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Message: 4925 - Contents - Hide Contents Date: Thu, 30 May 2002 08:02:28 Subject: Optimal bases for 7-limit planar temperaments From: Gene W Smith I found these by first taking the Hermite form of the generator matrix, and then finding a basis which minimized the 7-limit complexity (sums of squares of lattice distances.) These should be very useful in finding scales associated to these temperaments. 25/24 <5/4, 21/20> 28/27 <4/3, 6/5> 250/243 <7/6, 10/9> 36/35 <4/3, 6/5> 49/48 <8/7, 35/32> 50/49 <5/4, 6/5>; 1/2 octave period 64/63 <4/3, 5/4> 875/874 <6/5, 25/24> 81/80 <4/3, 9/7> 2048/2025 <4/3, 8/7> 1/2 octave period 245/243 <4/3, 9/7> or <9/7, 7/6> 126/125 <5/4, 6/5> 4000/3969 <4/3, 80/63> 1728/1715 <7/6, 49/48> 1029/1024 <8/7, 35/32> 225/224 <4/3, 16/15> or <5/4, 16/15> 3136/3125 <6/5, 28/25> 5120/5103 <4/3, 27/20> 6144/6125 <6/5, 35/32> 2401/2400 <7/5, 60/49> 4375/4374 <6/5, 27/25> 250047/250000 <4/3, 6/5> or <4/3, 10/9> 1/3 octave period

Message: 4927 - Contents - Hide Contents Date: Fri, 31 May 2002 23:22:20 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: Carl Lumma>> >e get in the 72-et version a 9595959597 pattern, or the scale >> [0,9,14,23,28,37,42,51,56,65]. This has the following number of >> consonant intervals and triads at these odd limits: /.../ >> Since I don't keep very good track of scales, I wonder if Carl or >> Paul can tell us if they've seen this one before? >>That's a mode of this scale: > >10-tone scale, e=24 c=4, in 72-tET >(0 5 14 19 28 33 42 49 58 63)I actually asked about this only two days ago...>What's this: > >10-tone scale, e=24 c=4, in 72-tET >(0 5 14 19 28 33 42 49 58 63 72)...how's that for coincidences? -Carl

Message: 4928 - Contents - Hide Contents Date: Fri, 31 May 2002 08:16:41 Subject: A 1029/1024 (385/384) planar temperament scale From: genewardsmith I suggested my data on bases for planar temperaments could help locate scales, and I tried it out on the 1029/1024 planar temperament, which extends to the closely related 11-limit planar temperament tempering out 385/384 and 441/440. If we take two chains of four 8/7s separated by a 35/32, we get in the 72-et version a 9595959597 pattern, or the scale [0,9,14,23,28,37,42,51,56,65]. This has the following number of consonant intervals and triads at these odd limits: 7: 24, 16 9: 24, 16 11: 34, 46 Since I don't keep very good track of scales, I wonder if Carl or Paul can tell us if they've seen this one before?

Message: 4929 - Contents - Hide Contents Date: Fri, 31 May 2002 08:25:28 Subject: A ten-tone, 2401/2400 scale From: Gene W Smith It occurred to me that one way to get a smooth scale might be to enforce the epimorphic property for a suitable mapping, meaning one consistent with 2401/2400 and as close to being a good et mapping as possible. The h11 mapping is too irregular for this to work well, but h10 is excellent. We get from it the following scale in 612-et: [0, 43, 104, 179, 222, 297, 358, 401, 476, 537] Step sizes are 75, 61, and 43, and it has 19 intervals and 6 triads in the 7-limit, 23 intervals and 14 triads in the 9-limit. This is a fair amount of harmony for something which is effectively JI.

