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Message: 11225 - Contents - Hide Contents Date: Sat, 03 Jul 2004 18:30:31 Subject: Re: NOT tuning From: Carl Lumma>>> >eantone >>> >>> 5-limit: 698.0187 (43, 55, 98, 153, 251, 404) >>> >>> 7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43) >>>> Hmm... I dunno, this seems a bit far from the old-style rms >> optimum. >> >> -Carl >>Carl, when Graham investigated this same question here a few months >ago, he concluded that pure-octaves TOP would be a uniform stretching >or compression of TOP,That seems obvious for ETs....>except where TOP already had pure octaves, in >which case it would actually change!That's impossible given the criterion of NOT. Maybe I don't comprehend you. -Carl

Message: 11226 - Contents - Hide Contents Date: Sat, 03 Jul 2004 20:25:09 Subject: Re: from linear to equal From: Carl Lumma>> >-limit should also be considered when you're going "poptimal". >>True enough. Alas, even though we have the same wedgie, commas and >tuning map, the poptimal range need not even overlap. Orwell is a >typical example--there seems to be no overlap from 7 to 9, and none >between 11 and 9, but the others are OK. So, 5 and 9 overlap, and >have 43/190 as a common generator, but 7 and 9, no.This is AWESOME. Seriously, if you had come to me in a past life and asked me to imagine the most heinously interesting thing ever, for torturing curious folks in purgatory or something, I wouldn't have come up with the half of this temperaments thing. How annoying, that there doesn't seem to be any really good way to famlify the temperaments. By the way, Gene, how does poptimal relate to TOP? If the commas dictate the TOP tuning, is there necessarily a generator/ period pair that give it? And if there is, is it not specified exactly by the TOP tuning, for a given map? Maybe what I'm asking is, "could you walk me through the functions you'd call to find a generator/period pair for a 7-limit linear temperament?". I can look in your code. Though I guess what I have doesn't cover TOP... -Carl

Message: 11227 - Contents - Hide Contents Date: Sat, 03 Jul 2004 20:27:10 Subject: Re: NOT tuning From: Carl Lumma>>> >xcept where TOP already had pure octaves, in >>> which case it would actually change! >>>> That's impossible given the criterion of NOT. >> >> Maybe I don't comprehend you. >>Some examples of this method of tuning would be nice, and >a definition even better.Which method? Graham's? I think he gave examples. Graham, what's a good word to search for? I know I have that post. I think I replied to it. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 11228 - Contents - Hide Contents Date: Sun, 04 Jul 2004 06:24:31 Subject: Re: from linear to equal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> 9-limit should also be considered when you're going "poptimal". >>>> True enough. Alas, even though we have the same wedgie, commas and >> tuning map, the poptimal range need not even overlap. Orwell is a >> typical example--there seems to be no overlap from 7 to 9, and none >> between 11 and 9, but the others are OK. So, 5 and 9 overlap, and >> have 43/190 as a common generator, but 7 and 9, no. >> This is AWESOME. Seriously, if you had come to me in a past > life and asked me to imagine the most heinously interesting > thing ever, for torturing curious folks in purgatory or > something, I wouldn't have come up with the half of this > temperaments thing.Har. You think that is bad, try this: two different 11-limit linear temperaments are the meantone variants meantone or meanpop (sharing the same TOP tuning with the 7-limit temperament) and huygens (sharing the same NOT tuning with the 7-limit temperament.) The poptimal ranges for these two theoretically distinct temperaments, one of which adds 385 to the 7-limit comma set {81/80, 126/125} and the other of which adds 99/98, actually overlap. Unsurprisingly, 31 is poptimal for both, and 31 is what you get by adding *both* 99.98 and 385/384 to 7-limit meantone. However, 31 is not the only poptimal possibility; the allegedly universal 198 will work also. This, of course, uses two different possible versions of 11 for huygens and meantone, with the huygens version being more accurate, since 198 has a meantone fifth very slightly (0.1955 cents) sharper than 31, where the two temperaments are identical. If we used 267 equal instead, also poptimal for both temperaments, the 11 would be more accurate in the meantone version rather than the huygens version. What it really boils down to is that for the 11-limit we should just give it up, and use 31 equal. That way we've got both 99/98 and 385/384 to play with, commas also happy with 11-limit orwell, by the way.> By the way, Gene, how does poptimal relate to TOP?Not very well, apparently. If the> commas dictate the TOP tuning, is there necessarily a generator/ > period pair that give it?You've lost me.> Maybe what I'm asking is, "could you walk me through the > functions you'd call to find a generator/period pair for a > 7-limit linear temperament?". I can look in your code. > Though I guess what I have doesn't cover TOP...Do you mean in terms of cents, or a generator period pair in terms of p-limit intervals which temper to the generator and period whatever tuning you use? In terms of cents, the easy ones to find are TOP, NOT, rms, and minimax, but each of these is different; rms involves least squares, and the rest I set up as a linear programming problem and solve using Maple's simplex method implementation.

