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Message: 4350 - Contents - Hide Contents Date: Fri, 22 Mar 2002 06:07:25 Subject: A group "best" listt From: genewardsmith This was done in a way exactly comparable to the most recent 5-limit search. 567/512 map [[0, -1, 4], [1, 2, 1]] generators 535.897694119756 1200. badness 670.137142647868 rms 27.2570556594034 g_w 2.90778025274862 243/224 map [[0, -1, -5], [1, 2, 5]] generators 528.248377247002 1200. badness 534.715045110858 rms 21.7489179735939 g_w 2.90778025274862 343/324 map [[0, 3, 4], [1, 1, 2]] generators 240.309212849092 1200. badness 493.563423167166 rms 19.3483699528574 g_w 2.94374053640771 256/243 map [[0, 0, -1], [5, 8, 15]] generators 222.151593963576 240. badness 378.244754205085 rms 12.7597412545895 g_w 3.09487905565715 28/27 map [[0, -1, -3], [1, 2, 4]] generators 475.558962037126 1200. badness 91.4596348840926 rms 16.8270093610171 g_w 1.75822388550304 16807/16384 map [[0, -1, 0], [5, 10, 14]] generators 502.457952369172 240. badness 550.435696708892 rms 6.24085831443580 g_w 4.45130571562582 19683/19208 map [[0, -4, -9], [1, 3, 6]] generators 425.724527133587 1200. badness 626.695380808806 rms 3.82888181257598 g_w 5.47009468947274 49/48 map [[0, -2, -1], [1, 2, 3]] generators 249.022499567306 1200. badness 67.8093075322853 rms 14.5731625037549 g_w 1.66946954361805 64/63 map [[0, -1, 2], [1, 2, 2]] generators 488.307823491718 1200. badness 39.6049568556885 rms 7.28663503411012 g_w 1.75822388550304 531441/524288 map [[0, 0, -1], [12, 19, 36]] generators 232.151593963576 100. badness 566.495842771669 rms 1.38239436914426 g_w 7.42770973357716 537824/531441 map [[0, 5, 12], [1, 0, -1]] generators 380.751294592580 1200. badness 524.893579245064 rms 1.39991667198979 g_w 7.21090363091141 65536/64827 map [[0, 4, -3], [1, 0, 4]] generators 476.124945200378 1200. badness 206.568540713651 rms 2.18910247435947 g_w 4.55266851389582 177147/175616 map [[0, -3, -11], [1, 3, 8]] generators 566.505581347493 1200. badness 281.673403211886 rms 1.07889655377572 g_w 6.39129227512076 1605632/1594323 map [[0, 2, 13], [1, 1, -1]] generators 351.476961726092 1200. badness 314.893972274088 rms .713664767134272 g_w 7.61301520051980 67108864/66706983 map [[0, -7, 4], [1, 3, 2]] generators 242.458056889570 1200. badness 316.903243699017 rms .762466103296028 g_w 7.46280420851729 13841287201/13759414272 map [[0, 3, 2], [4, 3, 9]] generators 334.046136423623 300. badness 705.581672022668 rms .686251415126464 g_w 10.0930249317794 1029/1024 map [[0, 3, -1], [1, 1, 3]] generators 233.444441296619 1200. badness 42.1870934730102 rms 1.65379250859746 g_w 2.94374053640771 118098/117649 map [[0, 3, 5], [2, 1, 2]] generators 433.782535424977 600. badness 155.357296461064 rms .534890245526000 g_w 6.62250546977042 70632088586703/70368744177664 map [[0, 13, -6], [1, -1, 4]] generators 238.589951176615 1200. badness 643.457389814532 rms .271819428385752 g_w 13.3274306917460 94450499584/94143178827 map [[0, -8, -23], [1, 2, 4]] generators 62.2228613205755 1200. badness 487.989402670565 rms .197276010332117 g_w 13.5242141244344 4760622968832/4747561509943 map [[0, 15, 19], [1, -3, -3]] generators 366.784885146694 1200. badness 579.210669002310 rms .193857182595423 g_w 14.4029532849298 34451725707/34359738368 map [[0, 4, -15], [1, 1, 5]] generators 175.423395680988 1200. badness 253.815642297510 rms .188649236250683 g_w 11.0396281212545 33554432/33480783 map [[0, -1, 14], [1, 2, -3]] generators 497.783581217437 1200. badness 135.621744048494 rms .185179785930646 g_w 9.01388298147804 282429536481/281974669312 map [[0, 5, 24], [1, 1, 0]] generators 140.366054018771 1200. badness 244.087415237322 rms .899689577077160e-1 g_w 13. 9471102220917 68719476736/68641485507 map [[0, 2, -1], [5, 6, 15]] generators 231.033669889318 240. badness 117.928158169825 rms .105083431144134 g_w 10.3918871916168 40353607/40310784 map [[0, -1, -1], [9, 18, 29]] generators 497.942879666076 133.333333333333 badness 74.2853761882641 rms .144418737385944 g_w 8.01235028812647 5559060566555523/5556003412779008 map [[0, 14, 33], [1, -3, -8]] generators 392.994636775254 1200. badness 184.673444690140 rms .234734105085574e-1 g_w 19. 8888383853365 1153470752371588581/1152921504606846976 map [[0, 5, -29], [1, 0, 12]] generators 380.385861981302 1200. badness 145.368173525748 rms .183370867638466e-1 g_w 19. 9394418821855 11399736556781568/11398895185373143 map [[0, 19, 4], [1, -7, 1]] generators 542.208280827679 1200. badness 22.2288637273498 rms .520794634506544e-2 g_w 16. 2212227246308

