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Message: 5600 Date: Thu, 20 Dec 2001 04:50:28 Subject: [tuning] Re: great explanation [periodicity block] From: paulerlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: > Right--the 250/243 "major chroma". I think I can do this without the > lattice, in fact that's the whole point! Well seeing it on a lattice would convince us of this . . .
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Message: 5602 Date: Thu, 20 Dec 2001 05:06:46 Subject: Re: Badness with gentle rolloff From: clumma >Tell me how to calculate real-number Hahn consistency. According to Paul, it's 1/(max_error*1200*steps), but I don't see this coming from the algorithm I've always used, given by Paul Hahn: | consistency_level(ET_number, limit): | max <- 0 | min <- 0 | FOR loop <- 3 TO limit BY 2 | exact_steps <- ET_number * log2(loop) | error <- exact_steps - round(exact_steps) | IF error > max THEN | max <- err | ELSEIF error < min THEN | min <- err | ENDIF | ENDFOR | RETURN integer_part(0.5 / (max - min)) >Do you mean you want to see both Hahn consistency and steps*cents >badness /.../ plotted against steps? Yes. >(or do you want 1/(steps*cents) goodness? I don't know why I'd care. >>But, right, since consistency is just steps*max_error... I guess >>I was just wondering how this looked, over the ETs, compared to >>steps*max_rms_error. > >You mean steps*rms_error? Yes. >>Is there still periodicity at good ets? > >I'll check it out, but I bet there is. 5-limit or 7-limit? 7, of course. -Carl
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Message: 5610 Date: Thu, 20 Dec 2001 05:29:46 Subject: Re: 55-tET From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 19, 2001 1:36 PM > Subject: [tuning-math] Re: 55-tET > > > Well, I still hold strongly to the views that I expressed, but that > doesn't mean that there isn't some mathematics that could be useful > to you for fleshing out _your_ views, nor that I would be averse to > helping you with such mathematics. Thanks! Much appreciated. > ... 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: > > <table snipped> Thanks, Paul! I haven't checked yet, but my guess is that the ratios you provided here should be the same as the lattice that could be extended from the one on my webpage (except for the commatic duplicated tones on my lattice), yes? lattices comparing various Meantone Cycles, (c)2001 by Joseph L. Monzo * love / peace / harmony ... -monz http://www.monz.org * "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 5611 Date: Thu, 20 Dec 2001 02:16:31 Subject: supernatural superparticulars? From: jpehrson2 Yahoo groups: /tuning/message/31663 * > >Curiously, J Gill > > No -- but what seems to be the case very often, is that when one > comes up with such a scale in the form of a periodicity block, one > has quite a few arbitrary choices to make as to which version of a > particular scale degree one wants (the different versions differing > by a unison vector), and then _one such set_ of arbitrary choices > does lead to a scale with superparticular step sizes. Hi Paul... Well, that's pretty *mysterious* isn't it? Why does that happen that the superparticular step sizes result? Is it just the way the system is set up. Spooky stuff! (If we can't believe in "magic primes" that surely is something a little weird... yes?) Joseph
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Message: 5612 Date: Thu, 20 Dec 2001 05:39:31 Subject: Re: 55-tET From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 19, 2001 2:02 PM > Subject: [tuning-math] Re: 55-tET > > > There's an even smaller unison vector you can use, which comes from > subtracting these two from one another: > > (47 15 10) = 7.54 cents. > > Now, combining this with the syntonic comma, we get the following > Fokker periodicity block, which should be even closer to 55-tET: > > <table snipped> > > Meanwhile, combining the two smallest so far, (-51 19 9) and (47 15 > 10), leads to this, closer still to 55-tET, but more unlikely from a > JI standpoint: > > <table snipped> OK, now I have checked, and yes indeed, all the tables you've provided are related to the lattice on my webpage. lattices comparing various Meantone Cycles, (c)2001 by Joseph L. Monzo * In fact, this is quite interesting... as your tables describe successively closer approximations to 55-tET, they also successively eliminate the "commatic-duplicate" pitches on my lattice! In other words, the implied ratios of the lattice *as a group* huddle closer and closer to the linear axis which represents the actual meantone. Hmmm.... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 5613 Date: Fri, 21 Dec 2001 20:43:04 Subject: Re: Four funky ones From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > > Many theorists associate this with Pelog. > > Hmmm...anyone tried 23-et gamelan music? Lots of people have. > Would you mind calculating > > the optimal version of this where the octave is _not_ constrained to > > be exactly 1200 cents? > > This is what I got fitting to {2,3,5,3/2,5/2,5/3}: > > a = .43763, b = 1.0113 > > errors: > > 2: 13.6 > 3: 0 (exactly) > 3/2: -13.6 > 5/4: -24.4 > 6/5: 10.8 > 4/3: 27.1 > 5/3: 2.7 Thanks a lot, Gene!
