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Message: 7750 Date: Sun, 26 Oct 2003 14:26:57 Subject: Re: comma search (was Re: Polyphonic notation) From: Carl Lumma >> > where badness is defined as log_2(ratio)^2 * prime-limit(ratio)... >> >> Any comments on this badness measure? > >You haven't defined it yet. What's lacking in the above definition? -Carl
Message: 7751 Date: Sun, 26 Oct 2003 15:02:16 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma >> Since every prime limit contains an infinite number of ratios, and >> neither size nor complexity behave smoothly as one searches farther >> out, it seems we'll never know the top 10 lowest-badness ratios at >> any prime limit.... > >For any limit, zero will be an accumulation point of log2(q)^2, since >p-limit commas are arbitrarily small; but this hardly matters, since >whatever it is you are calculating, it clearly isn't log2(q)^2 >primelimit(q). Can we start over? Whoops, I wasn't actually taking the log2 of q. The formula used was... q^2 * primelimit(q) Probably I should use (log2(q) + 1)^2 * primelimit(q). In this case the 10 lowest-scoring ratios <= 600 cents with denominator <= 500 are... (badness, primelimit, ratio) ((3.468084457207407 3 256/243) (4.106173526999384 3 9/8) (4.6509153968061785 3 32/27) (5.180825053903934 5 81/80) (5.348010729259556 5 128/125) (5.385594090689787 3 81/64) (5.418111244777691 5 250/243) (5.6062792235873795 5 25/24) (5.797659150259687 5 135/128) (5.974440849845497 5 16/15)) -Carl
Message: 7752 Date: Mon, 27 Oct 2003 17:03:57 Subject: comma search (was Re: Polyphonic notation) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > Paul, any thoughts on a badness heuristic > > log(d) * |n-d|/log(d) = |n-d| > > ? > > Thanks, > > -Carl it's a good one, but how is it derived? it almost looks like the term between the '*' and the '=' is the error heuristic, but it's missing a factor of d in the denominator.
Message: 7753 Date: Mon, 27 Oct 2003 17:49:45 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote: > > > Maybe the original exposition can just be updated a bit, and > > then monz or I could host it, certainly. > > You might want to add to > > complexity ~ log(d) > > error ~ log(n-d)/(d log(d)) > > a badness heursitic of > > badness ~ log(n-d) log(d)^e / d > > where e = pi(prime limit)-1 = number of odd primes in limit. gene, you too got the error heuristic wrong, it's error ~ |n-d|/(d log(d)) and what kind of temperaments was this badness heuristic meant to apply to?
Message: 7754 Date: Mon, 27 Oct 2003 10:44:47 Subject: Re: comma search (was Re: Polyphonic notation) From: Carl Lumma >> Paul, any thoughts on a badness heuristic >> >> log(d) * |n-d|/log(d) = |n-d| >> >> ? >> >> Thanks, >> >> -Carl > >it's a good one, but how is it derived? it almost looks like the term >between the '*' and the '=' is the error heuristic, but it's missing >a factor of d in the denominator. Drat! Ok, howabout this... log(d) * |n-d|/d*log(d) = |n-d|/d -Carl
Message: 7755 Date: Mon, 27 Oct 2003 19:02:51 Subject: comma search (was Re: Polyphonic notation) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >> Paul, any thoughts on a badness heuristic > >> > >> log(d) * |n-d|/log(d) = |n-d| > >> > >> ? > >> > >> Thanks, > >> > >> -Carl > > > >it's a good one, but how is it derived? it almost looks like the term > >between the '*' and the '=' is the error heuristic, but it's missing > >a factor of d in the denominator. > > Drat! Ok, howabout this... > > log(d) * |n-d|/d*log(d) = |n-d|/d > > -Carl another decent badness measure -- complexity times error.
