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Message: 4325 - Contents - Hide Contents Date: Tue, 19 Mar 2002 08:42:02 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> So Gene, > > How's the 5-limit list coming along.Here's what I did for weighting: The rms weights could be regarded as factors of 3^(-1/2) times each of the errors of 3,5, and 5/3; when squared this becomes dividing the sum of squares by three. The new proposed weights were 1/ln(3), 1/ln(5) and 1/ln(5), suggesting we adjust things by a factor of 3*3^(-1/2)/(ln(3)^-2 + 2 ln(5)^-2). That's the origin of the mysterious weight factor. What do you think should be done?

Message: 4327 - Contents - Hide Contents Date: Wed, 20 Mar 2002 19:56:47 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> [Secor]>>> That's something that I don't like about the Sims notation -- down >>> arrows used in conjunction with sharps, and up arrows with flats. > > [Keenan]>> I think Manuel exempts sharps and flats from this criticism. >> Yes indeed, for example, Eb/ is always the nearest tone to 6/5 > as E\ is always nearest to 5/4. > > ManuelMy objection is to alterations used in conjunction with sharps and flats that alter in the opposite direction of the sharp or flat by something approaching half of a sharp or flat. For example, 3/7 or 4/7 is much clearer than 1 minus 4/7 or 1 minus 3/7, but I would not object to 1 minus 2/7 (instead of 5/7). I have been dealing with this issue in evaluating ways to notate ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded that the alterations for these should not be in the opposite direction from an associated sharp or flat. In other words, relative to C, I would prefer to see these as varieties of E-semiflat rather than E- flat with varieties of semisharps. But (for other intervals) something no larger than 2 Didymus commas (~43 cents or ~3/8 apotome) altering in the opposite direction would be okay with me. --George

Message: 4328 - Contents - Hide Contents Date: Wed, 20 Mar 2002 20:40:34 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> The comma is 4095:4096 (~0.423 cents). ... > >> It does not vanish in 19, 27, >> 50, 58, 72, 149, 159, or 198, and this seems to be due to >> inconsistencies in those ET's. >> No. Every one of these (except I didn't check 198) is > {1,3,5,7,9,13}-consistent. >>> For those systems under 100 in which >> it does not vanish, I don't think that ratios of 13 will be used to >> define their notation, so this should not be a problem. >> Well Gene and I are already using the 13-comma (1024:1053) to notate > 27-tET, and it looks like it would be pretty useful in 50-tET too. > There are many others under 100-tET where 4095:4096 doesn't vanish. > 37-tET is another such, where I was planning to use the 13-comma for > notation. 37-tET is {1,3,5,7,13} consistent.I spent some time wrestling with 27-ET last night, and it proved to be a formidable opponent that severely limited my options. There is one approach that allows me to do it justice (using 13 -- what else is there?) that also takes the following into account: Yahoo groups: /tuning-math/message/3768 * [with cont.] With this it looks as if I am going to be stopping at the 17 limit, with intervals measurable in degrees of 183-ET. Once I have made a final decision regarding the symbols, I hope to have something to show you in about a week or so. --George

Message: 4329 - Contents - Hide Contents Date: Wed, 20 Mar 2002 03:34:36 Subject: Re: A common notation for JI and ETs From: genewardsmith --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> Good work. Did you (or can you) prove that the smallest on each of John's > lists is the smallest there is?Baker's theorem would give an effective bound, but not a good one. Proving that the seemingly smallest one is in fact the smallest one looks to me, as a number theorist, to be a difficult problem in number theory, though certainly not hard in the 3-limit. Maybe I should try showing 81/80 is the smallest in the 5-limit.

Message: 4330 - Contents - Hide Contents Date: Wed, 20 Mar 2002 21:38:04 Subject: Re: Gould Article From: paulerlich --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:> It appears that I need to join Tuning Math, as for some unknown reason, > pleople have mentioned a certain article.unknown reason . . . well, you asked on the tuning list for reactions to your article, and since most of the generalizing-diatonicity work goes on over here, it's not too surprising that it first got mentioned here. if you're interested in generalizing diatonicity to higher harmonic limits, you're in the right place. hope you will participate and be patient during the weeks/months/years it often takes people to learn one another's languages.

