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Message: 2601 - Contents - Hide Contents Date: Thu, 20 Dec 2001 00:00:12 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> Gene seems to be saying that for any given badness cutoff, >> Only if it's low enough, or if you start far enough away from zero.This is all sounding a little contrived. How low is low enough? How far from 1-tET is far enough? If we have to go out beyond what is musically relevant, then what's the point?>> the number >> less than that should be about the same in every decade (1 to 9- > tET,>> 10 to 99-tET, 100 to 999-tET, etc). Could you check that with > Matlab>> Paul, for both steps^(4/3)*cents and steps*cents? For various > cutoffs? >> Sure -- just tell me how far out I should go, what cutoffs to use -- > perhaps Gene would like to weigh in on these decisions to help guide > us toward something that will make the distinction more clear . . .Go out to (10^7 - 1)-tET and use whatever badness cutoff gives 10 ETs in the second decade (i.e. from 10-tET to 99-tET). Then tell us how many we get in the other 6 decades, using steps^(4/3)*cents in one case and steps*cents in the other.
Message: 2602 - Contents - Hide Contents 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 . . .
Message: 2603 - Contents - Hide Contents Date: Thu, 20 Dec 2001 00:01:30 Subject: Re: Two conditions on temperaments From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> (1) Weak condition--no element of the tonality diamond is allowed to be a unison (exluding 1/1) > > (2) Strong condition--all elements of the tonality diamond are distinct (including 1/1)
Message: 2604 - Contents - Hide Contents 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
Message: 2605 - Contents - Hide Contents Date: Thu, 20 Dec 2001 00:07:44 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> This is all sounding a little contrived. How low is low enough? How > far from 1-tET is far enough? If we have to go out beyond what is > musically relevant, then what's the point?Why not just exlude the inconsistent ets? The point is to get rid of the crap parade at the very beginning, and this looks like a use of consistency I could go for.> Go out to (10^7 - 1)-tET and use whatever badness cutoff gives 10 ETs > in the second decade (i.e. from 10-tET to 99-tET). Then tell us how > many we get in the other 6 decades, using steps^(4/3)*cents in one > case and steps*cents in the other.Why not just fit a line to log n vs rank, in the sense of the m-th item on a tops list? Paul's top 75 list should work, if you sort it from smallest to largest.
Message: 2606 - Contents - Hide Contents Date: Thu, 20 Dec 2001 05:07:50 Subject: My top 5--for Paul From: genewardsmith #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, because of the 730. Other ets consistent with this are 84, 323, 407, 1053 and 1460. badness: 34 rms: .000763 g: 35.5 errors: [-.000234, -.001029, -.000796] #2 32805/32768 Schismic badness=55 #3 25/24 Neutral thirds badness=82 #4 15625/15552 Kleismic badness=97 #5 81/80 Meantone badness=108 It looks pretty flat so far as this method can show, I think.
Message: 2607 - Contents - Hide Contents Date: Thu, 20 Dec 2001 01:16:11 Subject: Meantone & co From: genewardsmith Here are four more: 81/80 Map: [ 0 1] [-1 2] [-4 4] Generators: a = 20.9931/50; b = 1 badness: 108 rms: 4.22 g: 2.944 errors: [-5.79, -1.65, 4.14] Nothing left to say about this one. :) 2048/2025 Map: [ 0 2] [-1 4] [ 2 3] Generators: 14.0123/34 (~4/3); b = 1/2 badness: 211 rms: 2.613 g: 4.32 errors: [3.49, 2.79, -.70] A good way to take advantage of the 34-ets excellent 5-limit harmonies is two gothish 17-et chains of fifths a sqrt(2) apart. 78732/78125 = 2^2 3^9 5^-7 Map: [ 0 1] [ 7 -1] [ 9 -1] Generators: 23.9947/65 (~9/7); b = 1 badness: 346 rms: 1.157 g: 6.68 errors: [-1.1, 0.5, 1.6] 393216/390625 = 2^17 3 5^-8 Map: [ 0 1] [ 8 -1] [ 1 2] Generators: a = 31.9951/99 (~5/4); b = 1 Works with 31,34,65,99,164 badness: 251 rms: 1.072 g: 6.16 error: [.602, 1.506, .904]
Message: 2608 - Contents - Hide Contents Date: Thu, 20 Dec 2001 10:30:48 Subject: Re: Meantone & co From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Would you call this one a "unique facet of 65-tET"? Is this the kind > of thing that Graham's searching pairs of ETs is likely to miss?No, he'd be bound to hit it if he did things right.