Message: 7756 Date: Mon, 27 Oct 2003 11:04:37 Subject: Re: comma search (was Re: Polyphonic notation) From: Carl Lumma >Drat! Ok, howabout this... > >log(d) * |n-d|/d*log(d) = |n-d|/d Difficult to see how we could get some of our favorites like 135/128 to come out of this, without some kind of restriction on prime limit. -Carl
Message: 7757 Date: Mon, 27 Oct 2003 19:48:11 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > <genewardsmith@j...>" <genewardsmith@j...> wrote: > > --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" > <clumma@y...> wrote: > > > > > Maybe the original exposition can just be updated a bit, and > > > then monz or I could host it, certainly. > > > > You might want to add to > > > > complexity ~ log(d) > > > > error ~ log(n-d)/(d log(d)) > > > > a badness heursitic of > > > > badness ~ log(n-d) log(d)^e / d > > > > where e = pi(prime limit)-1 = number of odd primes in limit. > > gene, you too got the error heuristic wrong, it's > > error ~ |n-d|/(d log(d)) > > and what kind of temperaments was this badness heuristic meant to > apply to? if i correct the error and use 5-limit linear temperaments (of course you meant single-comma temperaments, duh), and thus use e=2, and cut off the numerator and denominator at about 10^50, but don't cut off for error (i just insist the size of the comma is under 600 cents), i get the following for lowest badness: numerator denominator ( 1 1) 2.92300327466181e+048 2.92297733949268e+048 atomic 32805 32768 schismic 1.77635683940025e+034 1.77630864952823e+034 pirate 81 80 meantone 4.5035996273705e+017 4.50283905890997e+017 monzismic 4 3 - 25 24 dicot 1.7179869184e+047 1.7179250691067e+047 raider 15625 15552 kleismic 7629394531250 7625597484987 ennealimmal 9.01016235351562e+015 9.00719925474099e+015 kwazy 16 15 father 6 5 - 5 4 - 9 8 - 10 9 - 274877906944 274658203125 semithirds 128 125 augmented 3.81520424476946e+029 3.814697265625e+029 senior 2048 2025 diaschismic 1600000 1594323 amity 27 25 beep 1.16450459770592e+023 1.16415321826935e+023 whoosh 250 243 porcupine 1.62285243890121e+032 1.62259276829213e+032 fortune 5.00315450989997e+016 5e+016 minortone 1076168025 1073741824 UNNAMED!!!!!!!! 6115295232 6103515625 semisuper 78732 78125 semisixths 3125 3072 magic 393216 390625 würschmidt 2109375 2097152 orwell 135 128 pelogic 10485760000 10460353203 vulture 68719476736000 68630377364883 tricot 4.44089209850063e+035 4.44002166576103e+035 egads 1224440064 1220703125 parakleismic 2.23007451985306e+043 2.22975839456296e+043 gross 19073486328125 19042491875328 enneadecal 648 625 diminished 20000 19683 tetracot 256 243 blackwood 2.47588007857076e+027 2.47471500188112e+027 astro 6561 6400 - 32 27 - 2.02824096036517e+035 2.02755595904453e+035 - 531441 524288 aristoxenean 2.95431270655083e+021 2.95147905179353e+021 counterschismic 31381059609 31250000000 - 5.82076609134674e+023 5.81595589965365e+023 - 4294967296 4271484375 escapade 75 64 - 16875 16384 negri 27 20 - 95367431640625 95105071448064 - 2.25283995449392e+034 2.25179981368525e+034 - 32 25 - 25 18 - 129140163 128000000 - 125 108 - 390625000 387420489 - 2.9557837600708e+020 2.95147905179353e+020 vavoom 625 576 - 35303692060125 35184372088832 - 3.4359738368e+030 3.43368382029251e+030 - 67108864 66430125 misty 244140625 241864704 - etc. etc. etc. etc. etc. etc. etc. etc. etc. etc. etc. etc. etc. gene, did 1076168025:1073741824 not make your geometric badness cutoff, or did i mistakenly skip over it when i was working from your list? 67108864:66430125 made it onto your list, does that have lower geometric badness? if so, why is 1076168025:1073741824 so unusual from the point of view of heuristic vs. geometric badness?