Message: 4331 - Contents - Hide Contents Date: Wed, 20 Mar 2002 04:50:04 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: >>> Good work. Did you (or can you) prove that the smallest on each of John's >> lists is the smallest there is? >> Baker's theorem would give an effective bound, but not a good one. > Proving that the seemingly smallest one is in fact the smallest one > looks to me, as a number theorist, to be a difficult problem innumber theory, though certainly not hard in the 3-limit. Maybe I should try showing 81/80 is the smallest in the 5-limit.>If it looks hard, forget it. I'm sure you've got better things to do with your time.

Message: 4332 - Contents - Hide Contents Date: Wed, 20 Mar 2002 07:01:15 Subject: Re: A common notation for JI and ETs From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> If it looks hard, forget it. I'm sure you've got better things to do > with your time.If I *could* prove it, it might be enough for a paper.

Message: 4333 - Contents - Hide Contents Date: Thu, 21 Mar 2002 22:51:59 Subject: Re: Hemithirds no 3s From: Herman Miller On Thu, 21 Mar 2002 08:47:09 -0000, "genewardsmith" <genewardsmith@xxxx.xxx> wrote:>If anyone wants to experiment with a well-in-tune, 12 or 13 or even 6 note scale, they might try septimal hemithirds, with 3136/3125 >as its comma. The deal is that the third (of course) takes two steps, and the 7/4 five steps. A comparable no 5s system would be no 5s >miracle, with 1029/1024 as a comma, I suppose.Hmm, it seems to be compatible with 37-ET as well. I'm really going to have to start playing around with 37, if I can find the time; it looks like it has some good commas.

Message: 4334 - Contents - Hide Contents Date: Thu, 21 Mar 2002 23:09:29 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: Herman Miller On Thu, 21 Mar 2002 20:47:00 -0000, "paulerlich" <paul@xxxxxxxxxxxxx.xxx> wrote:>--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >> 16875/16384 >>>> map [[0, -4, 3], [1, 2, 2]] >> >> generators 126.2382718 1200 >> >> badness 716.2095928 rms 5.942562596 weighted_g 4.939565964 >>someone tell me why i should care about this one.It could have melodic uses in 19-ET, dividing major thirds into three equal parts. A 9-note scale is a pretty useful size for a basic scale. C Db D# E F F# G Ab A# B C

Message: 4335 - Contents - Hide Contents Date: Thu, 21 Mar 2002 00:43:32 Subject: Re: Gould Article From: dkeenanuqnetau --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:> It appears that I need to join Tuning Math, as for some unknown reason, > pleople have mentioned a certain article.Paul, I think Mark is being facetious here. :-)> Please feel free to lambast/berate/vilify etc, for any inaccuracies in the > theory within. Hi Mark,It seems we must first berate you for failing to remove the quote of an entire tuning-math digest from the end of your post. :-) Regards, -- Dave Keenan

Message: 4336 - Contents - Hide Contents Date: Thu, 21 Mar 2002 01:20:16 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> My objection is to alterations used in conjunction with sharps and > flats that alter in the opposite direction of the sharp or flat by > something approaching half of a sharp or flat. For example, 3/7 or > 4/7 is much clearer than 1 minus 4/7 or 1 minus 3/7, but I would not > object to 1 minus 2/7 (instead of 5/7).Aha! I'm glad you've clarified that. I assume then that you would just barely allow 1 minus 1/3, since that's closer to 2/7 than 3/7.> I have been dealing with this issue in evaluating ways to notate > ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded that > the alterations for these should not be in the opposite direction > from an associated sharp or flat. In other words, relative to C, I > would prefer to see these as varieties of E-semiflat rather than E- > flat with varieties of semisharps. But (for other intervals) > something no larger than 2 Didymus commas (~43 cents or ~3/8 apotome) > altering in the opposite direction would be okay with me.I totally agree, when the 242:243 or 507:512 vanishes (as in many ETs), but I don't see how it is possible when notating strict ratios. If two of the above come out as E] and E}, then the other two must be Eb{ and Eb[, where ] and } represent an increase, and { and [ a decrease, by the 11 and 13 comma respectively. Also which ever have no sharp or flat from F,C or G _will_ have a sharp or flat from A, E or B and vice versa.