Message: 11229 - Contents - Hide Contents Date: Sun, 04 Jul 2004 16:40:26 Subject: Re: from linear to equal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Isn't TOP a minimax (p=inf.) method? Oh, but in the definition above > you restrict poptimal to octaves. . . .TOP is a weighted method, not restricted, which can be regarded as minimax applied to just the primes.

Message: 11230 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:23:23 Subject: Re: A chart of syntonic comma temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> http://www.io.com/~hmiller/png/syntonic.png - Type Ok * [with cont.] (Wayb.) > > This is a chart of 7-limit temperaments that temper out the syntonic > comma 81;80. The horizontal axis is deviation from 3:1 and the vertical > axis is the deviation from 7:1. This time I limited the list to 7- limit > consistent ET's.What you call Mothra, I call Cynder, since it's basically the same as the Slendric or Wonder temperament, but with 5 thrown into the primal mix. What you call Hemifourths, I call Semaphore.

Message: 11231 - Contents - Hide Contents Date: Sun, 04 Jul 2004 06:37:27 Subject: Re: from linear to equal From: Kalle Aho --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> For pure octave tunings, a system I sometimes use is to close at a > "poptimal" generator. A generator is "poptimal" for a certain set of > octave-eqivalent consonances if there is some exponent p, 2 <= p <= > infinity, such that the sum of the pth powers of the absolute value of > the errors over the set of consonances is minimal.This is quite an interesting approach. What makes poptimal generators good? And why can't p be 1?> This is convenient > for Scala score files, since the notes are now represented by > (reasonably small) integers. I also sometimes use it when cooking up a > Scala scl file (just did, in fact, over on the tuning list) though in > that case it makes little difference. > > If you follow this system, 5-limit meantone closes for 81, 7-limit > meantone for 31, and 11-limit meantone for 31. 5 and 7 taken together > are 1/4-comma exactly, which doesn't close; 5 and 11 taken together > closes at 112, and 7 and 11 of course also at 31. One rarely > encounters problems; even a microtemperament like ennealimmal closes > at 1053, which is perfectly reasonable for Scala applications; one > does, however, need to ensure the division is divisible by 9.These results are interesting. Do these poptimal generators make these linear temperaments close exactly at these ETs?!> A different naming convention than using TOP tuning would be to give > the same name iff the poptimal ranges intersect. This isn't very > convenient in practice, due to the difficulty of computing the > poptimal range, but clearly it leads to quite different results. > Miracle, for instance, has the same TOP tuning in the 5, 7 and 11 > limits, but while the 5 and 7 limit poptimal ranges intersect, the 5 > and 11 or 7 and 11 ranges apparently do not, though as I say computing > these is a pain, so I may have the range too small. In any case, > miracle closes at 175 in the 5 and 7 limits, and at 401 in the 11- limit.