Message: 7758 Date: Mon, 27 Oct 2003 01:06:26 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma I wrote... >...each triple is (badness, prime-limit, ratio). The search took >less than 10 minutes on a P3 600 laptop (code available). Performance >would be drastically better by using anything other than the slowest >conceivable factoring algorithm, which I chose for expediency. I measured the slowdown due to factoring by comparing the prime-limit complexity mode with the n*d complexity mode. It's there, but the main cause of slowness was sorting all the results just to get the top r of them, using insertsort which is generally O(n^2). So I cooked up a procedure that just gets the top r results and leaves the rest unsorted in O(n*r). The above search only takes a few seconds now. Anyway, the point of posting this here is to find out how you guys (Gene, Paul, Graham) cook up commas. Gene, is there a particular maple function I should look at? I see lists of commas doped into the code in various places... -Carl
Message: 7759 Date: Mon, 27 Oct 2003 19:51:12 Subject: comma search (was Re: Polyphonic notation) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >Drat! Ok, howabout this... > > > >log(d) * |n-d|/d*log(d) = |n-d|/d > > Difficult to see how we could get some of our favorites like > 135/128 to come out of this, without some kind of restriction > on prime limit. > > -Carl you just need to penalize complexity more.
Message: 7761 Date: Mon, 27 Oct 2003 20:15:02 Subject: 1076168025:1073741824 From: Paul Erlich a web search on 1076168025 took me to this rameau article:!!!!!!!!! RAMNOU TEXT * 1073741824 is just 2^30, so maybe rameau actually considered this interval?
Message: 7762 Date: Mon, 27 Oct 2003 09:55:14 Subject: Re: Polyphonic notation From: Graham Breed Carl Lumma wrote: > Anyway, the point of posting this here is to find out how you guys > (Gene, Paul, Graham) cook up commas. Gene, is there a particular > maple function I should look at? I see lists of commas doped into > the code in various places... I don't cook up commas, because it looks like a difficult problem that I'll leave for those who care about it. If you have commas, I can find temperaments from them, but it may take a very long time. This is because the number of commas per temperament increases the more prime numbers you consider. I cook up linear temperaments by combining linear temperaments. This is roughly O(n**2) in the number of equal temperaments, and although it can be slow, is never intolerably so, given the investment required to actually make music in a linear temperament. For 5-limit linear temperaments it doesn't make any difference, as there is only one comma. But then the 5-limit case is easy however you go about it. Graham
Message: 7764 Date: Mon, 27 Oct 2003 13:50:20 Subject: Re: heuristic and straightness From: Carl Lumma >> > You might want to add to >> > >> > complexity ~ log(d) >> > >> > error ~ log(n-d)/(d log(d)) >> > >> > a badness heursitic of >> > >> > badness ~ log(n-d) log(d)^e / d >> > >> > where e = pi(prime limit)-1 = number of odd primes in limit. >> >> gene, you too got the error heuristic wrong, it's >> >> error ~ |n-d|/(d log(d)) >> >> and what kind of temperaments was this badness heuristic meant to >> apply to? > >if i correct the error Giving (n-d)log(d)^e / d ? I don't get the point of the log(d)^e term. -Carl
Message: 7765 Date: Mon, 27 Oct 2003 22:00:02 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > > > 1076168025 1073741824 UNNAMED!!!!!!!! > > Unnamed since it is a schisma squared. OOPS!!!!!!!! (i can feel that torsion in my gut)
Message: 7766 Date: Mon, 27 Oct 2003 22:00:25 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >> > You might want to add to > >> > > >> > complexity ~ log(d) > >> > > >> > error ~ log(n-d)/(d log(d)) > >> > > >> > a badness heursitic of > >> > > >> > badness ~ log(n-d) log(d)^e / d > >> > > >> > where e = pi(prime limit)-1 = number of odd primes in limit. > >> > >> gene, you too got the error heuristic wrong, it's > >> > >> error ~ |n-d|/(d log(d)) > >> > >> and what kind of temperaments was this badness heuristic meant to > >> apply to? > > > >if i correct the error > > Giving (n-d)log(d)^e / d ? > > I don't get the point of the log(d)^e term. > > -Carl that gives you a log-flat badness measure.
