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Message: 5100 - Contents - Hide Contents Date: Mon, 15 Jul 2002 18:59:29 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4532]:> [a lengthy reply] ... > > Whew! > > With that I must sadly inform you that I will not be able tocontribute to this discussion again for quite some time. I need to get seriously involved in an electronic design project for some months now. The trouble is I'm a tuning theory addict. I can't have just a little.> > George, I strongly encourage you to present what we've agreed uponso far, to the wider community for comment.> > Regards, > -- Dave Keenan Dave,Thank you for your latest comments and ideas. It will take me some time to digest and thoughtfully consider all of what you discussed. I have also been busy with other things for the past few weeks and will not be looking at this in detail for at least a few more, at which time I will be able to review all of this with a fresher perspective. So I expect that it will be at least a month before I present anything about what we have accomplished. And once that's started, I imagine that it's going to take a while to cover, given that there will probably be a lot of questions. So let's both enjoy our summer break. Best regards, --George
Message: 5101 - Contents - Hide Contents Date: Mon, 15 Jul 2002 16:46:41 Subject: Yet another arrgangement of 7-limit linear temperaments From: Gene W Smith This time I used symmetrical geometric complexity; the result probably seems less plausible, but in fact there is a good reason to look at this if our interest is strictly 7 (not 9) limit and we are allowing ourselves the freedom to construct scales along non-MOS lines. [18, 27, 18, -34, 22, 1] Ennealimmal bad 200.7612789 comp 39.23009049 rms .1304491741 [16, 2, 5, 6, 37, -34] Hemiwuerschmidt bad 534.8468679 comp 24.71841419 rms .8753631224 [6, -7, -2, 15, 20, -25] Miracle bad 568.1796030 comp 18.62793601 rms 1.637405196 [4, 2, 2, -1, 8, -6] Decimal bad 766.2480478 comp 5.656854249 rms 23.94525150 [10, 9, 7, -9, 17, -9] Small diesic bad 810.1208287 comp 15.62049935 rms 3.320167332 [1, 4, 10, 12, -13, 4] Meantone bad 890.6035608 comp 15.58845727 rms 3.665035228 [7, -3, 8, 27, 7, -21] Orwell bad 890.6976986 comp 18.54723699 rms 2.589237496 [4, 4, 4, -2, 5, -3] Diminished bad 918.5756443 comp 6.928203230 rms 19.13699259 [2, 25, 13, -40, -15, 35] Hemififth bad 931.5691063 comp 39.89987469 rms .5851564738 [6, 5, 3, -7, 12, -6] Kleismic bad 1031.000003 comp 9.165151390 rms 12.27380956 [9, 5, -3, -21, 30, -13] Quartaminorthirds bad 1039.361025 comp 18.41195264 rms 3.065961726 [2, -4, -4, 2, 12, -11] Pajara bad 1177.543176 comp 10.39230485 rms 10.90317755 [0, 5, 0, -14, 0, 8] Quintal bad 1186.151431 comp 8.660254038 rms 15.81535241 [4, -3, 2, 13, 8, -14] Tertiathirds bad 1304.177048 comp 10.34408043 rms 12.18857055 [8, 18, 11, -25, 5, 10] Octafifths bad 1376.914655 comp 25.82634314 rms 2.064339812 [3, 0, -6, -14, 18, -7] Tripletone bad 1385.216081 comp 13.07669683 rms 8.100678834 [8, 6, 6, -3, 13, -9] Double wide bad 1459.046339 comp 12.00000000 rms 10.13226624 [5, 1, 12, 25, -5, -10] Magic bad 1473.502081 comp 18.86796226 rms 4.139050792 [3, 12, -1, -36, 10, 12] Supermajor seconds bad 1503.290103 comp 20.49390153 rms 3.579262150 [1, 4, -2, -16, 6, 4] Dominant seventh bad 1512.246113 comp 8.660254038 rms 20.16328150 [5, 13, -17, -76, 41, 9] Amt bad 1633.393513 comp 43.94314509 rms .8458796028 [3, 0, 6, 14, -1, -7] Augmented bad 1643.269165 comp 9.949874371 rms 16.59867843 [15, -2, -5, -6, 50, -38] Hemithird bad 1648.130712 comp 30.85449724 rms 1.731229740 [1, -8, -14, -10, 25, -15] Schismic bad 1724.179823 comp 24.55605832 rms 2.859336356 [6, 5, 22, 37, -18, -6] Catakleismic bad 1757.115994 comp 33.03028913 rms 1.610555448 [2, -9, -4, 16, 12, -19] Neutral thirds bad 1767.424388 comp 16.82260384 rms 6.245315858 [7, 9, 13, 5, -1, -2] Semisixths (tiny diesic) bad 1793.790515 comp 18.84144368 rms 5.052931030 [1, 9, -2, -30, 6, 12] Superpythagorean bad 1794.928339 comp 16.73320053 rms 6.410458352 [3, 5, -6, -28, 18, 1] Porcupine bad 1879.273475 comp 16.61324773 rms 6.808961862 [13, -10, 6, 42, 27, -46] bad 1911.832046 comp 33.74907406 rms 1.678518039 [2, 8, 1, -20, 4, 8] bad 1966.962149 comp 12.44989960 rms 12.69007837 [2, 6, 6, -3, -4, 5] Supersharp bad 2037.299988 comp 10.39230485 rms 18.86388876 [9, 10, -3, -35, 30, -5] bad 2042.562846 comp 22.44994432 rms 4.052704060 [12, 10, -9, -49, 48, -12] Hemikleismic bad 2144.955624 comp 33.63034344 rms 1.896512488 [5, -11, -12, 3, 33, -29] bad 2255.013430 comp 28.91366459 rms 2.697384486 [4, -8, 14, 55, -11, -22] Shrutar bad 2259.485358 comp 31.68595904 rms 2.250483424 [12, -2, 20, 52, 2, -31] bad 2265.152215 comp 35.94440151 rms 1.753213789 [3, 17, -1, -50, 10, 20] bad 2278.812132 comp 28.89636655 rms 2.729116326 [2, 8, 8, -4, -7, 8] Injera bad 2288.664030 comp 14.28285686 rms 11.21894132 [0, 12, 24, 22, -38, 19] bad 2371.077791 comp 39.79949748 rms 1.496892545 [1, -3, 5, 20, -5, -7] Hexadecimal bad 2434.569514 comp 11.44552314 rms 18.58450012 [5, 1, -7, -19, 25, -10] bad 2609.423437 comp 17.29161647 rms 8.727168682 [2, 8, -11, -48, 23, 8] bad 2817.934201 comp 27.47726333 rms 3.732363180 [1, 4, -9, -32, 17, 4] Flattone bad 2877.300282 comp 19.39071943 rms 7.652394368 [2, -4, -16, -26, 31, -11] Diaschismic bad 2980.871818 comp 27.92848009 rms 3.821630536
Message: 5102 - Contents - Hide Contents Date: Mon, 15 Jul 2002 06:32:24 Subject: Re: Temperaments sorted by "geometric badness" From: genewardsmith --- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:> [2, 25, 13, -40, -15, 35] > comp 46.45156501 rms .5851564738 bad 1262.620148Because of its high rank (#4) on my list of 45, it might be time to give this more-or-less-microtemperament a little respect, and an actual name. Below I give a Fokker block for the commas <64/63, 5120/5103, 2401/2400>, and a tempering by the 239-et. [1, 28/27, 21/20, 49/45, 9/8, 7/6, 189/160, 49/40, 80/63, 1029/800, 4/3, 441/320, 10/7, 196/135, 3/2, 14/9, 63/40, 49/30, 27/16, 343/200, 16/9, 147/80, 40/21, 3087/1600] 12 7-limit intervals, no triads, not connected :( [0, 12, 17, 29, 41, 53, 58, 70, 82, 87, 99, 111, 123, 128, 140, 152, 157, 169, 181, 186, 198, 210, 222, 227] 59 intervals, 24 triads. It may not win a prize, but it does show the importance of tempering in some cases. Since 239 wants to be an 11-limit system, I also checked the 11-limit numbers: 97 intervals, 100 triads. Smokin'!