Message: 11232 - Contents - Hide Contents Date: Sun, 04 Jul 2004 17:11:47 Subject: Re: NOT tuning From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> You can always define the method to give the same answer for pure-odd > ratios. But yes, for the 5-limit it should give quarter comma meantone, > because the 81:80 is shared between the four factors of 3 in the > numerator. It's clearly doing something different. > > I haven't defined the 7-limit result because I don't generally know how > to do 7-limit linear TOP. What I do have is: > > Minimax 696.58 > RMS (7) 696.65 > RMS (9) 696.44 > PORMSWE 697.22 > > The last one you may recall is my alternative to TOP. Here, the octave > is stretched by 1.24 cents. I can't generalize it to the > octave-equivalent case (which is why I switched to odd limits in the > first place). But you can always unstretch the octave, which here gives > a fifth of 696.49 cents. > > Either there's a systematic error in all my calculations, or Gene's > result is perverse.I don't see why you think that; my results are a consequence of my definition, and your results I presume of yours. NOT tunings are NOT5: 698.02 error in 3, over log 3, is equal to error in 5, over log 5, is equal to error in 5/3, over 15 NOT7: 697.65 error in 3, over log 3, is error in 7, over log 7, is error in 7/3, over log 21

Message: 11233 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:30:59 Subject: Re: from linear to equal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Har. You think that is bad, try this: two different 11-limit linear >> temperaments are the meantone variants meantone or meanpop (sharing >> the same TOP tuning with the 7-limit temperament) and huygens > (sharing>> the same NOT tuning with the 7-limit temperament.) >> Isn't that a ridiculous name for an 11-limit temperament?You'd maybe prefer Fokker?

Message: 11234 - Contents - Hide Contents Date: Sun, 04 Jul 2004 07:08:39 Subject: Re: from linear to equal From: Kalle Aho --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> Kalle Aho wrote: >>> For TOP tempered linear temperaments I suggest closing the circle >> when you start getting better approximations to the primes for which >> the tuning is optimized. >> >> What are your thoughts about this? >> That's pretty much my opinion, although you can go some ways beyond that > point if you try to avoid the better approximations, and there may be > other reasons to stick with the temperaments in particular cases. TOP > father (g = 447.3863410, p = 1185.869125) has a better approximation of > 5 after only 7 iterations of the generator, so you might want to switch > to 8-ET. This works out fine for the 5-limit, since TOP 5-limit 8- ET > <1185.032536, 1925.677871, 2814.452272] is pretty close to TOP 5- limit > father <1185.869125, 1924.351908, 2819.124590]. But 8-ET isn't 7- limit > consistent, so if you're using 7-limit father temperament <1185.869125, > 1924.351908, 2819.124589, 3401.317477], you're probably better off > sticking with the 8-note father MOS rather than going to one of the > versions of TOP 8-ET.Wasn't consistency supposed to be a nonissue with TOP tunings? But I understand why you say these things.

Message: 11235 - Contents - Hide Contents Date: Sun, 04 Jul 2004 10:14:22 Subject: Re: from linear to equal From: Carl Lumma>> >sn't TOP a minimax (p=inf.) method? Oh, but in the definition above >> you restrict poptimal to octaves. . . . >>TOP is a weighted method, not restricted, which can be regarded as >minimax applied to just the primes.Oh, and why should I believe that p=inf gives minimax again? -Carl

Message: 11236 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:35:12 Subject: Re: bimonzos, and naming tunings (was: Gene's mail server)) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>>>> Are you expecting people to read the comma values >>>> off of the bimonzo? >>>>>> No. But as long as we're on the subject here, it might >>> be worth reviewing here for list memmbers how you do that. >>> Not in the paper. >>> yes, please do review it! >> If you have a bimonzo ||a1 a2 a3 a4 a5 a6>> then you can read off the > odd comma of the temperament (the comma which is a ratio of odd > integers) by taking out the common factor if needed of a1, a2, a3 to > get b1, b2, b3, and then the comma is 3^b1 5^b2 7^b3, or its > reciprocal if you need to make it bigger than 1. > > In general, however, reading the commas from a bimonzo is no easier > than reading them from a bival, and in fact probably harder,Why harder? Can you show this?> and I > don't think this makes much of a reason for using bimonzos. I really > would like to know why Paul insists on this so stubbornly.I welcome constructive suggestions for making the paper go bival, without adding to its math-heaviness.> If you have a bival <<a1 a2 a3 a4 a5 a6||, then > > 2^a4 3^(-a2) 5^a1 gives the 5-limit comma.You don't need to remove common factors?> 2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the power > of the (3,7) coefficient, 3 to minus the power of the (2,7) > coefficient, and 7 to the power of the (2,3) coefficient;How is this easier than the bimonzo case?