Message: 7767 Date: Mon, 27 Oct 2003 14:08:36 Subject: Re: heuristic and straightness From: Carl Lumma >> >> > a badness heursitic of >> >> > >> >> > badness ~ log(n-d) log(d)^e / d >> >> > >> >> > where e = pi(prime limit)-1 = number of odd primes in limit. >> >> >> >> gene, you too got the error heuristic wrong, it's >> >> >> >> error ~ |n-d|/(d log(d)) // >> >if i correct the error >> >> Giving (n-d)log(d)^e / d ? >> >> I don't get the point of the log(d)^e term. >> >> -Carl > >that gives you a log-flat badness measure. Aha. Can we get results with e = 7 (19-limit)? You said I just needed to penalize complexity, but: () Wouldn't this ruin the log-flatness? () Here you are restricting yourself to the 5-limit! -Carl
Message: 7768 Date: Mon, 27 Oct 2003 22:20:58 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >> >> > a badness heursitic of > >> >> > > >> >> > badness ~ log(n-d) log(d)^e / d > >> >> > > >> >> > where e = pi(prime limit)-1 = number of odd primes in limit. > >> >> > >> >> gene, you too got the error heuristic wrong, it's > >> >> > >> >> error ~ |n-d|/(d log(d)) > // > >> >if i correct the error > >> > >> Giving (n-d)log(d)^e / d ? yes, if n>d. > >> I don't get the point of the log(d)^e term. > >> > >> -Carl > > > >that gives you a log-flat badness measure. > > Aha. Can we get results with e = 7 (19-limit)? sure, but then we'd be talking about 6-dimensional temperaments. i *might* attempt the calculation if you give me a nice low cutoff for numerator and denominator . . . > You said I just needed to penalize complexity, penalize it more, yes. > but: > > () Wouldn't this ruin the log-flatness? no, it would get you closer to it. > () Here you are restricting yourself to the 5-limit! yes, though a few of the commas are 3-limit too. so?
Message: 7769 Date: Mon, 27 Oct 2003 14:37:25 Subject: Re: heuristic and straightness From: Carl Lumma >> >> Giving (n-d)log(d)^e / d ? > >yes, if n>d. I thought you might say that! >> >> I don't get the point of the log(d)^e term. >> >> >> >> -Carl >> > >> >that gives you a log-flat badness measure. >> >> Aha. Can we get results with e = 7 (19-limit)? > >sure, but then we'd be talking about 6-dimensional temperaments. Saints preserve us! >i *might* attempt the calculation if you give me a nice low cutoff >for numerator and denominator . . . Well, 10^50 would send my code to the sun. I could probably do this with an imperative style, but since you apparently already have done so, I thought I'd ask you. What I don't get is why upping the prime limit from 5 to 19 would make it any harder. The way I'd do it, is for each d < 10^50, run n until n/d > 600 cents, kicking out any ratios where n*d has a factor greater than 19. The factoring algorithm I'm using walks up from 2, so aborting it after 19 or 5 wouldn't make much difference. >> You said I just needed to penalize complexity, > >penalize it more, yes. > >> but: >> >> () Wouldn't this ruin the log-flatness? > >no, it would get you closer to it. You mean without the log(d)^e term? Because if that term gives flatness, and then I put an exponent on d, wouldn't I be ruining the flatness? >> () Here you are restricting yourself to the 5-limit! > >yes, though a few of the commas are 3-limit too. so? I asked if these searches could be done without a restriction on prime-limit, and you said yes. -Carl