Message: 5103 - Contents - Hide Contents Date: Wed, 17 Jul 2002 18:20:14 Subject: Geometric comma measures From: Gene W Smith If we want to consider temperaments of codimension one, which is to say, ones using a single comma, we need a way of measuring comma complexity, and then of putting that together with the rms value of the comma. If we use a size hueristic in place of the rms value, we then want a reasonable way of putting together complexity (in particular, geometric complexity) with size to get a comma goodness measure. One way to do this is to appeal to Baker's theorem, which implies that if L(q) is a Euclidean metric on the p-limit group (turning it into a lattice), then good(q) = -ln(ln(q)/ln(L(q)) is bounded above, so there are infinite sets of commas with good(q) > A for a suitable choice of A. Here is a list of all 7-limit intervals of size less than 50 cents, within a radius of 10 of the unison, and such that good(q) > 2.6: [36/35, 49/48, 50/49, 64/63, 81/80, 245/243, 126/125, 4000/3969, 1728/1715, 1029/1024, 225/224, 10976/10935, 3136/3125, 5120/5103, 6144/6125, 65625/65536, 32805/32768, 2401/2400, 4375/4374]
Message: 5104 - Contents - Hide Contents Date: Thu, 18 Jul 2002 16:31:57 Subject: Re: geometric complexity From: Gene W Smith On Thu, 18 Jul 2002 18:45:41 -0400 "Paul H. Erlich" <PErlich@xxxxxxxxxxxxx.xxx> writes:> Anyway, why aren't we closing in on finality for the project? What > exactly > is Euclidean geometric complexity going to mean to a musician that > our > previous measures don't capture well?I want a measure which applies to all temperaments, not just linear ones.
Message: 5105 - Contents - Hide Contents Date: Thu, 18 Jul 2002 00:39:27 Subject: Small diesic scales From: Gene W Smith These are related to the samll diesic (126/125 and 1728/1715) linear temperament. The first is a Fokker block, from the commas <10/9, 126/125, 1728/1715>; the Scala file I present here may be copied and pasted. ! smalldi11.scl ! Small diesic 11-note block, <10/9, 126/125, 1728/1715> commas ! 11 ! 36/35 7/6 6/5 216/175 7/5 10/7 175/108 5/3 12/7 35/18 2/1 We have 18 intervals, 8 triads, and no tetrads; more specifically we get 1-6/5-7/5 and 1-7/6-7/5 chords: roots on degrees 0,6,8,9 1-6/5-7/5-5/3 and 1-7/6-7/5-5/3 chords on degrees 0,9 If we temper this by the 120-et (which is does a good job for small diesic and makes the scale degrees into nice round numbers in terms of cents) we get the 11-note small diesic MOS, with generator 31/120: ! smalldimos11.scl ! Small diesic 11-note MOS, 31/120 version ! 11 ! 40.0 270.0 310.0 350.0 580.0 620.0 850.0 890.0 930.0 1160.0 2/1 We now have 34 intervals, 33 triads, and 2 tetrads; the tetrads occur on degree 7, which might serve as a tonic. ! smalldi19a.scl ! Small diesic 19-note block, <16/15, 126/125, 1728/1715> commas ! 19 ! 36/35 25/24 8/7 7/6 6/5 175/144 5/4 48/35 7/5 10/7 35/24 8/5 288/175 5/3 12/7 7/4 48/25 35/18 2/1 52 intervals, 44 triads, 8 tetrads ! smalldi19b.scl ! Small diesic 19-note block, <16/15, 126/125, 2401/2400> commas ! 19 ! 50/49 21/20 8/7 7/6 6/5 49/40 5/4 48/35 7/5 10/7 35/24 8/5 80/49 5/3 12/7 7/4 40/21 49/25 2/1 50 intervals, 40 triads, 6 tetrads. There are four more tetrads if we are willing to count those off by 2401/2400, which is less than a cent. Either of these, when tempered by the 12-et. gives us the 19-note small diesic MOS: ! smalldimos19.scl ! Small diesic 19-note MOS, 31/120 version ! 19 ! 40.0 80.0 230.0 270.0 310.0 350.0 390.0 540.0 580.0 620.0 660.0 810.0 850.0 890.0 930.0 970.0 1120.0 1160.0 2/1 82 intervals, 105 triads and 18 tetrads Here is variant 19-note scale containing glumma: ! smalldi19c.scl ! Small diesic 19-note scale containing glumma ! 19 49/48 21/20 15/14 35/32 6/5 49/40 5/4 9/7 21/16 10/7 35/24 3/2 49/32 5/3 12/7 7/4 9/5 35/18 2/1 53 intervals, 45 triads, 8 tetrads Tempering this gives the following: ! smalldiglum19.scl ! Small diesic "glumma" variant of 19-note MOS, 31/120 version ! 19 ! 40.0 80.0 120.0 160.0 310.0 350.0 390.0 430.0 470.0 620.0 660.0 700.0 740.0 890.0 930.0 970.0 1010.0 1160.0 2/1 78 intervals, 94 triads, 16 tetrads
Message: 5106 - Contents - Hide Contents Date: Thu, 18 Jul 2002 02:25:05 Subject: Re: Compton/Erlich temperament From: Gene W Smith On Thu, 18 Jul 2002 01:22:19 -0700 Carl Lumma <carl@xxxxx.xxx> writes:> How do we classify the Compton/Erlich scheme of tuning multiple > 12-et keyboards 15 cents apart? Some sort of planar temperament > with the following commas? > > 531441/524288 (pythagorean comma) > 5120/5103 (difference between syntonic comma and 64/63) > > Is this right?I think it's another system, discussed below. The wedgie you find from the pyth comma and 5120/5103 gives what we are calling a linear temperament. It is [0,12,12,-6,-19,19], and has a TM reduced basis <50/49, 3645/3584>. The mapping is [[12, 19, 28, 34], [0, 0, -1, -1]] However, the rms optimum is 23.4 cents apart, not 15. I think what you want is the linear temperament with wedgie [0,12,12,-6,-19,19], TM reduced basis <225/224, 250047/250000> and mapping [[12,19,28,34],[0,0,-1,-1]]. You can use the 72 or 84 ets for this. By the way, is 250047/250000 not deserving of a little recognition?
