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Message: 2550 - Contents - Hide Contents Date: Wed, 19 Dec 2001 09:03:09 Subject: The best 5-limit temperaments From: genewardsmith I did a search for commas which lead to temperaments such that if g is the rms of generator steps, and e is rms error in cents, then g < 50, e < 50, and e g^3 < 500; the last being the 5-limit logarithmically flat badness measure. I obtained the following 32 commas, sorted by size: 5^49/2^90/3^15 3^62/2^17/5^35 2^144/3^22/5^47 3^47*5^14/2^107 2^54*5^2/3^37 5^51/2^36/3^52 2^37*3^25/5^33 3^10*5^16/2^53 2^91/3^12/5^31 2*5^18/3^27 3^35/2^16/5^17 2^38/3^2/5^15 5^34/2^52/3^17 3^8*5/2^15 5^19/2^14/3^19 2^23*3^6/5^14 2^8*3^14/5^13 2^9*5^5/3^13 5^6/2^6/3^5 3^3*5^7/2^21 2^17*3/5^8 2^2*3^9/5^7 2^11/3^4/5^2 3^4/2^4/5 5^5/2^10/3 2^7/5^3 2*5^3/3^5 2^3*3^4/5^4 5^2/2^3/3 3^3*5/2^7 2^4/3/5 3^3/5^2 Though it was slightly outside my generator range (I searched quite a bit farther than the above limits), I will also mention 2^161 3^(-84) 5^(-12); as a temperament this consists of 12-ets stacked by fourths and fifths, a sort of Pythagorean/12-et system. It is, of course, absurdly well in tune, and can also be thought of as 12-ets separated by schismas.
Message: 2551 - Contents - Hide Contents Date: Wed, 19 Dec 2001 20:14:28 Subject: Re: 55-tET From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> In fact, I've been wondering if there's an analogue to Minkowski > reduction which, instead of finding the simplest commatic unison > vectors, finds the _smallest_ ones that work, without torsion > (or "potential torsion"). This could be valuable to JI-oriented > theorists like Monz and Kraig. For example, it seems that for > Blackjack in the 7-limit, the answer might be 2401:2400 and > 16875:16807 . . . Gene, does this make any sense?It makes sense, but I don't think it defines a unique interval.
Message: 2552 - Contents - Hide Contents Date: Wed, 19 Dec 2001 10:18:07 Subject: Four funky ones From: genewardsmith Here are the four largest commas on my list of 32, and the temperaments they lead to. 27/25 Map: [ 0 1] [ -2 2] [ -3 3] Generators: a = 2.0104 / 9; b = 1 badness: 359 rms: 35.6 cents g: 2.16 errors: [-38.1, 9.5, 47.6] 16/15 Map: [ 0 1] [ -1 2] [ 1 2] Generators: a = 2.9479 / 8; b = 1 badness: 129 rms: 45.6 cents g: 1.414 errors: [55.9, 55.9, 0] 135/128 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 Map: [ 0 1] [ 2 1] [ 1 2] Generators: a = 4.9722 / 17; b = 1 badness: 81.6 rms: 28.9 g: 1.414 errors: [0, -35.3, -35.3] Neutral thirds
Message: 2553 - Contents - Hide Contents Date: Wed, 19 Dec 2001 20:17:12 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> 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 . . .You could fit a line to log n vs ets which pass a badness test.
Message: 2554 - Contents - Hide Contents Date: Wed, 19 Dec 2001 10:29:34 Subject: Re: Four funky ones From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 135/128 > badness: 46.1Should be 461.