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Message: 5614 Date: Fri, 21 Dec 2001 20:45:00 Subject: Re: Meantone & co From: paulerlich --- In tuning-math@y..., graham@m... wrote: > The 24 note periodicity block produced by a comma and diesis is the > traditional 22 shrutis plus two extra notes. That's good enough for me. > The confusion only comes in when you try and generate a temperament. Naah . . . I would say it's already confusing beforehand. Why would a comma squared be considered an equivalence if a comma itself isn't?
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Message: 5615 Date: Fri, 21 Dec 2001 22:23:55 Subject: Re: Four funky ones From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > > Many theorists associate this with Pelog. > > Hmmm...anyone tried 23-et gamelan music? > > Would you mind calculating > > the optimal version of this where the octave is _not_ constrained to > > be exactly 1200 cents? > > This is what I got fitting to {2,3,5,3/2,5/2,5/3}: > > a = .43763, b = 1.0113 > > errors: > > 2: 13.6 > 3: 0 (exactly) > 3/2: -13.6 > 5/4: -24.4 > 6/5: 10.8 > 4/3: 27.1 > 5/3: 2.7 What if you include 4, 4/3, and 5/4 as well? At least including 4 and 4/3 would seem logical. I'd even include 4/2 also!
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Message: 5616 Date: Fri, 21 Dec 2001 22:54:30 Subject: Re: My top 5--for Paul From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > #1 > > 2^-90 3^-15 5^49 > > This is not only the the one with lowest badness on the list, it is the smallest comma, which suggests we are not tapering off, and is evidence for flatness. > > Map: > > [ 0 1] > [49 -6] > [15 0] > > Generators: a = 275.99975/1783 = 113.00046/730; b = 1 > > 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? I'd favor reeling in the cutoff . . . schismic is pretty complex already . . . as long as Ennealimmal makes it into the 7- limit list, we're searching out far enough, as far as I'm concerned. Also, I'm thinking a badness cutoff around 300 might be good, but I'll hold off until I see more results. Finally, I'd like to reinstate my strong belief that the "g" measure should be _weighted_.
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Message: 5617 Date: Fri, 21 Dec 2001 22:58:15 Subject: Re: Four funky ones From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > 135/128 > > Map: > > [ 0 1] > [-1 2] > [ 3 1] > > Generators: a = 10.0215 / 23; b = 1 > > badness: 46.1 Is this a typo? Should this be 461? I might revise my badness cutoff now . . .
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Message: 5618 Date: Sat, 22 Dec 2001 11:42:36 Subject: 55-tET & 1/6-comma meantone From: monz > From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, December 18, 2001 8:25 PM > Subject: [tuning-math] Re: 55-tET > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > > The next "closure" size for 1/6-comma meantone is a 67-note set. > > The 8ve-invariant 67th generator is ~9.168509182 [cents] lower > > (narrower) than the starting pitch, and its tuning is > > 3^(67/3) * 5^(67/6). > > > > The ratio it implies acoustically most closely is 3^23 * 5^11. > > The unison-vector would therefore be described, in my matrix > > notation, as (-61 23 11). > > > > Gene, does this agree with your program's output? > > > I'm not sure what your question means; however I can make > the following comments: > > (1) Presumably you meant the comma 2^62 3^(-23) 5^(-11) Yes... I simply got the exponents from my lattice, and inadvertently referenced them in the wrong direction. > (2) This is a 67-et comma; however, and much more significantly, > it is a 65-et comma. It really doesn't work very well for > anything *but* 65-et, in fact. > > (3) For the associated linear temperament, we have a map > > [ 0 1] > [-11 7] > [ 23 9] > > The generator is 31.997/65, so this can be more or less equated > with 32/65. OK, here's what I really meant: The 1/6-comma meantone generator = (3/2) / ( (81/80)^(1/6) ). This is approximately equal to the following ET generators, listed in order of increasing proximity of the ET generators to the meantone one: 2^(~32.00865338 / 55) 2^(~38.99235958 / 67) 2^(~71.00101296 / 122) 2^(~322.9964114 / 555) 2^(~393.9974244 / 677) 2^(~464.9984373 / 799) 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). I would consequently suppose that the 2^(71/122) generator results in a periodicity-block which is even closer to my 1/6-comma implied ratios lattice, and that 2^(323/555) is closer still, etc. Yes? 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. ...? Totally perplexed. love / peace / harmony ... -monz http://www.monz.org * "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 5619 Date: Sat, 22 Dec 2001 12:21:03 Subject: coordinates from unison-vectors (was: 55-tET) From: monz > From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > 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. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 5620 Date: Sat, 22 Dec 2001 13:18:31 Subject: I don't understand (was: inverse of matrix --> for what?) From: monz > From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > 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. 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"? > The period matrix is > > [ 0 5] > [ -2 11] > [ 3 7] ?? -- what does this mean? > 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? > 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? What is the unit of measurement for these "errors"? Help! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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Message: 5624 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-matrix.gif * (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 <The Indian sruti system as a periodicity-block *>, to get the actual lattice coordinates? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
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