Message: 11237 - Contents - Hide Contents Date: Sun, 04 Jul 2004 07:24:51 Subject: Re: from linear to equal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> For pure octave tunings, a system I sometimes use is to close at a >> "poptimal" generator. A generator is "poptimal" for a certain set of >> octave-eqivalent consonances if there is some exponent p, 2 <= p <= >> infinity, such that the sum of the pth powers of the absolute value > of>> the errors over the set of consonances is minimal. >> This is quite an interesting approach. What makes poptimal generators > good? And why can't p be 1?It could be 1. In fact, Paul thinks it should be 1. My feeling is that we accept 2 and infinity, so anything in between is OK, but I'm not confident with 1. What makes them good is that they approximate a given list of target consonances in an optimal way, for some sense of optimal.> These results are interesting. Do these poptimal generators make > these linear temperaments close exactly at these ETs?!Right. While I don't try to compute what exponent p they close for, I can prove it must exist.

Message: 11238 - Contents - Hide Contents Date: Sun, 04 Jul 2004 17:36:00 Subject: Re: from linear to equal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Isn't TOP a minimax (p=inf.) method? Oh, but in the definition above >>> you restrict poptimal to octaves. . . . >>>> TOP is a weighted method, not restricted, which can be regarded as >> minimax applied to just the primes. >> Oh, and why should I believe that p=inf gives minimax again?If you have a>b, then a^p+b^p is dominated by a as p goes to infinity, since (b/a)^p --> 0. Hence (|a|^p+|b|^p)^(1/p) --> max(|a|, |b|) as p --> infinity. (If a and b are of the same size, the doubling makes no difference, since 2^(1/p)-->1)

Message: 11239 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:35:23 Subject: Re: from linear to equal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> This is quite an interesting approach. What makes poptimal >> generators >> good? >> Not much, IMHO -- the "true" value of p in any situation will be some > number, not an infinite range of numbers.What in the world does this mean? What allegedly "true" value?>> And why can't p be 1? >> My graphs show p going even slightly below 1, and I think this is > more than appropriate when you look at the kinds of discordance > curves Bill Sethares predicts and George Secor prefers. Very sharp > spikes at the simple ratios.If you go below 1 your don't even get a corresponding metric, but you can go as far as 1 and have a metric. Let's at least keep the triangle inequality, please. As for 1, I think a lot of people would find the supposedly optimal tunings not really very optimal in some cases.>> These results are interesting. Do these poptimal generators make >> these linear temperaments close exactly at these ETs?! >

Message: 11240 - Contents - Hide Contents Date: Sun, 04 Jul 2004 00:27:37 Subject: Re: from linear to equal From: Carl Lumma>> >y the way, Gene, how does poptimal relate to TOP? >>Not very well, apparently. > > If the>> commas dictate the TOP tuning, is there necessarily a >> generator/period pair that give it? >>You've lost me.I meant, for a given TOP-tuned linear temperament, does it not stand to reason that there is at least one generator/period pair (in cents) that produces scales in said tuning?>> Maybe what I'm asking is, "could you walk me through the >> functions you'd call to find a generator/period pair for a >> 7-limit linear temperament?". I can look in your code. >> Though I guess what I have doesn't cover TOP... >>Do you mean in terms of cents, Yes. >or a generator period pair in terms of >p-limit intervals which temper to the generator and period whatever >tuning you use?You lost me here; sounds interesting but I don't think I meant this.>In terms of cents, the easy ones to find are TOP, NOT, >rms, and minimax, but each of these is different; rms involves least >squares, and the rest I set up as a linear programming problem and >solve using Maple's simplex method implementation.Ok. I know roughly what this means. But it'd still be nice to know what data you feed into which processes. You need a map at some point, I'd think, so you can specify whether to find, say, fifths or fourths for meantone... I'm trying to build a picture of what kinds of things I need to know to get what kinds of answers out. -Carl