Message: 4930 - Contents - Hide Contents Date: Fri, 31 May 2002 10:11:13 Subject: Two 3136/3125 planar scales From: genewardsmith If we take two chains of 28/25s a minor third apart, we get the following 10 note scales, in their 99, 118 and 130-et versions: sa99 := [0, 16, 26, 32, 42, 48, 58, 64, 74, 90] sa118 := [0, 19, 31, 38, 50, 57, 69, 76, 88, 107] sa130 := [0, 21, 34, 42, 55, 63, 76, 84, 97, 118] We also get these twelve note scales: sb99 := [0, 7, 16, 26, 32, 42, 48, 58, 64, 74, 80, 90] sb118 := [0, 8, 19, 31, 38, 50, 57, 69, 76, 88, 95, 107] sb130 := [0, 9, 21, 34, 42, 55, 63, 76, 84, 97, 105, 118] The ten-note scales have 20 consonant intervals and 10 triads, while the twelve-note scales have 29 intervals and 20 triads, all in the 7-limit.

Message: 4931 - Contents - Hide Contents Date: Fri, 31 May 2002 12:16:37 Subject: A 25-note, 2401/2400 scale From: genewardsmith If you think a scale can be so large, at least. I went to this many notes not because a sufficient amount of harmony does not appear until then, but because the smaller rectangles I looked at had rather uneven step sizes. A closer look involving non-rectangular scales might be interesting. This one is a 5x5 square, with generators 7/5 and 60/49, in the 612-et version: [0, 25, 43, 68, 86, 104, 143, 161, 179, 204, 222, 279, 297, 322, 340, 358, 383, 401, 458, 476, 501, 519, 537, 576, 594] It has 91 intervals and 90 triads in the 7-limit, and 110 intervals and 149 triads in the 9-limit. A comparison to Blackjack, Canasta, and 27-note Ennealimmal might be in order; of course the tuning is of Ennealimmal-style exactness, much better than Miracle.

Message: 4932 - Contents - Hide Contents Date: Fri, 31 May 2002 08:42:04 Subject: Re: Two 3136/3125 planar scales From: Carl Lumma>sa99 := [0, 16, 26, 32, 42, 48, 58, 64, 74, 90] >sa118 := [0, 19, 31, 38, 50, 57, 69, 76, 88, 107] >sa130 := [0, 21, 34, 42, 55, 63, 76, 84, 97, 118] /.../ >The ten-note scales have 20 consonant intervals and 10 triads, while >the twelve-note scales have 29 intervals and 20 triads, all in the >7-limit.This is so cool. -Carl

Message: 4933 - Contents - Hide Contents Date: Fri, 31 May 2002 08:43:49 Subject: Re: A 25-note, 2401/2400 scale From: Carl Lumma>If you think a scale can be so large, at least.I think melodic material can be subsetted on the fly with excellent results. -Carl

Message: 4934 - Contents - Hide Contents Date: Fri, 31 May 2002 08:45:55 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: Carl Lumma>we get in the 72-et version a 9595959597 pattern, or the scale >[0,9,14,23,28,37,42,51,56,65]. This has the following number of >consonant intervals and triads at these odd limits: > >7: 24, 16 >9: 24, 16 >11: 34, 46 > >Since I don't keep very good track of scales, I wonder if Carl or >Paul can tell us if they've seen this one before?That's a mode of this scale: 10-tone scale, e=24 c=4, in 72-tET (0 5 14 19 28 33 42 49 58 63) Connectivity seems so good, I'm not sure why we're not using it more often. Gene, can we get _tetrads_ for 7 and greater limits from now on? Triadic music is such a bore. -Carl