Message: 5107 - Contents - Hide Contents Date: Thu, 18 Jul 2002 21:46:40 Subject: Re: geometric complexity From: Gene W Smith On Fri, 19 Jul 2002 00:36:20 -0400 "Paul H. Erlich" <PErlich@xxxxxxxxxxxxx.xxx> writes:> > First of all, i don't know where you're getting just linear ones > from.What's your definition of complexity in general?> Secondly, i don't see what there is about a Euclidean, as opposed to > a > triangular-taxicab, metric that is going to be reflective of how we > hear. In > fact, it would seem especially important at the 9-limit and above to > deviate > from Euclid.I was proposing using a Euclidean metric which did not give the same size to all prime numbers; prime p would have length ln(p), and if p and q are odd primes, with q>p, then length p/q = length q/p = ln(q). This uniquely determines a Euclidean metric.
Message: 5109 - Contents - Hide Contents Date: Thu, 18 Jul 2002 03:18:02 Subject: 38 linear 7-limit temperaments compatible with 12-et From: Gene W Smith Here is raw material for exotic 12-tone tunings and multiple keyboard experiments, not to mention a lot of old friends. I think I've seen this one before: [1, 4, 10, 12, -13, 4] [[1, 0, -4, -13], [0, 1, 4, 10]] comp 5.322447240 rms 3.665035228 bad 103.8247475 Here's a microtemperament for multi-keyboard enthusiasts: [3, -24, -54, -58, 94, -45] [[3, 0, 45, 94], [0, 1, -8, -18]] comp 31.51783075 rms .1469057415 bad 145.9322934 [2, -4, -4, 2, 12, -11] [[2, 0, 11, 12], [0, 1, -2, -2]] comp 3.938677761 rms 10.90317755 bad 169.1429833 [3, 0, -6, -14, 18, -7] [[3, 0, 7, 18], [0, 1, 0, -2]] comp 4.631825456 rms 8.100678834 bad 173.7904007 [4, 4, 4, -2, 5, -3] [[4, 0, 3, 5], [0, 1, 1, 1]] comp 3.144366918 rms 19.13699259 bad 189.2082747 [1, 4, -2, -16, 6, 4] [[1, 0, -4, 6], [0, 1, 4, -2]] comp 3.128478105 rms 20.16328150 bad 197.3456024 It worked for Helmholtz [1, -8, -14, -10, 25, -15] [[1, 0, 15, 25], [0, 1, -8, -14]] comp 8.612526914 rms 2.859336356 bad 212.0930465 [3, 0, 6, 14, -1, -7] [[3, 0, 7, -1], [0, 1, 0, 2]] comp 3.675273386 rms 16.59867843 bad 224.2088808 This has possibilities [0, 12, 24, 22, -38, 19] [[12, 19, 0, -22], [0, 0, 1, 2]] comp 13.76571634 rms 1.496892545 bad 283.6535726 Another interesting one [3, -12, -30, -36, 56, -26] [[3, 0, 26, 56], [0, 1, -4, -10]] comp 17.83027719 rms .8942129314 bad 284.2870884 [2, 8, 8, -4, -7, 8] [[2, 0, -8, -7], [0, 1, 4, 4]] comp 5.343650829 rms 11.21894132 bad 320.3524287 [2, -4, -16, -26, 31, -11] [[2, 0, 11, 31], [0, 1, -2, -8]] comp 9.469818377 rms 3.821630536 bad 342.7141199 [4, -8, -20, -24, 43, -22] [[4, 0, 22, 43], [0, 1, -2, -5]] comp 12.75555760 rms 2.220377240 bad 361.2648129 [5, -4, -10, -12, 30, -18] [[1, 2, 2, 2], [0, 5, -4, -10]] comp 8.009157500 rms 6.041345016 bad 387.5317655 [6, 0, 0, 0, 17, -14] [[6, 0, 14, 17], [0, 1, 0, 0]] comp 4.716550378 rms 18.04292374 bad 401.3801294 [0, 0, 12, 28, -19, 0] [[12, 19, 28, 0], [0, 0, 0, 1]] comp 6.904855942 rms 9.840803062 bad 469.1803177 [4, -20, -44, -46, 81, -41] [[4, 0, 41, 81], [0, 1, -5, -11]] comp 26.35358322 rms .6908190406 bad 479.7816635 [2, -16, -40, -48, 69, -30] [[2, 0, 30, 69], [0, 1, -8, -20]] comp 23.01717204 rms .9641797248 bad 510.8129776 [5, 8, 2, -18, 11, 1] [[1, 2, 3, 3], [0, 5, 8, 2]] comp 5.083424305 rms 21.64417648 bad 559.3115508 [1, -8, -2, 18, 6, -15] [[1, 0, 15, 6], [0, 1, -8, -2]] comp 5.398824730 rms 19.66911204 bad 573.3016760 [3, 12, 18, 8, -20, 12] [[3, 0, -12, -20], [0, 1, 4, 6]] comp 10.19699576 rms 5.782918708 bad 601.3004996 [7, 4, 10, 12, 4, -10] [[1, 1, 2, 2], [0, 7, 4, 10]] comp 6.082925309 rms 16.44388527 bad 608.4563190 [2, -4, 8, 30, -7, -11] [[2, 0, 11, -7], [0, 1, -2, 4]] comp 6.058298120 rms 18.06996660 bad 663.2215524 [1, -8, -26, -38, 44, -15] [[1, 0, 15, 44], [0, 1, -8, -26]] comp 14.64779855 rms 3.106171476 bad 666.4539470 [0, 12, 12, -6, -19, 19] [[12, 19, 0, 6], [0, 0, 1, 1]] comp 8.845819922 rms 10.15948550 bad 794.9648068 [5, -16, -34, -34, 68, -37] [[1, 2, 1, 0], [0, 5, -16, -34]] comp 21.26748532 rms 1.787147240 bad 808.3372977 [6, 12, 12, -6, -2, 5] [[6, 0, -5, -2], [0, 1, 2, 2]] comp 7.833907871 rms 13.63960404 bad 837.0640348 [9, 0, -6, -14, 35, -21] [[3, 1, 7, 11], [0, 3, 0, -2]] comp 8.596026672 rms 11.94435731 bad 882.5885631 [8, -4, -4, 2, 29, -25] [[4, 1, 12, 14], [0, 2, -1, -1]] comp 8.101297218 rms 14.29026872 bad 937.8848637 [6, -12, -24, -22, 55, -33] [[6, 0, 33, 55], [0, 1, -2, -4]] comp 16.33270964 rms 3.600727660 bad 960.5207638 [7, -8, -14, -10, 42, -29] [[1, 1, 3, 4], [0, 7, -8, -14]] comp 11.74162535 