Message: 4351 - Contents - Hide Contents Date: Fri, 22 Mar 2002 06:17:03 Subject: Re: A group "best" listt From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> This was done in a way exactly comparable to the most recent 5-limit > search.I'll have a much better feel for this if we do the full 7-limit first. But have we agreed on the 5-limit list yet. Do you agree the weighted complexity cutoff can be reduced to 13, to give us a list of 20?

Message: 4352 - Contents - Hide Contents Date: Fri, 22 Mar 2002 10:44 +0 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a7eg9l+omel@xxxxxxx.xxx> Paul:>> a lower complexity seems desirable now but we'll regret it next year >> when we're working on the 19-limit.You don't plan on getting to the 19-limit until next year? Dave K:> 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.31&62 has an RMS complexity of 22.5 and it's the best one I've got below 25. It's got a worst error of 11 cents (or RMS of 5 cents) so I'm not sure it counts as 19-limit anything. There's also 31&72 with RMS complexity of 24.8 and 8.8 cent minimax, and 62&72 with a 23.5 complexity and 7.5 cent minimax. Graham

Message: 4353 - Contents - Hide Contents Date: Fri, 22 Mar 2002 11:49:48 Subject: A group "best" listt From: genewardsmith Here we find, of course, the 3136/3125 system I mentioned before, but I'd also like to draw attention to the last temperament on the list--a mathematically (if not practically) amazing temperament, associated to the rather amazing 789 et. 686/625 map [[0, 3, 4], [1, 1, 1]] generators 539.1060370 1200. badness 489.0718160 rms 31.61840515 g_w 2.491595620 343/320 map [[0, -3, -1], [1, 3, 3]] generators 274.0898101 1200. badness 232.0702794 rms 32.11518107 g_w 1.933316912 32768/30625 map [[0, 1, -2], [2, 4, 7]] generators 407.2232374 600. badness 643.0205101 rms 15.64725500 g_w 3.450872300 4375/4096 map [[0, 1, -4], [1, 2, 4]] generators 361.8678362 1200. badness 454.9331633 rms 17.60304342 g_w 2.956559423 2560/2401 map [[0, -4, -1], [1, 3, 3]] generators 201.2867627 1200. badness 392.3271620 rms 21.77086970 g_w 2.621747521 16807/16000 map [[0, 5, 3], [1, 1, 2]] generators 317.7111025 1200. badness 450.1531173 rms 13.81938198 g_w 3.193673737 8575/8192 map [[0, 3, -2], [1, 2, 3]] generators 123.9139044 1200. badness 328.0735082 rms 12.83254896 g_w 2.945909568 256/245 map [[0, 2, -1], [1, 2, 3]] generators 204.0189242 1200. badness 119.7408063 rms 20.32106851 g_w 1.806197400 823543/800000 map [[0, -7, -5], [1, 4, 4]] generators 287.3935752 1200. badness 536.0657103 rms 5.685477364 g_w 4.551456166 401408/390625 map [[0, 1, 4], [2, 4, 3]] generators 392.6597660 600. badness 470.1905708 rms 4.622664052 g_w 4.667960736 16807/16384 map [[0, 1, 0], [5, 10, 14]] generators 381.9007604 240. badness 281.4513766 rms 6.240857928 g_w 3.559478388 128/125 map [[0, 0, -1], [3, 7, 9]] generators 224.3309516 400. badness 69.03052049 rms 9.677665952 g_w 1.924967967 1071875/1048576 map [[0, 3, -5], [1, 1, 5]] generators 527.0889187 1200. badness 370.6464204 rms 3.843254849 g_w 4.585853840 268435456/262609375 map [[0, -5, 6], [1, 3, 2]] generators 162.0275880 1200. badness 708.1906883 rms 2.815872080 g_w 6.312175959 50/49 map [[0, -1, -1], [2, 5, 6]] generators 222.4301906 600. badness 35.69105715 rms 12.36574700 g_w 1.423791355 68359375/67108864 map [[0, 1, -10], [1, 2, 6]] generators 383.2901826 1200. badness 664.8390958 rms 2.145218464 g_w 6.767293431 4194304/4117715 map [[0, 7, -1], [1, 1, 3]] generators 226.9760748 1200. badness 445.7435327 rms 2.988171840 g_w 5.303466469 78125/76832 map [[0, -4, -7], [1, 3, 4]] generators 204.3976654 1200. badness 230.3456068 rms 3.358669391 g_w 4.093274712 40960000/40353607 map [[0, -9, -4], [1, 5, 4]] generators 357.0527370 1200. badness 