Message: 4935 - Contents - Hide Contents Date: Fri, 31 May 2002 17:06:11 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote [#4363]:> Good to hear from you George. > > I'm sorry I don't have time to respond to your latest posts at the > moment, but ... > > Here's a spreadsheet and chart that I have found very illuminating > with regard to notating ETs. It should be self-explanatory, except for > the mnemonic value of the markers chosen for the chart. > > Red is for left flags, Green is for right. Lighter shades are the > less favoured comma interpretations. Concave are Xs, Convex are Os, > Straight are triangles, Wavy are horizontal dashes. > > Yahoo groups: /tuning- * [with cont.] math/files/Dave/ETsByBestFifth.xls.zipPerhaps this could be taken a step further by taking the difference between the ET flag values and the rational flag values to get a deviation for each flag in each ET. Would low deviations then identify the most suitable flags for notating those ET's?> Also, it seems you may not yet have discovered this post of mine > Yahoo groups: /tuning-math/message/4298 * [with cont.]I did see it, but I figured that I had my hands full replying to your preceding posting. One thing that puzzles me is that you show ~|) for 2deg96. In one of your previous postings (#4297) you said the following: << The only maximal 1,3,9-consistent 19-limit set for 96-ET is {1, 3, 5, 9, 11, 13, 15, 17}. It is not 1,3,7-consistent so the |) flag should be defined as the 13-5 comma (64:65) if it's used at all. The 17 and 19 commas vanish, so we should avoid )| |( ~| and |~. So I end up with 96: /| |) /|) /|\ /|| ||) /||) /||\ >> So now you would *not* avoid it? Wait a minute! It just hit me that ~|) means ~|x -- *alternate* |x or 13-5 comma -- not w|x. We're getting tripped up by double meanings for single characters (for more fun, try using a symbol with a convex flag at the end of a parenthetical statement, such as (|). (That last symbol was meant to be x|, not x|x.) I think that your ASCII notation is okay; we'll just have to be more careful with it, perhaps employing [, ], and ' in place of (, ), and ~ in certain instances. I have my doubts about the merits of defining |) as the 13-5 comma. This would be of value only if it gives you a different number of degrees than defining it as the 7 comma (in 96 both are 2deg). (Otherwise, it is just an academic exercise.) Where would you use it under these circumstances? Yes, you did use it in 111-ET, but I got the same result by defining (| as the 11'-7 comma (which I discussed in posting #4346). Since I have said this much, I might as well address the rest of #4298. We do agree on the notation for 84-ET. We also agree on the symbols for 108-ET as far as you take them; I don't know exactly what you would do in the second half-apotome. I did give a solution in posting #4346. Our solutions for 132-ET are different only for 2deg. I used /| because I wanted that for any division in which 80:81 does not vanish; you avoided that, evidently because it didn't fit into the curve in your diagram. I believe that I would take a cue from the necessity to omit 120-ET to establish an upper boundary for strict adherence to the pattern that was indicated. [DK, #4298:] << In general, complement symbols are a pain in the posterior, and I'll leave it for you to wrestle with them. I'm starting to think that the only way to make them work is to make the second half- apotome the mirror image of the first, (with the addition of a second shaft to each symbol). >> I need to ask what you mean by "mirror image." Which of these would exemplify this: /| |) /|\ /|| ||) /||\ or /| |) /|\ (|| ||\ /||\ ? I think that it would be the first one, for which I would use the term "matching sequence" rather than "mirror image." I sense a little frustration in your statement, perhaps because I have seemed to be a bit capricious with the employment of the ||\ symbol while otherwise trying to achieve matched symbol sequences in the half-apotomes. I tried to address that in message #4346, a portion of which I will repeat here: << [DK:] > I agree, but how come you didn't want s|s for 1deg22? If you did that, then you wouldn't have the comma-up /| /|| and comma- down \! \!! symbols that are one of the principal features of this notation; this is something that I would want to have in every ET in which 80:81 does not vanish, even if that doesn't result in a completely matched sequence of symbols in the half-apotomes. I believe the matched sequence is more imporant once the number of tones gets above 100, by which point /| and |\ are usually a different number of degrees. Also, with the apotome divided into fewer than 5 parts, I would want to use /|\ only when it is exactly half of /||\. In essence, what I am proposing here is that, for the lower-numbered ET's, we should place a higher priority on the use of rational complements than on a matching sequence of symbols. (Note that virtually everything that we agree on below [i.e., 22, 50, 34, 41, 46, 53] follows this principle.) >> In borderline cases (around 100 tones), where the symbols start to get more numerous, but /| and |\ are not different numbers of degrees, I have tried to capitalize on that equivalence by having matching sequences for everything, *except* that the straight flag is laterally mirrored, for example: 87, 94: ~| /| ~|\ /|\ (|) ~|| ||\ ~||\ /||\ 99: |~ /| /|~ /|) (|~ (|\ ||~ ||\ /||~ /||\ 128 (64 ss.): )| ~| /| )|\ ~|\ /|\ (|) )|| ~|| ||\ )||\ ~||\ /||\ 135 (45 ss.): |( |~ /| (| /|~ /|\ (|) ||( ||~ ||\ (|| /||~ /||\ (The above contains a correction of what I gave near the end of #4346, in which my symbol for 3deg128 was given as ~|\, which was in error.) Notice that in the notation for 87 and 94 we can use ~| for 1deg (the simplest flag choice) and still retain some logic in the use of ~|\ as the 3deg symbol as a consequence of the equivalence of /| and |\. I hope that this clarifies how I have attempted to "wrestle" with these. --George