Message: 2555 - Contents - Hide Contents Date: Wed, 19 Dec 2001 21:36:36 Subject: Re: 55-tET From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> But you can indeed define 55-tET with the two unison vectors >> (-4 4 -1) and (-51 19 9), and in fact it's quite logical to do so, >> since these intervals are only 21.51 cents and 13.97 cents in JI. > >> Paul, I've been surprised all along that you disagree so strongly > with my interpretations of meantone systems. Is this perhaps > bringing our individual conceptions a bit closer together?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.>> In fact, I've been wondering if there's an analogue to Minkowski >> reduction which, instead of finding the simplest commatic unison >> vectors, finds the _smallest_ ones that work, without torsion >> (or "potential torsion"). This could be valuable to JI-oriented >> theorists like Monz and Kraig. For example, it seems that for >> Blackjack in the 7-limit, the answer might be 2401:2400 and >> 16875:16807 . . . Gene, does this make any sense? > >> Yes, this does sound interesting to me... but I really don't know > what "Minkowski reduction" is...That's means finding the _simplest_ (smallest numbers in the ratio) commatic unison vectors defining a system. For 55-tET, the simplest pair is . . . well, we gave it to you yesterday. One member of that pair was the syntonic comma, but the other member of that pair was 93 cents or something, which would make little sense in a JI view of the 55-tone system. Why would someone temper out a 93-cent interval if much smaller intervals abound at closer distances in the lattice? Well, my answer to that is that you're not using JI at all, you've already tempered out the 81:80, so that the "93-cent interval" in question is contracted to something far less than 93 cents. However, in your view, you're continuing to reference a JI pitch set with the 1/1 in common with the tempered pitch set you're actually using (very far-fetched, I think, but that is your view). So in your 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 -6 -4 -9 -3 -8 -3 -7 -3 -6 -3 -5 -3 -4 -3 -7 -2 -6 -2 -5 -2 -4 -2 -3 -2 -2 -2 -5 -1 -4 -1 -3 -1 -2 -1 -1 -1 0 -1 -3 0 -2 0 -1 0 0 0 1 0 2 0 3 0 0 1 1 1 2 1 3 1 4 1 5 1 2 2 3 2 4 2 5 2 6 2 7 2 4 3 5 3 6 3 7 3 8 3 9 3 6 4 7 4 8 4 9 4 10 4 11 4 And here are the pitch-heights, in cents, of these same 55 just tones: 0 19.553 39.105 72.626 92.179 111.73 131.28 164.8 184.36 203.91 223.46 243.02 276.54 294.13 296.09 313.69 333.24 366.76 386.31 405.87 425.42 458.94 478.49 498.04 517.6 537.15 570.67 590.22 609.78 629.33 662.85 682.4 701.96 721.51 741.06 774.58 794.13 813.69 833.24 866.76 886.31 903.91 905.87 923.46 956.98 976.54 996.09 1015.6 1035.2 1068.7 1088.3 1107.8 1127.4 1160.9 1180.4
Message: 2557 - Contents - Hide Contents Date: Wed, 19 Dec 2001 21:38:06 Subject: Re: 55-tET From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> In fact, I've been wondering if there's an analogue to Minkowski >> reduction which, instead of finding the simplest commatic unison >> vectors, finds the _smallest_ ones that work, without torsion >> (or "potential torsion"). This could be valuable to JI-oriented >> theorists like Monz and Kraig. For example, it seems that for >> Blackjack in the 7-limit, the answer might be 2401:2400 and >> 16875:16807 . . . Gene, does this make any sense? >> 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?
Message: 2558 - Contents - Hide Contents Date: Wed, 19 Dec 2001 21:40:05 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> 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 moreclear . . .> > You could fit a line to log n vs ets which pass a badness test.What badness test would you choose? As I showed, the "waves" seem to take over after the badness test is relaxed to allow only a reasonable number of ETs to pass.