439.4738808 rms 2.337786935 g_w 5.728523696 244140625/240945152 map [[0, 1, 2], [6, 12, 13]] generators 384.4129525 200. badness 514.3166613 rms 1.551964221 g_w 6.920162556 1977326743/1953125000 map [[0, 11, 12], [1, -3, -3]] generators 580.6686988 1200. badness 705.1161663 rms 1.307238490 g_w 8.140203868 2000000000/1977326743 map [[0, -11, -9], [1, 5, 5]] generators 292.2142738 1200. badness 544.3754097 rms 1.375242977 g_w 7.342427853 762939453125/755603996672 map [[0, 8, 17], [1, 1, 0]] generators 198.1639627 1200. badness 748.4465765 rms .8028678840 g_w 9.768748008 40353607/40000000 map [[0, 9, 7], [1, 0, 1]] generators 309.6535849 1200. badness 275.9060105 rms 1.316290184 g_w 5.940228799 97656250000/96889010407 map [[0, 13, 14], [1, -4, -4]] generators 583.5195442 1200. badness 623.8314928 rms .7137704924 g_w 9.560990833 69206436005/68719476736 map [[0, 12, -1], [1, 0, 3]] generators 232.1473890 1200. badness 478.2656671 rms .6898731204 g_w 8.850476649 1638400000000000/1628413597910449 map [[0, 18, 11], [1, 1, 2]] generators 88.12377826 1200. badness 726.3196762 rms .4762208496 g_w 11.51082544 823543/819200 map [[0, -7, -2], [1, 3, 3]] generators 116.2911944 1200. badness 97.72611226 rms 1.036476664 g_w 4.551456166 8589934592/8544921875 map [[0, 1, -13], [1, 2, 7]] generators 386.9847164 1200. badness 311.5353743 rms .4754474072 g_w 8.685636061 19073486328125/18990246039772 map [[0, -15, -19], [1, 8, 10]] generators 454.2650388 1200. badness 540.6271280 rms .3086102069 g_w 12.05486958 3136/3125 map [[0, 2, 5], [1, 2, 2]] generators 193.7971982 1200. badness 23.18483462 rms .9868326492 g_w 2.864091121 65712362363534280139543/65536000000000000000000 map [[0, 3, 2], [9, 10, 18]] generators 484.3391018 133.3333333 badness 727.8282850 rms .1381631578 g_w 17.39985221 186759498717153574912/186264514923095703125 map [[0, -19, -29], [1, 9, 13]] generators 421.7657192 1200. badness 669.4337469 rms .1273318109 g_w 17.38829083 7450580596923828125/7430984482854797312 map [[0, 4, 9], [3, 1, -5]] generators 596.5348276 400. badness 510.7162100 rms .1375962522 g_w 15.48309146 80000000000000000/79792266297612001 map [[0, -5, -4], [4, 19, 19]] generators 582.7533331 300. badness 406.7075158 rms .1736411704 g_w 13.28035949 591363588909912109375/590295810358705651712 map [[0, -13, 14], [1, 4, 1]] generators 154.9079648 1200. badness 354.5227097 rms .9459396236e-1 g_w 15.53317425 2100875/2097152 map [[0, 5, -3], [1, 2, 3]] generators 77.19380915 1200. badness 33.21811689 rms .3101843093 g_w 4.748812924 134217728/133984375 map [[0, -3, 8], [1, 3, 1]] generators 271.1304136 1200. badness 56.35222399 rms .2162902226 g_w 6.386918744 37778931862957161709568/37714514598846435546875 map [[0, 11, -19], [1, -3, 12]] generators 580.5834970 1200. badness 404.7327302 rms .7947342068e-1 g_w 17.20476745 22539340290692258087863249/22517998136852480000000000 map [[0, -3, -1], [10, 29, 30]] generators 231.2326670 120. badness 316.7381823 rms .4383199824e-1 g_w 19.33316912 3909821048582988049/3906250000000000000 map [[0, -22, -21], [1, 13, 13]] generators 582.4387327 1200. badness 189.2472322 rms .5198637212e-1 g_w 15.38319960 23303567338232644370432/23283064365386962890625 map [[0, 7, 16], [2, 0, -5]] generators 398.0518660 600. badness 239.3677518 rms .3878099806e-1 g_w 18.34350066 59604644775390625/59553411580724992 map [[0, -17, -24], [1, 6, 8]] generators 259.6315753 1200. badness 155.9226447 rms .4924199004e-1 g_w 14.68445611 1342177280000000000000/1341068619663964900807 map [[0, 25, 13], [1, -5, -1]] generators 351.4524874 1200. badness 187.4404574 rms .4671147988e-1 g_w 15.89082956 281484423828125/281474976710656 map [[0, 8, -11], [1, 0, 6]] generators 348.2888153 1200. badness 3.206369973 rms .2486103265e-2 g_w 10.88507660