Message: 7770 Date: Mon, 27 Oct 2003 22:42:55 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >> >> Giving (n-d)log(d)^e / d ? > > > >yes, if n>d. > > I thought you might say that! > > >> >> I don't get the point of the log(d)^e term. > >> >> > >> >> -Carl > >> > > >> >that gives you a log-flat badness measure. > >> > >> Aha. Can we get results with e = 7 (19-limit)? > > > >sure, but then we'd be talking about 6-dimensional temperaments. > > Saints preserve us! > > >i *might* attempt the calculation if you give me a nice low cutoff > >for numerator and denominator . . . > > Well, 10^50 would send my code to the sun. I could probably do this > with an imperative style, but since you apparently already have done > so, I thought I'd ask you. i've only done it for prime limit 5. > What I don't get is why upping the prime limit from 5 to 19 would > make it any harder. The way I'd do it, is for each d < 10^50, run > n until n/d > 600 cents, kicking out any ratios where n*d has a > factor greater than 19. The factoring algorithm I'm using walks > up from 2, so aborting it after 19 or 5 wouldn't make much >difference. ok, so why don't you do it? (seriously -- my factoring algorithm refuses numbers higher than 2^32). see if you can reproduce my 5- limit results first. > >> You said I just needed to penalize complexity, > > > >penalize it more, yes. > > > >> but: > >> > >> () Wouldn't this ruin the log-flatness? > > > >no, it would get you closer to it. > > You mean without the log(d)^e term? i mean, you were using complexity * error, and that didn't penalize complexity enough, while a higher power on complexity would. > Because if that term gives > flatness, and then I put an exponent on d, wouldn't I be ruining > the flatness? > > >> () Here you are restricting yourself to the 5-limit! > > > >yes, though a few of the commas are 3-limit too. so? > > I asked if these searches could be done without a restriction > on prime-limit, and you said yes. i don't think i would have been referring to the same search. the exponent on complexity in the log-flat badness formula, at least according to gene, depends on the prime limit.
Message: 7771 Date: Mon, 27 Oct 2003 22:54:30 Subject: comma search From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > What I don't get is why upping the prime limit from 5 to 19 would > make it any harder. i did it this way: Searching Small Intervals *
Message: 7772 Date: Mon, 27 Oct 2003 17:38:49 Subject: Re: heuristic and straightness From: Carl Lumma >> What I don't get is why upping the prime limit from 5 to 19 would >> make it any harder. The way I'd do it, is for each d < 10^50, run >> n until n/d > 600 cents, kicking out any ratios where n*d has a >> factor greater than 19. The factoring algorithm I'm using walks >> up from 2, so aborting it after 19 or 5 wouldn't make much >> difference. > >ok, so why don't you do it? (seriously -- my factoring algorithm >refuses numbers higher than 2^32). see if you can reproduce my 5- >limit results first. Ok, maybe later tonight/this morning. But how'd you do 10^50 if you can't factor above 2^32? >> >> You said I just needed to penalize complexity, >> > >> >penalize it more, yes. >> > >> >> but: >> >> >> >> () Wouldn't this ruin the log-flatness? >> > >> >no, it would get you closer to it. >> >> You mean without the log(d)^e term? > >i mean, you were using complexity * error, and that didn't penalize >complexity enough, while a higher power on complexity would. Ok. >> Because if that term gives >> flatness, and then I put an exponent on d, wouldn't I be ruining >> the flatness? >> >> >> () Here you are restricting yourself to the 5-limit! >> > >> >yes, though a few of the commas are 3-limit too. so? >> >> I asked if these searches could be done without a restriction >> on prime-limit, and you said yes. > >i don't think i would have been referring to the same search. the >exponent on complexity in the log-flat badness formula, at least >according to gene, depends on the prime limit. You were referring to |n-d|/d. I get it now. -Carl
Message: 7773 Date: Mon, 27 Oct 2003 18:23:16 Subject: Re: Linear temperament names? From: Carl Lumma >Graham, have you ever thought of spelling it "majic" since it's >generated by MAJor thirds? I thought it was an acronym. -Carl
