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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: 7760 Date: Mon, 27 Oct 2003 09:31:53 Subject: [tuning] Re: Polyphonic notation From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> 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...

Extremely small commas are easy to find using integer relation algorithms. The trick comes if you want a complete list of them satisfying certain conditions. One way to do that is via what I call notations, where you pass to notations using progressively smaller commas (or equivalently, larger equal temperaments) taking care while doing so to ensure you don't miss anything fulfilling your conditions.

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 * [with cont.] (Wayb.) 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: 7763 Date: Mon, 27 Oct 2003 21:12:36 Subject: Re: heuristic and straightness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> 1076168025 1073741824 UNNAMED!!!!!!!!

Unnamed since it is a schisma squared.

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 * [with cont.] (Wayb.)

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"?

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7900
7950

**7750 -**
7775 -