Message: 4354 - Contents - Hide Contents Date: Fri, 22 Mar 2002 12:16:13 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <a7eg9l+omel@e...> > Paul:>>> a lower complexity seems desirable now but we'll regret it next year >>> when we're working on the 19-limit. >> You don't plan on getting to the 19-limit until next year? > > Dave K:>> 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. >> 31&62 has an RMS complexity of 22.5 and it's the best one I've got below > 25. It's got a worst error of 11 cents (or RMS of 5 cents) so I'm not > sure it counts as 19-limit anything. There's also 31&72 with RMS > complexity of 24.8 and 8.8 cent minimax, and 62&72 with a 23.5 complexity > and 7.5 cent minimax.Thanks. So we'll definitely be going _beyond_ weighted rms complexity of 25 for 19-limit. Or maybe it just says that 19-limit really isn't worth the trouble.

Message: 4355 - Contents - Hide Contents Date: Fri, 22 Mar 2002 12:23:11 Subject: Re: 25 best weighted generator steps 5-limit temperaments From: dkeenanuqnetau --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> 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)I find that I can also agree with Gene on a best 17, he just needs to change his complexity cutoff to 10 and we lose the last three off the list above, including septathirds which Paul objected to, and I agreed was boring. I would prefer this list of 17, to the list of 20. There is quite a big gap in complexity between quintelevenths and septathirds.

Message: 4356 - Contents - Hide Contents Date: Fri, 22 Mar 2002 22:16:50 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: >>>> 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.In the 17-limit approach that I outline below, you will see how I handle this. I am also outlining a 23-limit approach; I went for the 19 limit and got 23 as a bonus when I found that I could approximate it using a very small comma. The two approaches could be combined, in which case you could have the 11-13 semiflat varieties along with the 19 or 23 limit, but the symbols may get a bit complicated -- more about that below.>> 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.I thought more about this and now realize that the problem with 27-ET is not as formidable as it seemed. If we use the 80:81 comma for a single degree and the 1024:1053 comma for two degrees of alteration, we will do just fine, even if the *symbol* for 1024:1053 happens to be a combination of the 80:81 and 63:64 symbols (by conflating 4095:4096). For the 27-ET notation we can simply define the combination of the two symbols as the 13 comma alteration, and there would be no inconsistency in usage, since the 63:64 symbol is *never used by itself* in the 27-ET notation. The same could be said about 50-ET. Are there any troublesome divisions above 100 that we should be concerned about in this regard?>> 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.Yes, I can appreciate that, and there are other uses of ratios of 19 that I value. However, the 19 comma gets us down to less than 3.4 cents deviation (32/27 from 19/16), which is about the same as the minimax deviation for the Miracle tuning, which is not bad. However, I anticipate that you believe that the JI purists would still want to have this distinction, so we should go for 19.>>> with intervals measurable in degrees of 183-ET. >> I don't understand how this can work. See below.>> 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.I don't know what problems you are anticipating, but let me outline what I have, both in the way of semantics and symbols. As I said, I have both a 17-limit and 23-limit approach, although I expect that you will be interested in only the latter. Since there is not much difference between the two, it is not any trouble to give both. I have found that the semantics and symbols are so closely connected that I could not address one without the other, given the limitation of no more than one symbol in conjunction with a sharp or flat (possible in my 23-limit approach) or of no more than one symbol of any sort (accomplished in my 17-limit approach). For the sake of simplicity, I will outline only how the modifications to natural notes are accomplished by single symbols, leaving off the problem of how these are to be combined with sharps and flats for another time. Both the 17-limit and 23-limit approaches use 6 sizes of alterations. In the sagittal notation these are paired into left and right flags that are affixed to a vertical stem, to the top for upward alteration and to the bottom for downward alteration. These pairs of flags consist of straight lines, convex curved lines, and concave curved lines. With this arrangement there is a limitation that two left or two right flags cannot be used simultaneously. *17-LIMIT APPROACH* The 17-limit arrangement is: Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (3 degrees of 183) Straight right flag (sR): 54:55, ~31.8 cents (5deg183) Convex left flag (xL): 715:729 (3^6:5*11*13), ~33.6 cents (5deg183) Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (4deg183) Concave left flag (vL): 4131:4096 (3^5*17:2^12, the 17-as-flat comma), ~14.7 cents (2deg183) Concave right flag (vR): 2187:2176 (3^7:2^7*17, the 17-as-sharp comma), ~8.7 cents (1deg183) I was a bit hesitant to use two different 17-commas, but I saw that 16:17 could be used as either an augmented prime or a minor second (e.g., in a scale with a tonic triad of 14:17:21). Rather than choose between the two, I found that it is handy to have both intervals, especially when going to the 19 limit. With the above used in combination, the following useful intervals are available: sL+sR: 32:33 (the 11-as-semisharp comma), ~53.3 cents (8deg183) sL+xR: 35:36, ~48.8 cents, which approximates ~sL+xR: 1024:1053 (the 13-as-semisharp comma), ~48.3 cents (7deg183) sR+xL: 26:27 (the 13-as-semiflat comma), ~65.3 cents (10deg183) xL+xR: 5005:5184 (5*7*11*13:2^6*3^4), ~60.8 cents, which approximates ~xL+xR: 704:729 (the 11-as-semiflat comma), ~60.4 cents (9deg183) vL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates ~vL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents (6deg183) That is how I make the distinction between all the different ratios of 11 and 13. Note that ratios of 11 have like left and right flags, while ratios of 13 have dissimilar flags, making them easy to tell apart. With the single-symbol sagittal notation it is necessary to have alterations exceeding half an apotome, since there is no way to notate a (two-flag) sagittal sharp/flat *less* a (two-flag) ~semiflat/~semisharp alteration. The following combinations are probably not as useful, but are available anyway: xL+vR: 1555840:1594323 (2^7*5*11*13*17:3^3*13), ~42.3 cents, which also approximates ~xL+vR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents (6deg183) vL+vR: 524277:531441 (2^19:3^12, the Pythagorean comma), ~23.5 cents (3deg183) There are also two other combinations that I didn't see any use for: vL+sR, ~46.5 cents (7deg183) sL+vR, ~30.2 cents (4deg183) Why 183? Since I'm going only to the 17 limit here, it's one in which the 19 comma (512:513) vanishes, and it represents the building blocks of the notation rather nicely as approximate multiples of 7 cents. The apotome in 183 is 17 degrees. *23-LIMIT APPROACH* And here is the 23-limit arrangement, which correlates well with 217- ET (apotome of 21 degrees): Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (4 degrees of 217) Straight right flag (sR): 54:55, ~31.8 cents (6deg217) Convex left flag (vL): 4131:4096 (3^5*17:2^12, the 17-as-flat comma), ~14.7 cents (3deg217) Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (5deg217) Concave left flag (vL): 2187:2176 (3^7:2^7*17, the 17-as-sharp comma), ~8.7 cents (2deg217) Concave right flag (vR): 512:513 (the 19-as-flat comma), ~3.4 cents (1deg217) The difference between this and the 17-limit approach is that I have removed the 715:729 alteration and added the 512:513 alteration, while reassigning the 17-commas to different flags. No combination of flags will now exceed half of an apotome. With the above used in combination, the following useful intervals are available: sL+sR: 32:33 (the 11-as-semisharp comma), ~53.3 cents (10deg217) sL+xR: 35:36, ~48.8 cents, which approximates ~sL+xR: 1024:1053 (the 13-as-semisharp comma), ~48.3 cents (9deg217) vL+sR: 4352:4455 (2^8*17:3^4*5*11), ~40.5 cents, which approximates ~vL+sR: 16384:16767 (2^14:3^6*23, the 23 comma), ~40.0 cents (8deg217) xL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates ~xL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents (8deg217) The vL+sR approximation of the 23 comma deviates by 3519:3520 (~0.492 cents). All of the above provide a continuous range of intervals in 217-ET, which I selected because it is consistent to the 21-limit and represents the building blocks of the notation as approximate multiples of 5.5 cents. --George