Message: 11241 - Contents - Hide Contents Date: Sun, 04 Jul 2004 11:21:49 Subject: Re: from linear to equal From: Carl Lumma>>> >OP is a weighted method, not restricted, which can be regarded as >>> minimax applied to just the primes. >>>> Oh, and why should I believe that p=inf gives minimax again? >>If you have a>b, then a^p+b^p is dominated by a as p goes to infinity, >since (b/a)^p --> 0. Hence (|a|^p+|b|^p)^(1/p) --> max(|a|, |b|) as >p --> infinity. (If a and b are of the same size, the >doubling makes no difference, since 2^(1/p)-->1)Of course. Thanks. -Carl

Message: 11242 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:35:54 Subject: Re: from linear to equal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>> Har. You think that is bad, try this: two different 11-limit linear >>> temperaments are the meantone variants meantone or meanpop (sharing >>> the same TOP tuning with the 7-limit temperament) and huygens >> (sharing>>> the same NOT tuning with the 7-limit temperament.) >>>> Isn't that a ridiculous name for an 11-limit temperament? >> You'd maybe prefer Fokker?Did Fokker have a particular route along the circle of fifths that he preferred to get 11?

Message: 11243 - Contents - Hide Contents Date: Sun, 04 Jul 2004 07:32:30 Subject: Re: from linear to equal From: Kalle Aho --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote: >> Hi, >>>> Linear temperaments (or 2-dimensional tunings) are infinitely >> extendable. Once you extend a linear temperament eonugh you'll > start>> getting different pitches that nevertheless are more or less >> indistinguishable from each other. Even before that you'll get >> approximations that are better than those the linear temperament is >> supposed to give. >> >> So what would be a good place to close the circle and go from > linear >> to equal? >> >> For TOP tempered linear temperaments I suggest closing the circle >> when you start getting better approximations to the primes for > which>> the tuning is optimized. >> Not a bad idea. I don't think any of my horagrams go further than > this, although 5:4 is slightly better in TOP Catler, and maybe > there's another similar example somewhere. > > You'd have to make your criterion a little more precise -- are you > assuming that the scales grow in one direction, or in both > directions, as you apply the generator more and more times?I would look at all intervals of the chain so it doesn't matter which way they grow. As soon as I get a better approximation to any one of the primes I would choose the nearest (in terms of cardinality) equal temperament that both supports the linear temperament and is better than the ones preceding it. Kalle

Message: 11244 - Contents - Hide Contents Date: Sun, 04 Jul 2004 18:46:03 Subject: 9&11 poptimal secor From: Gene Ward Smith The 9 and 11 poptimal range intersect only at the minimax for 9 and 11, which is the (18/5)^(1/19) secor. The continued fraction for this gives 10, 31, 41, 72, 329, 2046 ... as the et convergents. The 7/72 secor is between the poptimal range for 9 and and 11 and the range for 5 and 7, which makes it an all-purpose utility choice, and it's actually possible that the 11-limit poptimal range includes it, since it at least gets quite close. The 5 and 7 limit minimax tuning is (12/5)^(1/13), which defines the upper part of their range. Is either of these the official George Secor secor?

Message: 11245 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:40:22 Subject: Re: from linear to equal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>> This is quite an interesting approach. What makes poptimal >>> generators >>> good? >>>> Not much, IMHO -- the "true" value of p in any situation will be some >> number, not an infinite range of numbers. >> What in the world does this mean? What allegedly "true" value?If you're using this p-norm model in the first place, it's probably because you think it's true for some value of p. If you run over all possible 'p's, you're violating the assumptions of any such model.>>> And why can't p be 1? >>>> My graphs show p going even slightly below 1, and I think this is >> more than appropriate when you look at the kinds of discordance >> curves Bill Sethares predicts and George Secor prefers. Very sharp >> spikes at the simple ratios. >> If you go below 1 your don't even get a corresponding metric, but you > can go as far as 1 and have a metric. Let's at least keep the triangle > inequality, please.The behavior below 1 reflects the meaningful result that temperaments do not improve on JI tunings there. It's helpful to think of the bigger picture.> As for 1, I think a lot of people would find the supposedly optimal > tunings not really very optimal in some cases.And yet there is a significant bunch of composers who refuse to temper, clinging to their JI scales. Might they be modelled too? (no offense to them, of course.)