Message: 2609 - Contents - Hide Contents Date: Thu, 20 Dec 2001 01:21:03 Subject: Re: Meantone & co From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 2048/2025 > A good way to take advantage of the 34-ets excellent 5-limit harmonies > is two gothish 17-et chains of fifths a sqrt(2) apart.I should have added that this is the diaschismic temperament.
Message: 2610 - Contents - Hide Contents Date: Thu, 20 Dec 2001 11:10:32 Subject: Re: Meantone & co From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> No, he'd be bound to hit it if he did things right.Graham is more likely to include things I wouldn't than vice-versa--his methods can include more than one version of the same system, so that you have for instance meantone, and another meantone with a seemingly useless doubling of the generator steps. There are at least three version of kleismic on his list--one the usual, one with doubled generator steps, and one with a half-ocatave period.
Message: 2611 - Contents - Hide Contents Date: Thu, 20 Dec 2001 02:10:32 Subject: Re: 55-tET From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> It makes sense, but I don't think it defines a unique interval. >> Meaning you can always find smaller and smaller examples? Even if you > disallow "potential torsion"? What if you fix all the commas except > one, and just have to find the smallest candidate for the remaining > comma. Isn't that choice unique?That should do it, though it seems a little arbitary. If you don't fix all but one, I can prove easily enough you get arbitarily small commas, so this may be your best shot. For the 55-et, or anything else where there is a clear set of all-but-one keepers in the comma department, it would make sense.
Message: 2612 - Contents - Hide Contents 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? Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2613 - Contents - Hide Contents Date: Thu, 20 Dec 2001 02:16:31 Subject: supernatural superparticulars? From: jpehrson2 Yahoo groups: /tuning/message/31663 * [with cont.]>> 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
Message: 2614 - Contents - Hide Contents 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. Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) 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 - The best web-based email! * [with cont.] (Wayb.)
Message: 2615 - Contents - Hide Contents 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: 2.7Thanks a lot, Gene!
Message: 2616 - Contents - Hide Contents 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?
Message: 2617 - Contents - Hide Contents 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: 2.7What 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!
Message: 2618 - Contents - Hide Contents 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 isthe 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_.
Message: 2619 - Contents - Hide Contents 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: 46.1Is this a typo? Should this be 461? I might revise my badness cutoff now . . .
Message: 2620 - Contents - Hide Contents 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 Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 2621 - Contents - Hide Contents 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 - The best web-based email! * [with cont.] (Wayb.)
Message: 2622 - Contents - Hide Contents 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: -.040By "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 - The best web-based email! * [with cont.] (Wayb.)
Message: 2623 - Contents - Hide Contents Date: Sat, 22 Dec 2001 21:39:22 Subject: Re: 55-tET & 1/6-comma meantone From: genewardsmith --- 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 betterin the sense of simpler, though as commas the badness of the resulting temperaments is worse, since they are also quite a bit larger.> 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; from that point of view, the 67-et istoo far out of tune to use it to much advantage, whereas the 65-et represents it well. Of course this temperament has nothing really to do with meantone.
Message: 2624 - Contents - Hide Contents Date: Sat, 22 Dec 2001 21:51:06 Subject: Re: I don't understand (was: inverse of matrix --> for what?) From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> What's "badness"?In this case I was using the rms error in cents of approximating 3,5, and 5/3 times the cube of the rms of the generator steps to reach these, the higher the number the worse, roughly speaking, and hence "badness".>> The period matrix is >> >> [ 0 5] >> [ -2 11] >> [ 3 7]> ?? -- what does this mean?It has to do with a linear temperament, and gives how 2, 3 and 5 are represented by it.>>> 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?Because it isn't meantone or anything remotely like it. For one thing, it divides the octave into five equal parts as part of the system, which is whyit can only work for ets such as 55 or 65 which are divisible by five.> 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!Cents; and yes, the 65-et is quite a lot better in tune in the 5-limit thanthe 55-et. The errors in question were actually those of an optimized tuning of the temperament itself, which could equally well temper 55 as 65 notes.
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