Message: 4357 - Contents - Hide Contents Date: Fri, 22 Mar 2002 22:56:54 Subject: List of 20 From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> I'll have a much better feel for this if we do the full 7-limit first. > But have we agreed on the 5-limit list yet. Do you agree the weighted > complexity cutoff can be reduced to 13, to give us a list of 20?It looks OK; hemithirds, ennealimmal and the 10485760000/10460353203 system are really best thought of as 7-limit temperaments.

Message: 4358 - Contents - Hide Contents Date: Sat, 23 Mar 2002 23:37:41 Subject: Re: Diatonics From: genewardsmith --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:> 5 0 7 2 9 4 11 reordered 0 2 4 5 7 9 11 (0, has no points at which 3 > adjacent PCs lie together.What do you mean by a "PC"?> I know I use the tterm generators differently, but in the context of the > group theoretic structures proposed initially by Balzano."Generators" is a term taken from mathematics, and I hope when we say "generator" or "group" we are all on the same page, and using the mathematical definitions. What are the two uses of "generator" you see around here? The looseness I've sometimes observed is "generator" as opposed to octaves or a division thereof, but that should be clear from context.

Message: 4360 - Contents - Hide Contents Date: Sat, 23 Mar 2002 23:41:19 Subject: Re: A common notation for JI and ETs From: genewardsmith If we have more than one comma per prime, we lose the very desireable property of uniqueness, and automatic translation becomes harder.