rms 7.012328960 bad 966.7601028 [3, 0, -18, -42, 37, -7] [[3, 0, 7, 37], [0, 1, 0, -6]] comp 11.16933529 rms 8.439018022 bad 1052.801683 [9, 0, 6, 14, 16, -21] [[3, 1, 7, 6], [0, 3, 0, 2]] comp 7.074825566 rms 21.62618964 bad 1082.459061 [6, -12, -36, -50, 74, -33] [[6, 0, 33, 74], [0, 1, -2, -6]] comp 22.11900171 rms 2.367078438 bad 1158.093686 [9, 12, 6, -20, 16, -2] [[3, 1, 2, 6], [0, 3, 4, 2]] comp 7.852968657 rms 21.26148578 bad 1311.177048 [10, 4, 16, 26, 3, -17] [[2, 4, 5, 7], [0, 5, 2, 8]] comp 9.546317939 rms 16.25734866 bad 1481.567725 [1, -8, 10, 46, -13, -15] [[1, 0, 15, -13], [0, 1, -8, 10]] comp 8.914766865 rms 18.78088561 bad 1492.574604 [2, -16, -16, 8, 31, -30] [[2, 0, 30, 31], [0, 1, -8, -8]] comp 12.60828383 rms 10.10704662 bad 1606.705286
Message: 5110 - Contents - Hide Contents Date: Fri, 19 Jul 2002 00:59:04 Subject: Re: 38 linear 7-limit temperaments compatible with 12-et From: genewardsmith --- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:> Here's a microtemperament for multi-keyboard enthusiasts: > [3, -24, -54, -58, 94, -45] [[3, 0, 45, 94], [0, 1, -8, -18]] > > comp 31.51783075 rms .1469057415 bad 145.9322934Let's see how this one might work. We can change the mapping given above to the equivalent [12 3] [19 5] [28 5] [34 4] The generators for this are well approximated by 14/171 and 1/171, and we can use this as a keyboard system for the 171-et, with the 12-note keyboards tuned to semitones of size 14/171 ocatves, and the keyboards separated by 1/171. This non-ocatave tuning (12*(14/171) = 168/171, a comma less than the octave) can be modified to one which tunes each rank of 12-note keyboards slightly unevenly, to the 14/171 MOS, namely [14 14 14 15] repeated three times. Of course this gives four different patterns for chords, but you pays your money and you makes your choice.
Message: 5111 - Contents - Hide Contents
Date: Sat, 20 Jul 2002 04:58:01
Subject: Hemiwuerschmidt, the Porcupine Complex, and transformations
From: Gene W Smith
These turn out to have some interesting properties.
Hemiwuerschmidt is the 7-limit linear temperament with wedgie [16, 2, 5,
6, 37, -34] and commas 2401/2400,
3136/3125, 6144/6125 and of course the wuerschmidt, 393216/390625.
13-note Fokker blocks
[1, 50/49, 28/25, 8/7, 5/4, 32/25, 7/5, 10/7, 25/16, 8/5, 7/4, 25/14,
49/25]
commas [15/14, 3136/3125, 2401/2400]
intervals 24 triads 12 tetrads 0
[1, 128/125, 28/25, 8/7, 5/4, 32/25, 7/5, 10/7, 25/16, 8/5, 7/4, 25/14,
125/64]
commas [15/14, 3136/3125, 6144/6125]
intervals 22 triads 10 tetrads 0
Both of these involve only the primes 2,5 and 7; and so may be thought of
as blocks for what I've called the "hemithirds no threes" 7-limit planar
temperament, whose defining comma is the 3-less 3136/3125. I suppose
"hemiwuerschmidt no threes" might be a better name.
[1, 49/48, 28/25, 8/7, 5/4, 32/25, 7/5, 10/7, 25/16, 8/5, 7/4, 25/14,
96/49]
[15/14, 2401/2400, 6144/6125]
intervals 22 triads 10 tetrads 0
If we temper by the 31, 68, 99, 130 or even (as I've done here) by the
[427, 677, 992, 1199] val, we get a 13-note MOS, which here I represent
by the 69/427 MOS of the 427-et:
[0, 13, 69, 82, 138, 151, 207, 220, 276, 289, 345, 358, 414]
intervals 31 triads 16 tetrads 0
This is almost, but not quite, the same as the "hemithirds no threes" 13
note MOS; it can't *exactly* be regarded as that since it has a couple of
poor but honest subminor thirds, though the actual notes are the same.
[1, 50/49, 35/32, 28/25, 8/7, 49/40, 5/4, 125/98, 343/250, 7/5, 10/7,
500/343, 196/125, 8/5, 80/49, 7/4, 25/14, 64/35,49/25]
commas [21/20, 3136/3125, 2401/2400]
intervals 38 triads 20 tetrads 0
The "0" does not count the four complete tetrads off by less than a cent
(by 2401/2400.)
[1, 128/125, 35/32, 28/25, 8/7, 49/40, 5/4, 32/25, 175/128, 7/5, 10/7,
256/175, 25/16, 8/5, 80/49, 7/4, 25/14,
64/35, 125/64]
comma [21/20, 3136/3125, 6144/6125]
intervals 38 triads 20 tetrads 0
Here the chord count (counting *only* theoretically exact chords) is the
same, and in fact the two characteristic polynomials are the same,
suggesting there is an isomorphism between the two graphs of the two
scales. This turns out to be the case: both scales are in the {2,3,7}
subgroup, and the mapping 5-->7, 7-->56/5 sends the first scale to the
second, and the inverse map 5->40/7, 7->5 sends it back again. This
transformation is of order 6, and induces a permutation of degree six on
the MOS. This MOS mapping is an automorphism of the "no threes" graph,
but not of the full hemiwuerschmidt graph.