Message: 4937 - Contents - Hide Contents Date: Fri, 31 May 2002 02:08:18 Subject: Re: A 7-limit best list From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> You haven't addressed my question: Why complexity^2 * error now, > when>> you used complexity^3 * error for 5-limit? >> the reason is log-flat badness, as usual!I really don't think so!> the formula depends on the > number of independent intervals you're trying to approximation -- > gene gave us the general formula, derived from diophantine > approximation theory, a while back.Yes. And I thought it involved fractional powers of complexity, and they didn't decrease by one for every additional prime. What will we have for 13-limit? complexity^0 ? And at the 17-limit will complexity suddenly become a good thing!?

Message: 4938 - Contents - Hide Contents Date: Fri, 31 May 2002 02:34:51 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau Good to hear from you George. I'm sorry I don't have time to respond to your latest posts at the moment, but ... Here's a spreadsheet and chart that I have found very illuminating with regard to notating ETs. It should be self-explanatory, except for the mnemonic value of the markers chosen for the chart. Red is for left flags, Green is for right. Lighter shades are the less favoured comma interpretations. Concave are Xs, Convex are Os, Straight are triangles, Wavy are horizontal dashes. Yahoo groups: /tuning-math/files/Dave/ETsByBes... * [with cont.] s.zip Also, it seems you may not yet have discovered this post of mine Yahoo groups: /tuning-math/message/4298 * [with cont.] Regards, -- Dave Keenan

Message: 4939 - Contents - Hide Contents Date: Fri, 31 May 2002 04:45:07 Subject: Re: hey gene! From: genewardsmith --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote: i remember you posted a bunch> of such examples a while back. however, i couldn't find them -- i > searched for "miracle-meantone" and "meantone-miracle" and found > nothing. do you remember doing this? (i know i'm not dreaming) . . .I did planar temperaments, and I did the Miracle-Magic system, which is <5/4, 16/15> and therefore one of the two optimal choices I just listed for 225/224. I'd need to seach more to find any other cross-temperment examples, except for the alternatives for 225/224 I dicussed. Messages 29901, 29987, 30017 and 30070 are what I found.

Message: 4940 - Contents - Hide Contents Date: Fri, 31 May 2002 04:47:13 Subject: Re: A 7-limit best list From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> You haven't addressed my question: Why complexity^2 * error now, when > you used complexity^3 * error for 5-limit?We've been over that--it's what is needed to make things log-flat.

Message: 4941 - Contents - Hide Contents Date: Sat, 01 Jun 2002 07:38:59 Subject: Re: A 7-limit best list From: dkeenanuqnetau Ok. I found it. I last asked questions about this in Yahoo groups: /tuning-math/message/1836 * [with cont.] It seems the sequence of exponents must be (n+1)/(n-1) where n is the number of odd primes being approximated. Reading the thread following this post, It looks like I either failed to understand Gene's answer to my question and gave up on empirically verifying the claims, or got distracted by something else. In Yahoo groups: /tuning-math/message/1853 * [with cont.] Gene wrote: "When you measure the size of an et n by log(n), and are at the critical exponent, the ets less than a certain fixed badness are evenly distributed on average; if you plotted numbers of ets less than the limit up to n versus log(n), it should be a rough line. If you go over the critical exponent, you should get a finite list. If you go under, it is weighted in favor of large ets, in terms of the log of the size." I think I understand this now, and could test it empirically given a big enough list of 7-limit linear temperaments with no additional cutoffs apart from badness. Correct me if I'm wrong, but I could do it by sorting them into bins according to complexity, where each bin corresponds to a doubling of complexity. i.e. bin 0 contains all those with complexity between 1 and 2, bin 1 has all those with complexity between 2 and 4, bin 2 between 4 and 8, and so on. I should find roughly an equal number of temperaments in every bin. As Paul observed, this is bound to fail for the lowest bins, and indeed there could be bin -1 having complexities between 0.5 and 1, and so on for increasingly negative bins, which will eventually all be empty. It also seems (from the to and fro between Gene and Paul in that thread) that the only justifications for using _log_-flat (and not something stronger) are that (a) it's easy to deal with mathematically, and (b) Gene likes it. Sorry to be difficult about this.