Message: 4361 - Contents - Hide Contents Date: Sat, 23 Mar 2002 18:08:21 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >>>>>> 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. >> In the 17-limit approach that I outline below, you will see how I > handle this.OK. So you have gone outside of one-comma-per-prime and one-(sub)symbol-per-prime. But you have given fair reasons for doing so in the case of 11 and 13.> I am also outlining a 23-limit approach; I went for the > 19 limit and got 23 as a bonus when I found that I could approximate > it using a very small comma. The two approaches could be combined, > in which case you could have the 11-13 semiflat varieties along with > the 19 or 23 limit, but the symbols may get a bit complicated -- more > about that below.You don't actually give more below about combining these approaches. But I had fun working it out for myself. I'll give my solution later.> I thought more about this and now realize that the problem with 27-ET > is not as formidable as it seemed. If we use the 80:81 comma for a > single degree and the 1024:1053 comma for two degrees of alteration, > we will do just fine, even if the *symbol* for 1024:1053 happens to > be a combination of the 80:81 and 63:64 symbols (by conflating > 4095:4096). For the 27-ET notation we can simply define the > combination of the two symbols as the 13 comma alteration, and there > would be no inconsistency in usage, since the 63:64 symbol is *never > used by itself* in the 27-ET notation. The same could be said about > 50-ET.You're absolutely right.> Are there any troublesome divisions above 100 that we should > be concerned about in this regard?Not that I can find on a cursory examination.> I anticipate that you believe that the JI purists would still want to > have this distinction, so we should go for 19.Correct. I'll skip the 183-ET based one.> I>> wouldn't spend too much time on the symbols yet. I expect serious >> problems with the semantics. >> I don't know what problems you are anticipating, ...Well none have materialised yet. :-)> I have found that the semantics and symbols are so closely connected > that I could not address one without the other,Yes. I see that now.> Both the 17-limit and 23-limit approaches use 6 sizes of > alterations. In the sagittal notation these are paired into left and > right flags that are affixed to a vertical stem, to the top for > upward alteration and to the bottom for downward alteration. These > pairs of flags consist of straight lines, convex curved lines, and > concave curved lines. With this arrangement there is a limitation > that two left or two right flags cannot be used simultaneously.Given these constraints I think your solution is brilliant.> *23-LIMIT APPROACH* > > And here is the 23-limit arrangement, which correlates well with 217- > ET (apotome of 21 degrees):I don't think you can call this a 23-limit notation, since 217-ET is not 23-limit consistent. But it is certainly 19-prime-limit.> Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (4 degrees > of 217) > Straight right flag (sR): 54:55, ~31.8 cents (6deg217) > Convex left flag (vL): 4131:4096 (3^5*17:2^12, the 17-as-flat comma), > ~14.7 cents (3deg217) > Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (5deg217) > Concave left flag (vL): 2187:2176 (3^7:2^7*17, the 17-as-sharp > comma), ~8.7 cents (2deg217) > Concave right flag (vR): 512:513 (the 19-as-flat comma), ~3.4 cents > (1deg217) > > The difference between this and the 17-limit approach is that I have > removed the 715:729 alteration and added the 512:513 alteration, > while reassigning the 17-commas to different flags. No combination > of flags will now exceed half of an apotome. > > With the above used in combination, the following useful intervals > are available: > > sL+sR: 32:33 (the 11-as-semisharp comma), ~53.3 cents (10deg217) > sL+xR: 35:36, ~48.8 cents, which approximates > ~sL+xR: 1024:1053 (the 13-as-semisharp comma), ~48.3 cents (9deg217) > vL+sR: 4352:4455 (2^8*17:3^4*5*11), ~40.5 cents, which approximates > ~vL+sR: 16384:16767 (2^14:3^6*23, the 23 comma), ~40.0 cents > (8deg217) > xL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates > ~xL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents > (8deg217) > > The vL+sR approximation of the 23 comma deviates by 3519:3520 (~0.492 > cents). > > All of the above provide a continuous range of intervals in 217-ET, > which I selected because it is consistent to the 21-limit and > represents the building blocks of the notation as approximate > multiples of 5.5 cents.Once I understood your constraints, I spent hours looking at the problem. I see that you can push it as far as 29-limit in 282-ET if you want both sets of 11 and 13 commas, and 31-limit in 311-ET if you can live with only the smaller 11 and 13 commas. But to make these work you have to violate what is probably an implicit constraint, that the 5 and 7 commas must correspond to single flags. Neither of them can map to a single flag in either 282-ET or 311-ET and so the mapping of commas to arrows is just way too obscure. 217-ET is definitely the highest ET you can use with the above additional constraint. I notice that left-right confusability has gone out the window. But maybe that's ok, if we accept that this is not a notation for sight-reading by performers. However, it is possible to improve the situation by making the left-right confusable pairs of symbols either map to the same number of steps of 217-ET or only differ by one step, so a mistake will not be so disastrous. At the same time as we do this we can reinstate your larger 11 and 13 commas, so you have both sizes of these available. The 13 commas will have similar flags on left and right, while the 11 commas will have dissimilar flags. It seems better that the 11 commas should be confused with each other than the 13 commas, since the 11 commas are closer together in size. To do this you make the following changes to your 217-ET-based scheme. 1. Swap the flags for the 17 comma and 19 comma (vL and vR) 2. Swap the flags for the 7 comma and 11-as-semisharp/5 comma. (xR and sR) 3. Make the xL flag the 11-as-semiflat/7 comma instead of the 17-as-flat comma. So we have Steps Flags Commas ------------------------ 1 vL 19-comma 512:513 2 vR 17-comma 2187:2176 3 vL+vR 17-as-flat-comma 4131:4096 4 sL 5-comma 80:81 5 sR 7-comma 63:64 6 xL or xR 7 vL+sR or xL+vR 8 vL+xR 9 sL+sR 13-as-semisharp comma 1024:1053 10 sL+xR 11-as-semisharp comma 32:33 11 xL+sR 11-as-semiflat comma 704:729 12 xL+xR 13-as-semiflat comma 26:27 21 apotome

