic? > >> 250/243 > > Maximal diesic? > > >> 128/125 >> Minor diesic? I think Paul named this; his names sounded good andmaybe I should dig them out and keep them handy!> >> 3125/3072 >> Magic--or Small Diesic for anyone who really digs this nomenclature.I'm going to die sick. I'm going to Diesic.>> 393216/390625 = 2^17 3 5^-8 >> This is Wuerschmidt's comma, so obviously the temperament is theWuerschmidt. Who the heck is Wuerschmidt? Do you read German? If so, you should seek out his research. He found some very interesting stuff, essentially looking into periodicity blocks before Fokker.
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