Message: 4337 - Contents - Hide Contents Date: Thu, 21 Mar 2002 02:05:08 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> I spent some time wrestling with 27-ET last night, and it proved to > be a formidable opponent that severely limited my options. There is > one approach that allows me to do it justice (using 13 -- what else > is there?) ...The only other option I could see was to notate it as every fourth note of 108-ET (= 9*12-ET), using a trinary notation where the 5-comma is one step, the 7-comma is 3 steps, and the apotome is 9 steps, but that would be to deny that it has a (just barely) usable fifth of its own.> With this it looks as if I am going to be stopping at the 17 limit,This might be made to work for ETs, but not JI. The 16:19:24 minor triad has a following.> with intervals measurable in degrees of 183-ET.I don't understand how this can work.> Once I have made a > final decision regarding the symbols, I hope to have something to > show you in about a week or so.I'm more interested in the sematics than the symbols at this stage. I wouldn't spend too much time on the symbols yet. I expect serious problems with the semantics.

Message: 4338 - Contents - Hide Contents Date: Thu, 21 Mar 2002 05:14:12 Subject: 25 best weighted generator steps 5-limit temperaments From: genewardsmith Yet another "best" list. This one uses weights of 1/ln(3), 1/ln(5) and 1/ln(5) times the steps to 3, 5, and 5/3. To get it to more or less correspond to the previous g, I adjusted by a factor of (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 sqrt(1/ln(5)) This isn't the only adjustment you might think of, so your milage may vary. I did a search for badness (adjusted) less than 750, rms less than 35 (as suggested by Dave) and adjusted g less than 20. Here's the result: 135/128 map [[0, -1, 3], [1, 2, 1]] generators 522.8623453 1200 badness 347.2957065 rms 18.07773298 weighted_g 2.678254330 256/243 map [[0, 0, 1], [5, 8, 10]] generators 395.3362123 240 badness 612.7594631 rms 12.75974156 weighted_g 3.634818387 25/24 map [[0, 2, 1], [1, 1, 2]] generators 350.9775007 1200 badness 134.9517606 rms 28.85189698 weighted_g 1.672379084 648/625 map [[0, -1, -1], [4, 8, 11]] generators 505.8656425 300 badness 536.5364242 rms 11.06006024 weighted_g 3.647096372 16875/16384 map [[0, -4, 3], [1, 2, 2]] generators 126.2382718 1200 badness 716.2095928 rms 5.942562596 weighted_g 4.939565964 250/243 map [[0, -3, -5], [1, 2, 3]] generators 162.9960265 1200 badness 363.8269146 rms 7.975800816 weighted_g 3.573058894 128/125 map [[0, -1, 0], [3, 6, 7]] generators 491.2018553 400 badness 198.0597092 rms 9.677665980 weighted_g 2.735322279 3125/3072 map [[0, 5, 1], [1, 0, 2]] generators 379.9679494 1200 badness 368.6051706 rms 4.569472316 weighted_g 4.320809550 20000/19683 map [[0, 4, 9], [1, 1, 1]] generators 176.2822703 1200 badness 565.4935338 rms 2.504205191 weighted_g 6.089559954 81/80 map [[0, -1, -4], [1, 2, 4]] generators 503.8351546 1200 badness 81.02783490 rms 4.217730124 weighted_g 2.678254330 2048/2025 map [[0, -1, 2], [2, 4, 3]] generators 494.5534684 600 badness 167.3579339 rms 2.612822498 weighted_g 4.001094414 78732/78125 map [[0, 7, 9], [1, -1, -1]] generators 442.9792974 1200 badness 412.2839142 rms 1.157498146 weighted_g 7.088571030 393216/390625 map [[0, 8, 1], [1, -1, 2]] generators 387.8196732 1200 badness 373.3878885 rms 1.071950166 weighted_g 7.036043951 2109375/2097152 map [[0, 7, -3], [1, 0, 3]] generators 271.5895996 1200 badness 340.7350960 rms .8004099292 weighted_g 7.522602841 4294967296/4271484375 map [[0, -9, 7], [1, 2, 2]] generators 55.27549315 1200 badness 687.5758107 rms .4831084292 weighted_g 11.24843211 15625/15552 map [[0, 6, 5], [1, 0, 1]] generators 317.0796754 1200 badness 146.7501955 rms 1.029625097 weighted_g 5.223559222 1600000/1594323 map [[0, -5, -13], [1, 3, 6]] generators 339.5088258 1200 badness 252.5491335 rms .3831037874 weighted_g 8.703150208 1224440064/1220703125 map [[0, -13, -14], [1, 5, 6]] generators 315.2509133 1200 badness 497.4863561 rms .2766026501 weighted_g 12.16115842 10485760000/10460353203 map [[0, 4, 21], [1, 0, -6]] generators 475.5422335 1200 badness 441.3527601 rms .1537673823 weighted_g 14.21152037 6115295232/6103515625 map [[0, 7, 3], [2, -3, 2]] generators 528.8539366 600 badness 313.0746567 rms .1940181460 weighted_g 11.72920349 19073486328125/19042491875328 map [[0, 1, 1], [19, 24, 38]] generators 386.2396202 1200/19 badness 544.7739924 rms .1047837215 weighted_g 17.32370777 32805/32768 map [[0, -1, 8], [1, 2, -1]] generators 498.2724869 1200 badness 39.20134011 rms .1616904714 weighted_g 6.235512615 274877906944/274658203125 map [[0, -15, 2], [1, 4, 2]] generators 193.1996149 1200 badness 178.5465796 rms .6082244804e-1 weighted_g 14.31844579 7629394531250/7625597484987 map [[0, -2, -3], [9, 19, 28]] generators 315.6754868 400/3 badness 203.0321497 rms .2559261582e-1 weighted_g 19.94420419 9010162353515625/9007199254740992 map [[0, -8, 5], [2, 9, 1]] generators 437.2581077 600 badness 116.1747349 rms .1772520822e-1 weighted_g 18.71429401