Message: 4362 - Contents - Hide Contents Date: Sat, 23 Mar 2002 18:16:40 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> So we have > Steps Flags Commas > ------------------------ > 1 vL 19-comma 512:513 > 2 vR 17-comma 2187:2176Oops! That should have been: Steps Flags Commas ------------------------ 1 vR 19-comma 512:513 2 vL 17-comma 2187:2176 3 vL+vR 17-as-flat-comma 4131:4096 4 sL 5-comma 80:81 5 sR 7-comma 63:64 6 xL or xR 7 vL+sR or xL+vR 8 vL+xR 9 sL+sR 13-as-semisharp comma 1024:1053 10 sL+xR 11-as-semisharp comma 32:33 11 xL+sR 11-as-semiflat comma 704:729 12 xL+xR 13-as-semiflat comma 26:27

Message: 4363 - Contents - Hide Contents Date: Sat, 23 Mar 2002 21:04 +0 Subject: Re: Diatonics From: graham@xxxxxxxxxx.xx.xx Mark Gould wrote:> 5 0 7 2 9 4 11 6 1 reordered 0 1 2 4 5 6 7 9 11 (0, has several places > where > three adjacent PCs lie together, Viz, 0 1 2 or 11 0 1 or 4 5 6 or 5 6 7.Ah, so this is relative to that base scale. And the equivalent 9 note scale from 19-equal would still pass. Doesn't this rule get taken care of by the later one rejecting "pentatonics"? At least along with another one rejecting "degenerate" scales which are simply an equal scale.> 2. The F F-sharp rule was discussed in section III, and from that > dicussion, > it is simply the result of transposing the diatonic generated by the > sum of > the two 'generators' (or the interval that behaves structurally like > the 3:2 > fifth, or 'generalised fifth', as Paul Zweifel dubs it) It must be noted > that the new tone in he diatonic shall have the following property: > > It should be the 'leading tone' of the new diatonic > > It can also be equivalent to adding 1 to the PC that lies a 'generalized > fifth' below the old tonic.But doesn't this always work for an MOS? Or is it this rule that specifies an MOS?> If the raised tone lies in the 'supertonic' position then the scale is > an > anhemitonic scale, in that it contains no tones that are separated by > the > difference between the two generators of the scale.And the 7 note diatonic in more than 12 notes would have this property.> I know I use the tterm generators differently, but in the context of the > group theoretic structures proposed initially by Balzano. I am merely > continuing to use the term, and did not invent it myself.It sounds consistent with Gene's ideas, so long as you can describe the octave in terms of the generators.> Incidentally, can anyone point me to Paul Ehrlich's decatonics, a web > address etc...?I have it as <A few items from Paul Erlich * [with cont.] (Wayb.)>. Only one h in Erlich. It's also in Xenharmonikon. Graham

Message: 4364 - Contents - Hide Contents Date: Sun, 24 Mar 2002 11:41:30 Subject: Hermite normal form From: genewardsmith The normal form I mentioned, which "fixed" my first try, turns out on reflection to be simply Hermite normal form. Hence, no special wedge products are required--one could, for instance, start with columns which are et maps. I propose therefore that we standardize the mapping to be the (column) Hermite normal form, and the generators to be the generators (in order) that this gives us.

Message: 4365 - Contents - Hide Contents Date: Mon, 25 Mar 2002 00:27:12 Subject: Re: Hermite normal form From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> The normal form I mentioned, which "fixed" my first try, turns outon reflection to be simply Hermite normal form. Hence, no special wedge products are required--one could, for instance, start with columns which are et maps. I propose therefore that we standardize the mapping to be the (column) Hermite normal form, and the generators to be the generators (in order) that this gives us.>I'm not familiar with this, and Mathworld was no help to me (again). Hermite Normal Form -- from MathWorld * [with cont.] It doesn't give me any clue as to what the HNF matrix looks like (except that it has a determinant of +-1) or how to compute it. Would you be so kind as to give as examples, the HNF matrices and consequent generators for Meantone, Diaschismic and Augmented, as 5-limit linear temperaments, Twin meantone and Half meantone fifth as degenerate 5-limit whatevers, and Starling as the 7-limit planar temperament where the 125:126 vanishes?

Message: 4366 - Contents - Hide Contents Date: Mon, 25 Mar 2002 09:38:29 Subject: Re: Pitch Class and Generators From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:My disguised followup has shown up yet, let's see if this works.