Message: 4942 - Contents - Hide Contents Date: Sat, 01 Jun 2002 07:40:27 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Gene, can we get _tetrads_ for 7 and greater limits from now on? > Triadic music is such a bore.I'd have to write some code, and the counting operation would take some time. Triads are much easier since they can be computed using the characteristic polynomial of the graph.

Message: 4943 - Contents - Hide Contents Date: Sat, 01 Jun 2002 08:04:40 Subject: Re: A 7-limit best list From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> It seems Gene is the only one who understands this derivation. I don't > feel very comfortable with this when we are putting so much store by > it, and against our own experience and what we've heard of others > experience with actually playing and listening. What if Gene has made > a mistake either in the derivation or in its relevance? We all make > mistakes.I don't see we are setting much store by it, and the fact is it seems to work, giving us an infinite list of possibilities, but one which does not choke us when we go to higher complexities. The only bad thing about it is that it might not be a good idea to stick the derivation into a paper.> So what _is_ the pattern. I guess you don't know either, or you would > have said.It's (n+1)/(n-1), where n is the number of odd primes; so for the 7-limit, n=3 and (3+1)/(3-1)=2.> Gene, > > We probably have gone over it before (although I think that was for > ETs, not linear temperaments), but I don't remember ^3 for 5-limit > and ^2 for 7-limit, I remember (improper)fractional powers.Yikes, I thought we went over it endlessly.

Message: 4944 - Contents - Hide Contents Date: Sat, 01 Jun 2002 08:51:21 Subject: Re: A 7-limit best list From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> It also seems (from the to and fro between Gene and Paul in that > thread) that the only justifications for using _log_-flat (and not > something stronger) are that > (a) it's easy to deal with mathematically, and > (b) Gene likes it.(a) It has a rational basis; what else does? (b) I tried g^3, leading to the grooviest 7-limit thread. I thought it showed a decided bias in favor of low complexities. (c) It works.

Message: 4945 - Contents - Hide Contents Date: Sat, 01 Jun 2002 09:06:57 Subject: Re: Two 3136/3125 planar scales From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> We also get these twelve note scales: > > sb99 := [0, 7, 16, 26, 32, 42, 48, 58, 64, 74, 80, 90] > sb118 := [0, 8, 19, 31, 38, 50, 57, 69, 76, 88, 95, 107] > sb130 := [0, 9, 21, 34, 42, 55, 63, 76, 84, 97, 105, 118] > The ten-note scales have 20 consonant intervals and 10 triads, while the twelve-note scales have 29 intervals and 20 triads, all in the > 7-limit.The twelve-note scale is also h12-epimorphic.

Message: 4946 - Contents - Hide Contents Date: Sat, 01 Jun 2002 09:14:14 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: genewardsmith --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> That's a mode of this scale: > > 10-tone scale, e=24 c=4, in 72-tET > (0 5 14 19 28 33 42 49 58 63) > > Connectivity seems so good, I'm not sure why we're not using it > more often.It's also h10-epimorphic. This is clearly an important scale, and needs a name. Does Joe know about it?