[1, 49/48, 35/32, 28/25, 8/7, 49/40, 5/4, 245/192, 48/35, 7/5, 10/7,
35/24, 384/245, 8/5, 80/49, 7/4, 25/14, 64/35, 96/49]
commas [21/20, 2401/2400, 6144/6125];
intervals 42 triads 26 tetrads 2
The tetrad count grows to four if we allow 2401/2400 relationships.
All of these scales have the same hemiwuerschmidt 19-tone MOS:
[0, 13, 56, 69, 82, 125, 138, 151, 194, 207, 220, 233, 276, 289,
302, 345, 358, 371, 414]
intervals 63 triads 50 tetrads 6
The 13 and 19 tone hemiwuerschmidt MOS can both be considered as "no
threes" MOS, and in this way a part of the Porcupine Complex of
transformations between the 250/243, 3125/3087 and 3136/3125 planar
temperaments I wrote about a while back; unfortunately Yahoo won't let me
use the archives just now, so I don't have the message number.
Message: 5112 - Contents - Hide Contents Date: Sat, 20 Jul 2002 20:54:17 Subject: Re: geometric complexity From: Gene W Smith On Fri, 19 Jul 2002 00:55:22 -0400 "Paul H. Erlich" <PErlich@xxxxxxxxxxxxx.xxx> writes: Just redo the 5-limit and see how everyone feels about the> rankings, and off we go . . . (but i'll keep harping on the question > of a > more elegant metric)Which 5-limit temperaments do you regard as essential? How would you rate 128/125, 135/128, 250/243, 78732/78125, 393216/390625, 3125/3072 or 648/625? How about the funky systems such as 25/24, 27/25, 16/15, 10/9, 9/8? Where do you draw that line?
Message: 5113 - Contents - Hide Contents Date: Sat, 20 Jul 2002 19:04:01 Subject: Re: geometric complexity From: Gene W Smith>What's your definition of complexity in general? >Just about the same as yours, but . . . What specifically?>> Secondly, i don't see what there is about a Euclidean, as opposed to >> a >> triangular-taxicab, metric that is going to be reflective of how we >> hear. In >> fact, it would seem especially important at the 9-limit and above to >> deviate > from Euclid.>> I was proposing using a Euclidean metric which did not give the same size >> to all prime numbers; prime p would have length ln(p), and if p and q are >> odd primes, with q>p, then >> length p/q = length q/p = ln(q). This uniquely determines a Euclidean >> metric.>Right, but first of all, do we or don't we have octave equivalence?We do; this is a metric on octave classes.>Secondly, the metric (if you replace "prime" with "odd") is inconsistent for >intervals like 9/5, right? You can't form a Euclidean figure for the 9-limit >pentad such that all the intervals obey this "odd" rule, can you?It doesn't treat 9 quite like a prime, but I don't think it does badly. In this case we have L(7/5) = ln 7 = 1.946 L(9/5) = sqrt(2 ln(3)^2 + ln(5)^2) = 2.237 L(11/5) = ln(11) = 2.398 The value 2.237 instead of ln(9) = 2.197 doesn't seem that horrible to me.
Message: 5114 - Contents - Hide Contents Date: Sun, 21 Jul 2002 21:37:27 Subject: A 5-limit, "geometric" temperament list From: Gene W Smith The following is a complete list of all 5-limit temperaments satisfying the requirements that rms error be less than 15, geometric complexity less than 40, and the badness calculated from these less than 3000. If people feel something valuable has been left off (e.g, 135/128, 25/24, or 16875/16384) we could raise the error limit. 81/80 (3)^4/(2)^4/(5) meantone [[1, 0, -4], [0, 1, 4]] comp 4.132030727 rms 4.217730828 bad 297.5565312 generators [1200., 1896.164845] 128/125 (2)^7/(5)^3 augmented [[3, 5, 7], [0, 1, 0]] comp 4.828313736 rms 9.677665780 bad 1089.323984 generators [400.0000000, 91.20185550] 256/243 (2)^8/(3)^5 quintal [[5, 8, 12], [0, 0, -1]] comp 5.493061445 rms 12.75974144 bad 2114.877638 generators [240.0000000, 84.66378778] 250/243 (2)*(5)^3/(3)^5 porcupine [[1, 2, 3], [0, -3, -5]] comp 5.948285733 rms 7.975800816 bad 1678.609846 generators [1200., 162.9960265] 2048/2025 (2)^11/(3)^4/(5)^2 diaschismic [[2, 3, 5], [0, 1, -2]] comp 6.271198982 rms 2.612821643 bad 644.4088670 generators [600.0000000, 105.4465315] 648/625 (2)^3*(3)^4/(5)^4 diminished [[4, 6, 9], [0, 1, 1]] comp 6.437751648 rms 11.06006024 bad 2950.938432 generators [300.0000000, 94.13435693] 3125/3072 (5)^5/(2)^10/(3) diesic (small diesic) [[1, 0, 2], [0, 5, 1]] comp 7.741412273 rms 4.569472316 bad 2119.954991 generators [1200., 379.9679493] 15625/15552 (5)^6/(2)^6/(3)^5 kleismic [[1, 0, 1], [0, 6, 5]] comp 9.338935129 rms 1.029625097 bad 838.6315482 generators [1200., 317.0796753] 32805/32768 (3)^8*(5)/(2)^15 shismic [[1, 0, 15], [0, 1, -8]] comp 9.459947973 rms .1616933186 bad 136.8857747 generators [1200., 1901.727514] 20000/19683 (2)^5*(5)^4/(3)^9 quadrafifths (minimal diesic) [[1, 