Message: 2559 - Contents - Hide Contents Date: Wed, 19 Dec 2001 22:02:12 Subject: Re: 55-tET From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> But you can indeed define 55-tET with the two unison vectors >>> (-4 4 -1) and (-51 19 9), and in fact it's quite logical to do so, >>> since these intervals are only 21.51 cents and 13.97 cents in JI.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: 3 5 --- --- -7 -5 -6 -5 -5 -5 -8 -4 -7 -4 -6 -4 -5 -4 -4 -4 -7 -3 -6 -3 -5 -3 -4 -3 -3 -3 -2 -3 -5 -2 -4 -2 -3 -2 -2 -2 -1 -2 -4 -1 -3 -1 -2 -1 -1 -1 0 -1 1 -1 -2 0 -1 0 0 0 1 0 2 0 -1 1 0 1 1 1 2 1 3 1 4 1 1 2 2 2 3 2 4 2 5 2 2 3 3 3 4 3 5 3 6 3 7 3 4 4 5 4 6 4 7 4 8 4 5 5 6 5 7 5 cents: 0 19.553 39.105 72.626 92.179 111.73 131.28 143.3 162.85 203.91 223.46 243.02 255.03 274.58 315.64 335.19 354.75 366.76 386.31 405.87 446.93 458.94 478.49 498.04 517.6 558.66 570.67 590.22 609.78 629.33 641.34 682.4 701.96 721.51 741.06 753.07 794.13 813.69 833.24 845.25 864.81 884.36 925.42 944.97 956.98 976.54 996.09 1037.1 1056.7 1068.7 1088.3 1107.8 1127.4 1160.9 1180.4 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: 3 5 --- --- -16 -9 -14 -8 -13 -7 -12 -7 -11 -6 -10 -6 -10 -5 -9 -5 -8 -5 -8 -4 -7 -4 -6 -4 -7 -3 -6 -3 -5 -3 -4 -3 -5 -2 -4 -2 -3 -2 -2 -2 -4 -1 -3 -1 -2 -1 -1 -1 0 -1 -2 0 -1 0 0 0 1 0 2 0 0 1 1 1 2 1 3 1 4 1 2 2 3 2 4 2 5 2 4 3 5 3 6 3 7 3 6 4 7 4 8 4 8 5 9 5 10 5 10 6 11 6 12 7 13 7 14 8 16 9 cents: 0 19.553 39.105 72.626 92.179 111.73 131.28 150.84 170.39 203.91 223.46 243.02 262.57 282.12 308.1 327.66 347.21 366.76 386.31 405.87 439.39 458.94 478.49 498.04 517.6 551.12 570.67 590.22 609.78 629.33 648.88 682.4 701.96 721.51 741.06 760.61 794.13 813.69 833.24 852.79 872.34 891.9 917.88 937.43 956.98 976.54 996.09 1029.6 1049.2 1068.7 1088.3 1107.8 1127.4 1160.9 1180.4
Message: 2560 - Contents - Hide Contents Date: Wed, 19 Dec 2001 22:09:08 Subject: The four before meantone From: genewardsmith Here are the second four 5-limit temperaments, which bring us up to meantone: 648/625 Map: [ 0 4] [ 1 5] [ 1 8] Generators: a = 21.0205/64; b = 1/4 badness: 385 rms: 11.06 g: 3.266 errors: [-7.82, 7.82, 15.64] 64-et, anyone? It could also be used to temper the 12-et. 250/243 Map: [ 0 1] [-3 2] [-5 3] Generators: a = 2.9883/22; b = 1 badness: 360 rms: 7.98 g: 3.559 errors: [9.06, -1.29, -10.35] One way to cure those 22-et major thirds of what ails them. 128/125 Map: [ 0 3] [-1 6] [ 0 7] Generators: a = 11.052/27 (~4/3); b = 1/3 badness: 142 rms: 9.68 g: 2.449 errors: [6.84, 13.69, 6.84] When extended to the 7-limit, this becomes the [ 0 3] [-1 6] [ 0 7] [ 2 6] system I've already mentioned in several contexts, such as the 15+12 system of the 27-et. Both as a 5-limit and a 7-limit system, it is good enough to deserve a name of its own. 3125/3072 Map: [ 0 1] [ 5 0] [ 1 2] Generators: a = 12.9822/41 (=6.016/19); b = 1 badness: 239 rms: 4.57 g: 3.74 errors: [-2.115, -6.346, -4.231] Graham has named this one: Magic.
Message: 2561 - Contents - Hide Contents Date: Wed, 19 Dec 2001 22:19:20 Subject: Re: The four before meantone From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Here are the second four 5-limit temperaments, which bring us up to > meantone: > > 648/625 > > Map: > > [ 0 4] > [ 1 5] > [ 1 8] > > Generators: a = 21.0205/64; b = 1/4 > > badness: 385 > rms: 11.06 > g: 3.266 > errors: [-7.82, 7.82, 15.64] > > 64-et, anyone? octo-diminished. > 250/243 > > Map: > > [ 0 1] > [-3 2] > [-5 3] > > Generators: a = 2.9883/22; b = 1 > > badness: 360 > rms: 7.98 > g: 3.559 > errors: [9.06, -1.29, -10.35] > > One way to cure those 22-et major thirds of what ails them.Huh? These major thirds are much worse than those of 22-tET. Oh wait - - the major third is the second entry under "errors"? So what on earth is the third entry? Oh, it's the minor third!> > > 128/125 > > Map: > > [ 0 3] > [-1 6] > [ 0 7] > > Generators: a = 11.052/27 (~4/3); b = 1/3 > > badness: 142 > rms: 9.68 > g: 2.449 > errors: [6.84, 13.69, 6.84] > > When extended to the 7-limit, this becomes the > > [ 0 3] > [-1 6] > [ 0 7] > [ 2 6] > > system I've already mentioned in several contexts, such as the > 15+12 system of the 27-et. Both as a 5-limit and a 7-limit system,it is good enough to deserve a name of its own. It's the augmented system, since the 6-tone MOS is commonly known as the augmented scale.