Message: 4339 - Contents - Hide Contents Date: Thu, 21 Mar 2002 08:47:09 Subject: Hemithirds no 3s From: genewardsmith If anyone wants to experiment with a well-in-tune, 12 or 13 or even 6 note scale, they might try septimal hemithirds, with 3136/3125 as its comma. The deal is that the third (of course) takes two steps, and the 7/4 five steps. A comparable no 5s system would be no 5s miracle, with 1029/1024 as a comma, I suppose.

Message: 4340 - Contents - Hide Contents Date: Thu, 21 Mar 2002 12:31 +0 Subject: Re: Gould Article From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <B8BE8615.3860%mark.gould@xxxxxxx.xx.xx> Mark Gould wrote:> It appears that I need to join Tuning Math, as for some unknown reason, > pleople have mentioned a certain article.It's an article on tuning and mathematics, isn't it? That suggests it's fairly relevant.> Please feel free to lambast/berate/vilify etc, for any inaccuracies in > the > theory within.As you're being negative, I'll suggest your use of "generated scale" is wrong, or at least inconsistent with Carey & Clampitt. You say something like generated scales will only have two step sizes, which is the extra criterion that makes a generated scale MOS or WF. I don't understand either of the second criteria you use for choosing diatonics. (That is the second numbered point on both the lists.) Can you clarify them? Graham

Message: 4341 - Contents - Hide Contents Date: Thu, 21 Mar 2002 20:47:00 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Yet another "best" list. This one uses weights of 1/ln(3), 1/ln(5) and > 1/ln(5) times the steps to 3, 5, and 5/3. To get it to more or lesscorrespond to the previous g, I adjusted by a factor of> > (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 sqrt(1/ln (5))thanks for doing this, gene.> This isn't the only adjustment you might think of, so your milagemay vary. I did a search for badness (adjusted) less than 750, rms less> than 35 (as suggested by Dave) and adjusted g less than 20. Here's the > result: > > 256/243 > > map [[0, 0, 1], [5, 8, 10]] > > generators 395.3362123 240 > > badness 612.7594631 rms 12.75974156 weighted_g 3.634818387blackwood's decatonic system. this is pretty essential, really. i can't believe we forgot all about it all this time. i'd like to see it in the paper.> 16875/16384 > > map [[0, -4, 3], [1, 2, 2]] > > generators 126.2382718 1200 > > badness 716.2095928 rms 5.942562596 weighted_g 4.939565964someone tell me why i should care about this one.> 4294967296/4271484375 > > map [[0, -9, 7], [1, 2, 2]] > > generators 55.27549315 1200 > > badness 687.5758107 rms .4831084292 weighted_g 11.24843211ditto. perhaps a weighted badness cutoff of 650, then.