1, 1], [0, 4, 9]] comp 9.785568434 rms 2.504205191 bad 2346.540676 generators [1200., 176.2822703] 78732/78125 (2)^2*(3)^9/(5)^7 hemisixths (tiny diesic) [[1, -1, -1], [0, 7, 9]] comp 12.19218236 rms 1.157498409 bad 2097.803242 generators [1200., 442.9792975] 393216/390625 (2)^17*(3)/(5)^8 wuerschmidt [[1, -1, 2], [0, 8, 1]] comp 12.54312332 rms 1.071949828 bad 2115.395301 generators [1200., 387.8196733] 2109375/2097152 (3)^3*(5)^7/(2)^21 orwell [[1, 0, 3], [0, 7, -3]] comp 12.77234114 rms .8004099292 bad 1667.723301 generators [1200., 271.5895996] 1600000/1594323 (2)^9*(5)^5/(3)^13 amt [[1, 3, 6], [0, -5, -13]] comp 13.79419993 rms .3831037874 bad 1005.555381 generators [1200., 339.5088256] 6115295232/6103515625 (2)^23*(3)^6/(5)^14 semisuper [[2, 4, 5], [0, -7, -3]] comp 21.20762522 rms .1940180530 bad 1850.624306 generators [600.0000000, 71.14606343] 1224440064/1220703125 (2)^8*(3)^14/(5)^13 parakleismic [[1, 5, 6], [0, -13, -14]] comp 21.32267248 rms .2766026501 bad 2681.521263 generators [1200., 315.2509133] 10485760000/10460353203 (2)^24*(5)^4/(3)^21 [[1, 0, -6], [0, 4, 21]] comp 21.73304916 rms .1537673823 bad 1578.433204 generators [1200., 475.5422333] 274877906944/274658203125 (2)^38/(3)^2/(5)^15 hemithird [[1, 4, 2], [0, -15, 2]] comp 24.97702150 rms .6082244804e-1 bad 947.7326423 generators [1200., 193.1996149] 68719476736000/68630377364883 (2)^39*(5)^3/(3)^29 trichotififths [[1, 0, -13], [0, 3, 29]] comp 30.55081228 rms .5750010064e-1 bad 1639.596150 generators [1200., 634.0119851] 19073486328125/19042491875328 (5)^19/(2)^14/(3)^19 enneadecal [[19, 30, 44], [0, 1, 1]] comp 30.57932033 rms .1047837215 bad 2996.244873 generators [63.15789474, 7.292252126] 9010162353515625/9007199254740992 (3)^10*(5)^16/(2)^53 [[2, 1, 6], [0, 8, -5]] comp 31.25573660 rms .1772520822e-1 bad 541.2283791 generators [600.0000000, 162.7418923] 7629394531250/7625597484987 (2)*(5)^18/(3)^27 ennealimmal [[9, 13, 19], [0, 2, 3]] comp 33.65327154 rms .2559250891e-1 bad 975.4269093 generators [133.3333333, 84.32451333] 50031545098999707/50000000000000000 (3)^35/(2)^16/(5)^17 [[1, -1, -3], [0, 17, 35]] comp 38.84548584 rms .2546649929e-1 bad 1492.763207 generators [1200., 182.4660891] 450359962737049600/450283905890997363 (2)^54*(5)^2/(3)^37 monzismic [[1, 2, 10], [0, -2, -37]] comp 39.66560308 rms .5738429624e-2 bad 358.1254995 generators [1200., 249.0184479]
Message: 5115 - Contents - Hide Contents Date: Sun, 21 Jul 2002 21:51:37 Subject: Re: A 5-limit, "geometric" temperament list From: Gene W Smith On Sun, 21 Jul 2002 21:37:27 -0700 Gene W Smith <genewardsmith@xxxx.xxx> writes:> 68719476736000/68630377364883 (2)^39*(5)^3/(3)^29 trichotififths > [[1, 0, -13], [0, 3, 29]] > > comp 30.55081228 rms .5750010064e-1 bad 1639.596150 > generators [1200., 634.0119851]Should be "trichototwelfths"
Message: 5116 - Contents - Hide Contents Date: Mon, 22 Jul 2002 22:02:15 Subject: A 7-limit planar temperaments list From: Gene W Smith Here is a list of 7-limit planar temperaments, with complexity and rms error less than 14, and badness calculated from that (as rms * complexity^4, which are the right dimensions to correspond to linear temperaments) less than 10000. I don't see much of interest in the ones with badness over 5000, but they are listed in terms of increasing badness and you can make your own judgments. Using Minkowski's basic theorem from his geometry of numbers would help us these to be used to get 7-limit temperaments. 4375/4374 (5)^4*(7)/(2)/(3)^7 [[1, 0, 0, 1], [0, 1, 0, 7], [0, 0, 1, -4]] comp 8.514085954 rms .3862236561e-1 bad 202.9509048 generators [1200., 1902.005884, 2786.302406] 250047/250000 (3)^6*(7)^3/(2)^4/(5)^6 [[3, 0, 0, 4], [0, 1, 0, -2], [0, 0, 1, 2]] comp 10.12074134 rms .2800947367e-1 bad 293.8693214 generators [400.0000000, 1901.922456, 2786.324561] 2401/2400 (7)^4/(2)^5/(3)/(5)^2 [[1, 1, 1, 2], [0, 2, 1, 1], [0, 0, 2, 1]] comp 7.001210638 rms .1255441309 bad 301.6400406 generators [1200., 350.9775007, 617.6844971] 225/224 (3)^2*(5)^2/(2)^5/(7) [[1, 0, 0, -5], [0, 1, 0, 2], [0, 0, 1, 2]] comp 4.026020476 rms 1.484083173 bad 389.9080111 generators [1200., 1899.812912, 2784.171625] 64/63 (2)^6/(3)^2/(7) [[1, 0, 0, 6], [0, 1, 0, -2], [0, 0, 1, 0]] comp 3.320882425 rms 5.949512815 bad 723.5947437 generators [1200., 1911.692178, 2792.156018] 81/80 (3)^4/(2)^4/(5) [[1, 0, -4, 0], [0, 1, 4, 0], [0, 0, 0, 1]] comp 4.132030727 rms 3.443762373 bad 1003.892815 generators [1200., 1896.164845, 3366.344411] 32805/32768 (3)^8*(5)/(2)^15 [[1, 0, 15, 0], [0, 1, -8, 0], [0, 0, 0, 1]] comp 9.459947973 rms .1320192291 bad 1057.285276 generators [1200., 1901.727514, 3368.705472] 126/125 (2)*(3)^2*(7)/(5)^3 [[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]] comp 4.396397238 rms 3.010264897 bad 1124.585345 generators [1200., 1899.984322, 2789.269735] 5120/5103 (2)^10*(5)/(3)^6/(7) [[1, 0, 0, 10], [0, 1, 0, -6], [0, 0, 1, 1]] comp 6.872898147 rms .5465065421 bad 1219.424726 generators [1200., 1902.888700, 2786.702752] 2460375/2458624 (3)^9*(5)^3/(2)^10/(7)^4 [[1, 0, 2, -1], [0, 1, 1, 3], [0, 0, 4, 3]] comp 11.82008947 rms .7718361083e-1 bad 1506.635331 generators [1200., 1901.831749, -378.8994468] 28/27 (2)^2*(7)/(3)^3 [[1, 0, 0, -2], [0, 1, 0, 3], [0, 0, 1, 0]] comp 3.320882425 rms 13.73919562 bad 1670.995600 generators [1200., 1924.441038, 2795.308127] 65625/65536 (3)*(5)^5*(7)/(2)^16 [[1, 0, 0, 16], [0, 1, 0, -1], [0, 0, 1, -5]] comp 9.483876888 rms .2200495161 bad 1780.180537 generators [1200., 1901.707688, 2785.942745] 420175/419904 (5)^2*(7)^5/(2)^6/(3)^8 [[1, 0, 3, 0], [0, 1, 4, 0], [0, 0, 5, -2]] comp 11.85680458 rms .9406410627e-1 bad 1859.065030 generators [1200., 1902.061943, -1684.389187] 6144/6125 (2)^11*(3)/(5)^3/(7)^2 [[1, 0, 1, 4], [0, 1, 1, -1], [0, 0, 2, -3]] comp 7.012333797 rms .7993271605 bad 1932.746493 generators [1200., 1902.491206, -157.4631745] 703125/702464 (3)^2*(5)^7/(2)^11/(7)^3 [[1, 0, 2, 1], [0, 1, 1, 3], [0, 0, 3, 7]] comp 11.00411811 rms .1342982709 bad 1969.207103 generators [1200., 1901.822082, -505.2414545] 1029/1024 (3)*(7)^3/(2)^10 [[1, 1, 0, 3], [0, 3, 0, -1], [0, 0, 1, 0]] comp 6.237538987 rms 1.350315689 bad 2044.035369 generators [1200., 233.4444416, 2785.016372] 49/48 (7)^2/(2)^4/(3) [[1, 0, 0, 2], [0, 2, 0, 1], [0, 0, 1, 0]] comp 3.733539376 rms 11.89893695 bad 2312.017448 generators [1200., 950.9775006, 2780.364245] 50/49 (2)*(5)^2/(7)^2 [[2, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 1]] comp 3.891820298 rms 10.09659043 bad 2316.252253 generators [600.0000000, 1901.955001, 2777.569810] 3136/3125 (2)^6*(7)^2/(5)^5 [[1, 0, 0, -3], [0, 1, 0, 0], [0, 0, 2, 5]] comp 7.348511491 rms .8057454012 bad 2349.607642 generators [1200., 1902.435257, 1393.797198] 245/243 (5)*(7)^2/(3)^5 [[1, 0, 0, 0], [0, 1, 1, 2], [0, 0, 2, -1]] comp 5.914949368 rms 1.987070580 bad 2432.301564 generators [1200., 1904.876579, 440.9272508] 4000/3969 (2)^5*(5)^3/(3)^4/(7)^2 [[1, 0, 1, 4], [0, 1, 0, -2], [0, 0, 2, 3]] comp 6.115596778 rms 1.784055755 bad 2495.535749 generators [1200., 1904.436192, 793.1568564] 10976/10935 (2)^5*(7)^3/(3)^7/(5) [[1, 0, 2, -1], [0, 1, 2, 3], [0, 0, 3, 1]] comp 8.434555688 rms .5771951599 bad 2921.268800 generators [1200., 1902.880572, -1139.661549] 5250987/5242880 (3)^7*(7)^4/(2)^20/(5) [[1, 0, 0, 5], [0, 1, 3, -1], [0, 0, 4, 1]] comp 11.71753460 rms .1695157602 bad 3195.619481 generators [1200., 1901.681064, -729.7186252] 2048/2025 (2)^11/(3)^4/(5)^2 [[2, 0, 11, 0], [0, 1, -2, 0], [0, 0, 0, 1]] comp 6.271198982 rms 2.133360336 bad 3299.639851 generators [600.0000000, 1905.446531, 3370.920823] 875/864 (5)^3*(7)/(2)^5/(3)^3 [[1, 0, 0, 5], [0, 1, 0, 3], [0, 0, 1, -3]] comp 5.590098759 rms 3.998747915 bad 3904.828359 generators [1200., 1904.145206, 2781.933305] 1728/1715 (2)^6*(3)^3/(5)/(7)^3 [[1, 0, 0, 2], [0, 1, 0, 1], [0, 0, 3, -1]] comp 6.389007079 rms 2.386896737 bad 3977.105469 generators [1200., 1900.647644, 929.2070233] 128/125 (2)^7/(5)^3 [[3, 0, 7, 0], [0, 1, 0, 0], [0, 0, 0, 1]] comp 4.828313736 rms 7.901781211 bad 4294.443852 generators [400.0000000, 1908.798145, 3375.669050] 525/512 (3)*(5)^2*(7)/(2)^9 [[1, 0, 0, 9], [0, 1, 0, -1], [0, 0, 1, -2]] comp 4.914886013 rms 7.556420940 bad 4409.302991 generators [1200., 1892.089470, 2774.475078] 19683/19600 (3)^9/(2)^4/(5)^2/(7)^2 [[1, 0, 0, -2], [0, 2, 0, 9], [0, 0, 1, -1]] comp 9.496383083 rms .5594506202 bad 4549.824682 generators [1200., 950.5282864, 2786.121192] 321489/320000 (3)^8*(7)^2/(2)^9/(5)^4 [[2, 0, 0, 9], [0, 1, 0, -4], [0, 0, 1, 2]] comp 9.685150414 rms .5990398573 bad 5270.856574 generators [600.0000000, 1901.017355, 2786.179764] 4096000/4084101 (2)^15*(5)^3/(3)^5/(7)^5 [[1, 0, 0, 3], [0, 1, 0, -1], [0, 0, 5, 3]] comp 11.01660696 rms .3957132811 bad 5828.704629 generators [1200., 1902.514624, 557.3000507] 686/675 (2)*(7)^3/(3)^3/(5)^2 [[1, 0, 2, 1], [0, 1, 0, 1], [0, 0, 3, 2]] comp 6.011672792 rms 4.481232704 bad 5853.004182 generators [1200., 1907.336788, 130.2063806] 33075/32768 (3)^3*(5)^2*(7)^2/(2)^15 [[1, 1, 0, 6], [0, 2, 0, -3], [0, 0, 1, -1]] comp 7.823970640 rms 1.622557336 bad 6080.074498 generators [1200., 349.7544516, 2784.112225] 405/392 (3)^4*(5)/(2)^3/(7)^2 [[1, 0, 