Message: 2562 - Contents - Hide Contents Date: Wed, 19 Dec 2001 22:24:46 Subject: Re: The best 5-limit temperaments From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> I did a search for commas which lead to temperaments such that if gis the rms of generator steps, and e is rms error in cents, then> g < 50, e < 50, and e g^3 < 500;I though we decided we were _not_ going to have a constraint on e, and simply use an upper bound on badness combined with both lower and upper bounds on g. What gives?> Though it was slightly outside my generator range (I searched quitea bit farther than the above limits), I will also mention> 2^161 3^(-84) 5^(-12);This just got a lot of discussion over at the tuning list.>as a temperament this consists of 12-ets stacked by fourths and >fifths, a sort of Pythagorean/12-et system. It is, of course, >absurdly well in tune, and can also be thought of as 12-ets >separated by schismas. aka 612-tET!
Message: 2563 - Contents - Hide Contents Date: Wed, 19 Dec 2001 00:03:19 Subject: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: dkeenanuqnetau Hey guys, Is there any chance someone could take the time to convince me that steps^(4/3)*cents is flat in some sense, for 7-limit ETs. It still looks to me like steps*cents is flatter. The purported reference to Diophantine approximation seems to have evaporated from Dave Benson's files, and my request for a steps*cents version of Paul's goodness chart (for comparison with steps^(4/3)*cents), seems to have been ignored. And surely someone can define "flatness" in such a way that I can make it a formula in my spreadsheet. I remind the reader that the value of this exponent becomes academic if one attempts to actually model human perception/cognition of 7-limit badness by adjusting values of k and r in badness = steps * exp((cents/k)^r), e.g. k = 3.7 cents, r = 0.5 Regards, -- Dave Keenan
Message: 2564 - Contents - Hide Contents Date: Wed, 19 Dec 2001 22:29:27 Subject: Re: Badness with gentle rolloff From: clumma>> >f you're talking about the maximum real-number consistency >> _level_ obeyed by an ET, this will be equal to >> 1/(max_error*1200*steps) wherever it's greater than 1.5. So for >> the good ETs, you'll just be plotting maximum error vs. rms error. >> Oops -- that's true if you plot maximum real-number consistency > level against 1/(steps*rms), which is what I'm guessing you really > had in mind anyway. . . ?Nnn... I had in mind plotting consistency against steps for all ETs. 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. Is there still periodicity at good ets? -C.
Message: 2565 - Contents - Hide Contents Date: Wed, 19 Dec 2001 00:04:27 Subject: 55-tET (was: Re: inverse of matrix --> for what?) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> So, Monz, according to Gene the simplest pair of unison vectors for > defining 55-tET is > > 81:80 > and > 6442450944:6103515625 > > The latter is the result of going 14 major thirds down and one > perfect fifth up. In JI, it's about 93.563 cents; and in 55-tET, it's > of course 0.000 ;)It's 14 major thirds down, a perfect fifth and four octaves up; in other words it tells us (5/4)^14 ~ 24 is an approximation of the 55-et, along with (3/2)^4 ~ 5.
Message: 2566 - Contents - Hide Contents Date: Wed, 19 Dec 2001 22:35:42 Subject: Re: The best 5-limit temperaments From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> I though we decided we were _not_ going to have a constraint on e, > and simply use an upper bound on badness combined with both lower and > upper bounds on g. What gives?I decided that pushing it farther was getting into garbage territory, but there aren't many more that we would add in this way, and they could be included. Do we really want to temper out the fifth?
Message: 2567 - Contents - Hide Contents Date: Wed, 19 Dec 2001 00:09:40 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Hey guys, > > Is there any chance someone could take the time to convince me that > steps^(4/3)*cents is flat in some sense, for 7-limit ETs. It still > looks to me like steps*cents is flatter.You or someone might try graphing log steps vs badness for both of these, or log steps vs number less than a cut-off value.> And surely someone can define "flatness" in such a way that I can make > it a formula in my spreadsheet.I'm not sure what you mean by this.