Message: 11246 - Contents - Hide Contents Date: Sun, 04 Jul 2004 00:34:01 Subject: Re: from linear to equal From: Carl Lumma>> >or pure octave tunings, a system I sometimes use is to close at a >> "poptimal" generator. A generator is "poptimal" for a certain set of >> octave-eqivalent consonances if there is some exponent p, 2 <= p <= >> infinity, such that the sum of the pth powers of the absolute value >> of the errors over the set of consonances is minimal.I guess I never understood how poptimal is different than 'all the error functions ever advocated here'.>> A different naming convention than using TOP tuning would be to give >> the same name iff the poptimal ranges intersect. This isn't very >> convenient in practice, due to the difficulty of computing the >> poptimal range, but clearly it leads to quite different results. >> Miracle, for instance, has the same TOP tuning in the 5, 7 and 11 >> limits, but while the 5 and 7 limit poptimal ranges intersect, >> the 5 and 11 or 7 and 11 ranges apparently do not, though as I say >> computing these is a pain, so I may have the range too small. In >> any case, miracle closes at 175 in the 5 and 7 limits, and at 401 >> in the 11-limit.Isn't TOP a minimax (p=inf.) method? Oh, but in the definition above you restrict poptimal to octaves. . . . -Carl

Message: 11247 - Contents - Hide Contents Date: Sun, 04 Jul 2004 18:58:07 Subject: 5, 7, and 9 copoptimal magic From: Gene Ward Smith A generator which is poptimal for 5, 7, and 9 limit magic is 2^(84/265). The 41-et major third is between the ranges for 5, 7, and 9, and that for 11, and just a hair under the 11 unless it is, in fact, poptimal. That puts it in pretty much the same position for magic as 72 is for miracle, but 265-equal seems like an interesting alternative.

Message: 11248 - Contents - Hide Contents Date: Sun, 04 Jul 2004 07:44:52 Subject: Re: from linear to equal From: Kalle Aho Carl wrote:> Hi Kalle, > > I wouldn't indicate such a hard-and-fast rule. If you reach those > notes (the better approximations) by modulating in a piece of music, > I'd say use them. If not, don't. Of course you're not allowed to > use them as direct approximations and still call it the same regular > temperament you started with. Maybe Gene will correct me but I > think changing the map in this fashion means you're using a different > temperament. There's nothing wrong with that of course -- or one > could remain faithful to the original map and keep the fine > distinctions of the extended progression -- or one could equalize. > All seem valid.Right and true. But I think it would be nice to have some systematic choice(s) when one wants to close the circle. For me the existence of better approximations is somewhat disturbing. Note that we don't have this "problem" in JI because every interval you reach will be worse than the generating primes.

Message: 11249 - Contents - Hide Contents Date: Sun, 04 Jul 2004 19:13:34 Subject: 9&11 limit copoptimal pajara From: Gene Ward Smith The 9 and 11 limit poptimal ranges intersect, and 78 or 100 equal work as copops for them. The 7-limit range is a tad higher, and 22 is excellent for that. The 5-limit temperament of course really isn't pajara at all, but it has 114 or 148 as possibles. The 78-et fifth is 707.692 cents, 5.737 cents sharp. The 100-et poptimal fifth is of course not the meantone fifth of 50-et of 696 cents, but the sharper fifth of exactly 708 cents, 6.045 cents sharp. The 100-et version of Paul's favorite 10 and 12 note scales come out exactly in terms of cents, like meantone does in 50, and 100-equal was one of the things which turned up when I went looking for something which did a good job for both meantone and pajara. It looks to me it's really better than 88 for that.

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