Message: 7774 Date: Tue, 28 Oct 2003 16:23:40 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: > > Have names been proposed for any of the linear microtemperaments below? > > > > The generators and errors are for minimax. The mappings given are > > octave equivalent: > > [gens_per_3 gens_per_5 gens_per_7 gens_per_11 gens_per_13; > > periods_per_3 periods_per_5 etc...] > > Graham convinced me to switch to his convention. Do you have a reason > for preferring [generator, period] over [period, generator]? I think > we should try for some degree of standardization. The article deals only with linear temperaments of octave-repeating octave-equivalent scales, so the reader is only interested in how many periods there are modulo the number in the octave. So when the period _is_ the octave these are all zero and I prefer to omit them. It's easier to omit them without confusion if they come _last_. I do not want to use any vector or matrix math in the article. It's pitched at an audience with more basic math skills. But I agree that for the greatest generality the period should come first, followed by the generator(s). > Noreover, you are > ignoring 2, and to me this is simply not acceptable. Oh blow it out your ear. :-) The article deals only with octave-repeating octave-equivalent scales so why should I bother saying that there are zero generators in the 1:2 every time. And the size of the period, given as a fraction of an octave, is a bit of a giveaway as to how many of _them_ are in the 1:2. Also I have limited space to fit many things about each LT in columns across the width of a page. But I certainly agree that for greatest generality 2 should be included in all the matrices and vectors. The main thing is that I explain the format I'm using. > > Limit Period Gen Max gens Max err Prime mapping Rep ET > > 7-limit 1 oct 193.87 c 16 1.4 c [16 2 5] > > Hemiwuerschmidt. You should give all of the mapping and give it in a > canonical reduced form, or a give a reduced comma basis, or a > wedgie--or best of all, all three. Commas and wedgies are utterly irrelevant to my article. My canonical generator is the smallest one (less than half the period), what's yours? > > 11-limit 1/2 oct 216.74 c 30 3.1 c [-6 -1 10 -3; 1 1 0 0] No name for this one? Is there any other LT more deserving of being called "twin thirds"? > > 11-limit 1/2 oct 183.21 c 30 2.4 c [-6 -11 2 3; 1 0 1 0] > > Unidec. Please explain. Why not call it "twin minortones" since the generator represents 9:10 in the temperament. > > 15-limit 1/3 oct 83.02 c 48 2.8 c [-6 -5 2 -3 -14; 0 2 2 2 2] > > Trikleismic. That makes sense, but I would have said "triple kleismic". If you use the prefix tri- to mean 3 equispaced chains of a generator then what would you use to mean a single chain of 1/3 of that generator? i.e. in the way that you use hemi- to mean a single chain of 1/2 a generator? > > 15-lm-wo-13 1 oct 193.24 c 35 2.8 c [-15 2 5 -22] > > For the 7-limit temperament, I have it listed as Hemithird. Makes sense too. But I don't understand why you use hemi- when it is already established that semi- is used to halve a musical interval, as in semitone and semisharp. I've asked you that before, but I don't remember a satisfactory answer. > > If you do propose a name, please also say why you think it is > > appropriate. > > The names I give are ones which have already been used; they are not > new proposals. So what? They are still only proposals as far as my article is concerned. I don't have to use the name you give me, and so I'd still like explanations for the less than obvious ones, like "unidec". > If I don't think the name makes much sense I may just > > include the temperament without a name. > > My preference is for you not to sow confusion by introducing new > names for already named and cataloged temperaments; That's why I'm asking. How about this one? Period Gen Max gens Max err Prime mapping (no 2s) 1 oct 351.45 c 10 1.9 c [2 25 13] "cata neutral thirds"?
7000 7050 7100 7150 7200 7250 7300 7350 7400 7450 7500 7550 7600 7650 7700 7750 7800 7850 7900 7950
7750 - 7775 -