Message: 4367 - Contents - Hide Contents Date: Mon, 25 Mar 2002 04:22:59 Subject: Re: Hermite normal form From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Would you be so kind as to give as examples, the HNF matrices and > consequent generators for Meantone, Diaschismic and Augmented, as > 5-limit linear temperaments, Twin meantone and Half meantone fifth > as degenerate 5-limit whatevers, and Starling as the 7-limit planar > temperament where the 125:126 vanishes?Do you want this if half-fourth doesn't work? Hermite form seems to allow twin meantone and schismic, and half-fifth meantome and schismic, but not the half-fourth versions.

Message: 4368 - Contents - Hide Contents Date: Mon, 25 Mar 2002 09:34:45 Subject: Followups From: genewardsmith Has anyone noticed that followups to other postings have been appearing, but things using the "Post" command have not? This is a followup disguised by removing what it follows up to--let's see if it appears. I've been posting a little about Hermite normal form, but if it doesn't appear, it slows the conversation down. How are other people doing?

Message: 4370 - Contents - Hide Contents Date: Mon, 25 Mar 2002 15:06:17 Subject: Re: Hermite normal form version of "25 best" From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Here's the same list, this time using Hermite normal form.How about dropping the g_w cutoff to 13 for the best 20 like we agreed, or even to 10 for the best 17? Who wants the 20 and who wants the 17? Paul, Graham, Carl, Herman, anyone?>The ideaof this is to have a standard form which generalizes to higher dimension temperaments and could allow us to measure badness for them. It also is conceptually not wedded to octave-equivalence, but works well in that context. The disadvantage is that you might not like it!>You're right. I don't like it. Generators bigger than an octave (in one case bigger than two octaves) and negative generators. This is not a form that allows a temperament's musical ramifications to be easily understood. However, I like the fact that the octave-related (or lowest-prime-related) generator comes first. If it facilitates badness calculations, well and good, but I wouldn't publish a list of temperaments in this form in a pink fit (unless it was not intended to be read by musicians). So far I stand by my earlier proposal that minimises (positive) generator sizes while having zeros in the lower left triangle of the matrix (with columns corresponding to increasing primes from left to right). Each generator is less than half the size of the preceeding one. This gives maximum information about the melodic structure of the scale.> I also changed the weighted "g" measure to one which is a weightedmean, since Dave complained bitterly that my adjustment wasn't one.>Thanks for doing that.

Message: 4371 - Contents - Hide Contents Date: Mon, 25 Mar 2002 09:31:36 Subject: Re: Pitch Class and Generators From: genewardsmith --- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:> Generators I have assumed is as follows: > > assume three equivalence classes a b c lie linearly adjacent, such that the > Next predicate follows this sequence > > b = Next (a) > c = Next (b) > > thus c = Next(Next(a)) or Next^2(a) > > If we (after Balzano), abbreviate Next to N, > > we can write N^i, representinh powers of N, where i is an integer mod n.This is a little strange, though mathematically correct. Normally you would write the cyclic group of order 12 additively unless it was something coming from a multiplicative structure, such as the multiplicative group of nonzero mod 13 classes. Writing it additively, we can indentify "N" with 1, and then b=a+1, c=b+1. Then N^i becomes simply i, and N^(jn+k) is jn+k. This raises the question of whether "j" is a generator for C12--which is the same as asking if j is relatively prime to 12.> A generator for Cn is an N^i, such that successive applications of > N^i to the starting tone will generate the full Cn.Or we could say, a generator for Cn is i, such that (i,n)=1, meaning i is relatively prime to n. It is an "n-unit"; an invertible element of Cn, taken as a ring.> That's my definition too.That's one usage; in general, a group generator for a finitely generated abelian group can mean more kinds of things, some of them not involving groups with torsion, and actually that's what's been discussed the most around here recently.

Message: 4372 - Contents - Hide Contents Date: Mon, 25 Mar 2002 20:49:49 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..., "dkeenanuqnetau" <d.keenan@u...> wrote: >> 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) >> I find that I can also agree with Gene on a best 17, he just needs to > change his complexity cutoff to 10 and we lose the last three off the > list above, including septathirds which Paul objected to, and I agreed > was boring. I would prefer this list of 17, to the list of 20. There > is quite a big gap in complexity between quintelevenths and > septathirds.let's use the name 'negri' for negri's system, shall we?

Message: 4374 - Contents - Hide Contents Date: Mon, 25 Mar 2002 11:34 +0 Subject: Re: Pitch Class and Generators From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <B8C4883F.38C4%mark.gould@xxxxxxx.xx.xx> Mark Gould wrote:> A generator for Cn is an N^i, such that successive applications of > N^i to the starting tone will generate the full Cn. > > For any n i=1 is a generator, as is i = -1 > > For C12, i = 5 and its mod 12 complement 7 are generators.But I thought the major and minor thirds were also the twin generators for a diatonic scale. They don't qualify in 12-equal by this definition. Graham

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