Message: 4947 - Contents - Hide Contents Date: Sat, 1 Jun 2002 05:59:23 Subject: Some 2401/2400 tempered 21-note blocks From: Gene W Smith It seemed to me that if I was going to enforce the epimorphic property, I might as well go all the way to blocks; this also allows one to dispense with the optimized bases. Here are some blocks, both in the JI form and with 2401/2400 tempered out by going to the 612-et. The two commas listed are those aside from 2401/2400 for the block. 405/392 36/35 [1, 21/20, 15/14, 49/45, 8/7, 7/6, 60/49, 9/7, 343/270, 4/3, 7/5, 10/7, 3/2, 540/343, 14/9, 49/30, 12/7, 7/4, 90/49, 28/15, 40/21, 2] [0, 43, 61, 75, 118, 136, 179, 222, 211, 254, 297, 315, 358, 401, 390, 433, 476, 494, 537, 551, 569]; 5: 25, 8 7: 61, 48 9: 86, 115 36/35 225/224 [1, 49/48, 15/14, 49/45, 8/7, 7/6, 60/49, 5/4, 98/75, 4/3, 7/5, 10/7, 3/2, 75/49, 8/5, 49/30, 12/7, 7/4, 90/49, 28/15, 96/49, 2] [0, 18, 61, 75, 118, 136, 179, 197, 236, 254, 297, 315, 358, 376, 415, 433, 476, 494, 537, 551, 594] 5: 29, 12 7: 69, 64 9: 78, 87 36/35 128/125 [1, 49/48, 15/14, 28/25, 8/7, 7/6, 60/49, 5/4, 98/75, 48/35, 7/5, 10/7, 35/24, 75/49, 8/5, 49/30, 12/7, 7/4, 25/14, 28/15, 96/49, 2] [0, 18, 61, 100, 118, 136, 179, 197, 236, 279, 297, 315, 333, 376, 415, 433, 476, 494, 512, 551, 594] 5: 29, 12 7: 71, 66 9: 71, 66 405/392 225/224 [1, 686/675, 15/14, 49/45, 8/7, 7/6, 60/49, 56/45, 21/16, 4/3, 7/5, 10/7, 3/2, 32/21, 45/28, 49/30, 12/7, 7/4, 90/49, 28/15, 675/343, 2] [0, 14, 61, 75, 118, 136, 179, 193, 240, 254, 297, 315, 358, 372, 419, 433, 476, 494, 537, 551, 598] 5: 21, 4 7: 51, 32 9: 64, 57 128/125 225/224 [1, 49/48, 15/14, 35/32, 8/7, 7/6, 60/49, 5/4, 98/75, 75/56, 7/5, 10/7, 112/75, 75/49, 8/5, 49/30, 12/7, 7/4, 64/35, 28/15, 96/49, 2] [0, 18, 61, 79, 118, 136, 179, 197, 236, 258, 297, 315, 354, 376, 415, 433, 476, 494, 533, 551, 594] 5: 25, 8 7: 63, 52 9: 63, 52

Message: 4948 - Contents - Hide Contents Date: Sat, 01 Jun 2002 00:01:08 Subject: Re: A 7-limit best list From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote: >>>>> gene gave us the general formula, derived from diophantine >>> approximation theory, a while back. >>>> Yes. And I thought it involved fractional powers of complexity, >> in general, yes.It seems Gene is the only one who understands this derivation. I don't feel very comfortable with this when we are putting so much store by it, and against our own experience and what we've heard of others experience with actually playing and listening. What if Gene has made a mistake either in the derivation or in its relevance? We all make mistakes.>> and >> they didn't decrease by one for every additional prime. >> only in this one instance, linear temperament, 5-limit to 7-limit. >>> What will we >> have for 13-limit? complexity^0 ? >> you're extrapolating the pattern as if it were linear in that > exponent. it isn't.So what _is_ the pattern. I guess you don't know either, or you would have said. Gene, We probably have gone over it before (although I think that was for ETs, not linear temperaments), but I don't remember ^3 for 5-limit and ^2 for 7-limit, I remember (improper)fractional powers. Can you please point me to the earlier posts where you went over it? Or explain it again? Or at least tell us what function generates the sequence of exponents.

Message: 4949 - Contents - Hide Contents Date: Sat, 01 Jun 2002 09:33:36 Subject: Re: A 1029/1024 (385/384) planar temperament scale From: Carl Lumma>> >ene, can we get _tetrads_ for 7 and greater limits from now on? >> Triadic music is such a bore. >>I'd have to write some code, and the counting operation would take some >time. Triads are much easier since they can be computed using the >characteristic polynomial of the graph.Eh? The biggest scales I'm interested in at the moment have 10 notes. That means at most 210 tetrads to test. -Carl

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