Message: 2568 - Contents - Hide Contents Date: Wed, 19 Dec 2001 22:42:04 Subject: Re: Badness with gentle rolloff From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:> 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?> Is there still periodicity at good ets?I'll check it out, but I bet there is. 5-limit or 7-limit?
Message: 2569 - Contents - Hide Contents Date: Wed, 19 Dec 2001 00:12:17 Subject: Re: Badness with gentle rolloff From: clumma> Yes. 404 Not Found * [with cont.] Search for http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip in Wayback Machine Got it!>> I'd love to see consistency plotted against steps*rms. >> This has been added at your request.Thanks! I wish I could say I knew what I was looking at. Cut-off badness on x and "is consistent" on y? I was thinking steps on x and real-number (unrounded) Hahn consistency on y (it looks like you're using boolean consistency). I'd just do it myself, but I can't wrap my head around Excel. I'm currently looking into a graphing calculator library/interface for Scheme... -Carl
Message: 2570 - Contents - Hide Contents Date: Wed, 19 Dec 2001 22:43:36 Subject: Re: The best 5-limit temperaments From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> I though we decided we were _not_ going to have a constraint on e, >> and simply use an upper bound on badness combined with both lower and >> upper bounds on g. What gives? >> I decided that pushing it farther was getting into garbageterritory, but there aren't many more that we would add in this way, and they could be included. Do we really want to temper out the fifth? Wouldn't the squared error of the fifth then take us outside our allowed bound on badness?
Message: 2571 - Contents - Hide Contents Date: Wed, 19 Dec 2001 00:17:52 Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >> Hey guys, >>>> Is there any chance someone could take the time to convince me that >> steps^(4/3)*cents is flat in some sense, for 7-limit ETs. It still >> looks to me like steps*cents is flatter. >> You or someone might try graphing log steps vs badness for both ofthese, or log steps vs number less than a cut-off value. Gene, I'm disappointed. Dave has produced tons of graphs, in case you didn't notice. Dave wants to understand how you use Diophantine approximation theory here.
Message: 2572 - Contents - Hide Contents Date: Wed, 19 Dec 2001 23:39:24 Subject: Re: The best 5-limit temperaments From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Wouldn't the squared error of the fifth then take us outside our > allowed bound on badness?Tempering out a fifth makes the error of a fifth equal to a fifth; since a major and minor third makes a fifth, it also makes the errors of the thirds a (neutral) third. The result, of course, makes absolutely no sense, but since you also don't need much in the way of generator steps to accomplish it, passes my badness limit. 3/2 Map: [ 0 1] [ 0 1] [ 1 2] Generators: a = .02945, b = 1 badness: 270 rms: 496 g: .8165 errors: [-702, -351, 351]
Message: 2573 - Contents - Hide Contents Date: Wed, 19 Dec 2001 00:34:50 Subject: Re: Badness with gentle rolloff From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> Yes. 404 Not Found * [with cont.] Search for http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip in Wayback Machine > > Got it! >>>> I'd love to see consistency plotted against steps*rms. >>>> This has been added at your request. >> Thanks! I wish I could say I knew what I was looking at. > Cut-off badness on x and "is consistent" on y? I was thinking > steps on x and real-number (unrounded) Hahn consistency on y > (it looks like you're using boolean consistency).If you're talking about the maximum real-number consistency _level_ obeyed by an ET, this will be equal to 1/(max_error*1200*steps) wherever it's greater than 1.5. So for the good ETs, you'll just be plotting maximum error vs. rms error.
Message: 2574 - Contents - Hide Contents Date: Wed, 19 Dec 2001 23:59:45 Subject: Two conditions on temperaments From: genewardsmith It seems to me there are two conditions it would make sense to put on something intended as an odd-limit temperament: (1) Weak condition--no element of the tonality diamond is allowed to be a unison (2) Strong condition--all elements of the tonality diamond are distinct Clearly (2)==>(1); in the 5-limit case, (1) excludes 6/5, 5/4, 4/3 and 3/2; and (2) also excludes 25/24 and 16/15, leaving 27/25 to be king of funkiness.
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