Message: 4342 - Contents - Hide Contents Date: Thu, 21 Mar 2002 20:48:01 Subject: Re: Hemithirds no 3s From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> If anyone wants to experiment with a well-in-tune, 12 or 13 or even6 note scale, they might try septimal hemithirds, with 3136/3125> as its comma. The deal is that the third (of course) takes twosteps, and the 7/4 five steps. A comparable no 5s system would be no 5s> miracle, with 1029/1024 as a comma, I suppose.do we call the latter 'slendric'?

Message: 4343 - Contents - Hide Contents Date: Fri, 22 Mar 2002 02:14:10 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Yet another "best" list. This one uses weights of 1/ln(3), 1/ln(5) and > 1/ln(5) times the steps to 3, 5, and 5/3. Thanks Gene. > To get it to more or lesscorrespond to the previous g, I adjusted by a factor of> > (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 sqrt(1/ln(5)) > > This isn't the only adjustment you might think of, so your milage may vary.Indeed. I see no justification for a weighted rms (or any kind of average or mean) that can produce a result which is _outside_ the range of its inputs. Feeding it 1, 1, 1 should give you 1 no matter what the weights are.>I did a search for badness (adjusted) less than 750, rms less > than 35 (as suggested by Dave) and adjusted g less than 20. Here's the > result:The fact that you uncovered for the first time that quintuple thirds temperament (256/243), that Paul tells us has actually been used by Blackwood, is I think justification enough for using Paul's weighting of the numbers of generators to favour fifths. Your list contains all the 5-limit temperaments that I think are of some interest (except of course it is missing the degenerates). However there are five temperaments in your list that I think are of no interest whatsoever. If you were to reduce your complexity cutoff so it is between that of parakleismic and 10485760000/10460353203 we would agree on a best 20. In your terms the g value would need to be between 12.2 and 14.2. In terms of the true weighted rms complexity it needs to be between 11.7 and 13.5.

Message: 4344 - Contents - Hide Contents Date: Fri, 22 Mar 2002 21:56:02 Subject: Re: A group "best" listt From: Herman Miller On Fri, 22 Mar 2002 06:07:25 -0000, "genewardsmith" <genewardsmith@xxxx.xxx> wrote:>This was done in a way exactly comparable to the most recent 5-limit >search.A suggestion: it might be easier to recognize some of the unison vectors if you supply the prime factorization in addition to the ratio. Here's one I didn't recognize right away:>531441/524288But expressed as 2^-19 * 3^12, it's clear that this is the familiar Pythagorean comma. As many times as I've seen "531441/524288", it somehow never stuck in my memory.

Message: 4345 - Contents - Hide Contents Date: Fri, 22 Mar 2002 03:15:59 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> blackwood's decatonic system. this is pretty essential, really. i > can't believe we forgot all about it all this time. i'd like to see > it in the paper.So would I. According to my badness it rates better than septathirds (4294967296/4271484375) or parakleismic (1224440064/1220703125).> >> 16875/16384 >>>> map [[0, -4, 3], [1, 2, 2]] >> >> generators 126.2382718 1200 >> >> badness 716.2095928 rms 5.942562596 weighted_g 4.939565964 >> someone tell me why i should care about this one.I'm calling this tertiathirds (was quadrafourths). By my badness it is better than: diminished, pelogic, twin-tertiatenths (semisuper), quintuple-thirds (Blackwood's decatonic) and parakleismic.>> 4294967296/4271484375 >> >> map [[0, -9, 7], [1, 2, 2]] >> >> generators 55.27549315 1200 >> >> badness 687.5758107 rms .4831084292 weighted_g 11.24843211 > > ditto.I'm calling this septathirds. By my badness it is marginally better than parakleismic. They both have rms error less than 0.5 c but tertiathirds needs one less generator (in weghted rms terms).> perhaps a weighted badness cutoff of 650, then.That won't work for me. The only way Gene and I can agree, when he is using sharp cutoffs and I'm using gentle rolloffs, is if there doesn't happen to be anything near the 3 upper/outer corners of Gene's "acceptance brick". I'm visualising here points in a 3D space where the horizontal dimensions are error and complexity and the vertical dimension is Gene's badness. There are very few choices for Gene's badness, error and complexity cutoffs, that agree with a single badness cutoff for me. Gene's already got badness and error cutoffs that work for me, all he needs to do now is lower his complexity cutoff from 20 to 14 (or 13 if he fixes up his weighted rms calculation). And "best 20" is a nice round number. Here they are by name, in order of weighted complexity. neutral thirds meantone pelogic augmented semiminorthirds (porcupine) quintuple thirds (Blackwood's decatonic) diminished diaschismic magic tertiathirds (quadrafourths) kleismic quartafifths (minimal diesic) schismic wuerschmidt semisixths (tiny diesic) subminor thirds (orwell) quintelevenths (AMT) septathirds (4294967296/4271484375) twin tertiatenths (semisuper) parakleismic (1224440064/1220703125) The fact that no-one's written any music in, or even heard of the last four (or is it the last six?) as 5-limit temperaments until recently, should indicate that we have gone far enough in the direction of increased complexity. As usual, you can experiment with agreement between Gene's and my lists, updated with Gene's latest temperaments, in http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.] (Wayb.)

Message: 4346 - Contents - Hide Contents Date: Fri, 22 Mar 2002 04:08:48 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> Yet another "best" list. This one uses weights of 1/ln(3), 1/ln (5) > and>> 1/ln(5) times the steps to 3, 5, and 5/3. > > > Thanks Gene. >>> To get it to more or less> correspond to the previous g, I adjusted by a factor of >>>> (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 > sqrt(1/ln(5)) >>>> This isn't the only adjustment you might think of, so your milage > may vary. >> Indeed. I see no justification for a weighted rms (or any kind of > average or mean) that can produce a result which is _outside_ the > range of its inputs. Feeding it 1, 1, 1 should give you 1 no matter > what the weights are. >>> I did a search for badness (adjusted) less than 750, rms > less>> than 35 (as suggested by Dave) and adjusted g less than 20. Here's > the >> result: >> The fact that you uncovered for the first time that quintuple thirds > temperament (256/243), that Paul tells us has actually been used by > Blackwood, is I think justification enough for using Paul's weighting > of the numbers of generators to favour fifths. > > Your list contains all the 5-limit temperaments that I think are of > some interest (except of course it is missing the degenerates). > However there are five temperaments in your list that I think are of > no interest whatsoever. If you were to reduce your complexity cutoff > so it is between that of parakleismic and 10485760000/10460353203 we > would agree on a best 20. In your terms the g value would need to be > between 12.2 and 14.2. In terms of the true weighted rms complexity it > needs to be between 11.7 and 13.5.a lower complexity seems desirable now but we'll regret it next year when we're working on the 19-limit.

Message: 4347 - Contents - Hide Contents Date: Fri, 22 Mar 2002 04:13:25 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: paulerlich --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:> On Thu, 21 Mar 2002 20:47:00 -0000, "paulerlich" <paul@s...> > wrote: >>> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>> 16875/16384 >>>>>> map [[0, -4, 3], [1, 2, 2]] >>> >>> generators 126.2382718 1200 >>> >>> badness 716.2095928 rms 5.942562596 weighted_g 4.939565964 >>>> someone tell me why i should care about this one. >> It could have melodic uses in 19-ET, dividing major thirds into three equal > parts. A 9-note scale is a pretty useful size for a basic scale. > > C Db D# E F F# G Ab A# B Cdave and you both? ok, then i think we should keep this in the paper - - it's john negri's system in 19-equal (looks a little better in the 7-limit).

Message: 4348 - Contents - Hide Contents Date: Fri, 22 Mar 2002 05:43:17 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> a lower complexity seems desirable now but we'll regret it next year > when we're working on the 19-limit.I have always assumed we would allow higher complexities for higher primes. We might well be looking at stuff with up to 25 rms generators for 19-limit. But 13 is plenty for 5-limit.

Message: 4349 - Contents - Hide Contents Date: Fri, 22 Mar 2002 06:00:41 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:>> On Thu, 21 Mar 2002 20:47:00 -0000, "paulerlich" <paul@s...> >> wrote:>>> someone tell me why i should care about this one. [tertiathirds] >>>> It could have melodic uses in 19-ET, dividing major thirds into > three equal>> parts. A 9-note scale is a pretty useful size for a basic scale. >> >> C Db D# E F F# G Ab A# B C >> dave and you both? ok, then i think we should keep this in the paper - > - it's john negri's system in 19-equal (looks a little better in the > 7-limit).If you keep this one (tertiathirds) you're also gonna have to keep the other one you asked about, septathirds (4294967296/4271484375), since septathirds is better than tertiathirds by Gene's badness. But I agree that it's extremely boring melodically, being essentially 22-tET.

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