1, -1], [0, 1, 0, 2], [0, 0, 2, 1]] comp 5.199938690 rms 8.419825102 bad 6155.962460 generators [1200., 1888.775891, 789.3913963] 3645/3584 (3)^6*(5)/(2)^9/(7) [[1, 0, 0, -9], [0, 1, 0, 6], [0, 0, 1, 1]] comp 6.872898147 rms 2.773231799 bad 6187.935855 generators [1200., 1897.216978, 2783.549864] 2100875/2097152 (5)^3*(7)^5/(2)^21 [[1, 0, 2, 3], [0, 1, 0, 0], [0, 0, 5, -3]] comp 12.52322351 rms .2532643498 bad 6229.290561 generators [1200., 1901.704334, 77.19380915] 1323/1280 (3)^3*(7)^2/(2)^8/(5) [[1, 0, 0, 4], [0, 1, 1, -1], [0, 0, 2, 1]] comp 5.199938690 rms 8.527334761 bad 6234.565687 generators [1200., 1888.607610, 445.9928964] 67108864/66976875 (2)^26/(3)^7/(5)^4/(7)^2 [[1, 0, 0, 13], [0, 2, 0, -7], [0, 0, 1, -2]] comp 13.68011660 rms .1803880478 bad 6317.815729 generators [1200., 951.1207079, 2786.557165] 15625/15552 (5)^6/(2)^6/(3)^5 [[1, 0, 1, 0], [0, 6, 5, 0], [0, 0, 0, 1]] comp 9.338935129 rms .8406854315 bad 6394.740953 generators [1200., 317.0796754, 3368.695142] 16875/16807 (3)^3*(5)^4/(7)^5 [[1, 0, 0, 0], [0, 1, 3, 3], [0, 0, 5, 4]] comp 9.567898928 rms .7767030640 bad 6509.105851 generators [1200., 1901.307749, -583.6772476] 102760448/102515625 (2)^21*(7)^2/(3)^8/(5)^6 [[2, 0, 0, -21], [0, 1, 0, 4], [0, 0, 1, 3]] comp 13.89893287 rms .2141059337 bad 7990.142647 generators [600.0000000, 1902.288028, 2786.613436] 250/243 (2)*(5)^3/(3)^5 [[1, 2, 3, 0], [0, 3, 5, 0], [0, 0, 0, 1]] comp 5.948285733 rms 6.512214090 bad 8152.596693 generators [1200., -162.9960265, 3371.413598] 458752/455625 (2)^16*(7)/(3)^6/(5)^4 [[1, 0, 0, -16], [0, 1, 0, 6], [0, 0, 1, 4]] comp 10.00226449 rms .8352030642 bad 8359.598450 generators [1200., 1903.280492, 2787.462472] 589824/588245 (2)^16*(3)^2/(5)/(7)^6 [[1, 0, 4, 2], [0, 1, 2, 0], [0, 0, 6, -1]] comp 11.93226425 rms .4664233339 bad 9455.221767 generators [1200., 1902.165950, -969.5993818] 256/243 (2)^8/(3)^5 [[5, 8, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]] comp 5.493061445 rms 10.41828523 bad 9485.365527 generators [240.0000000, 2795.336213, 3377.848405]
Message: 5117 - Contents - Hide Contents Date: Tue, 23 Jul 2002 02:39:56 Subject: Modified best 5-limit geometric list From: Gene W Smith Here is another list, with the cutoffs changed to complexity and rms error less than 20, and badness less than 3200: 32805/32768 (3)^8*(5)/(2)^15 shismic [[1, 0, 15], [0, 1, -8]] comp 9.459947973 rms .1616933186 bad 136.8857747 generators [1200., 1901.727514] 81/80 (3)^4/(2)^4/(5) meantone [[1, 0, -4], [0, 1, 4]] comp 4.132030727 rms 4.217730828 bad 297.5565312 generators [1200., 1896.164845] 2048/2025 (2)^11/(3)^4/(5)^2 diaschismic [[2, 3, 5], [0, 1, -2]] comp 6.271198982 rms 2.612821643 bad 644.4088670 generators [600.0000000, 105.4465315] 15625/15552 (5)^6/(2)^6/(3)^5 kleismic [[1, 0, 1], [0, 6, 5]] comp 9.338935129 rms 1.029625097 bad 838.6315482 generators [1200., 317.0796753] 1600000/1594323 (2)^9*(5)^5/(3)^13 amt [[1, 3, 6], [0, -5, -13]] comp 13.79419993 rms .3831037874 bad 1005.555381 generators [1200., 339.5088256] 128/125 (2)^7/(5)^3 augmented [[3, 5, 7], [0, 1, 0]] comp 4.828313736 rms 9.677665780 bad 1089.323984 generators [400.0000000, 91.20185550] 135/128 (3)^3*(5)/(2)^7 pelogic [[1, 0, 7], [0, 1, -3]] comp 4.132030727 rms 18.07773392 bad 1275.365360 generators [1200., 1877.137655] 2109375/2097152 (3)^3*(5)^7/(2)^21 orwell [[1, 0, 3], [0, 7, -3]] comp 12.77234114 rms .8004099292 bad 1667.723301 generators [1200., 271.5895996] 250/243 (2)*(5)^3/(3)^5 porcupine [[1, 2, 3], [0, -3, -5]] comp 5.948285733 rms 7.975800816 bad 1678.609846 generators [1200., 162.9960265] 78732/78125 (2)^2*(3)^9/(5)^7 hemisixths [[1, -1, -1], [0, 7, 9]] comp 12.19218236 rms 1.157498409 bad 2097.803242 generators [1200., 442.9792975] 256/243 (2)^8/(3)^5 quintal (blackwood?) [[5, 8, 12], [0, 0, -1]] comp 5.493061445 rms 12.75974144 bad 2114.877638 generators [240.0000000, 84.66378778] 393216/390625 (2)^17*(3)/(5)^8 wuerschmidt [[1, -1, 2], [0, 8, 1]] comp 12.54312332 rms 1.071949828 bad 2115.395301 generators [1200., 387.8196733] 3125/3072 (5)^5/(2)^10/(3) diesic [[1, 0, 2], [0, 5, 1]] comp 7.741412273 rms 4.569472316 bad 2119.954991 generators [1200., 379.9679493] 20000/19683 (2)^5*(5)^4/(3)^9 quadrafifths [[1, 1, 1], [0, 4, 9]] comp 9.785568434 rms 2.504205191 bad 2346.540676 generators [1200., 176.2822703] 648/625 (2)^3*(3)^4/(5)^4 diminished [[4, 6, 9], [0, 1, 1]] comp 6.437751648 rms 11.06006024 bad 2950.938432 generators [300.0000000, 94.13435693] 4294967296/4271484375 (2)^32/(3)^7/(5)^9 septathirds [[1, 2, 2], [0, -9, 7]] comp 18.57395503 rms .4831084292 bad 3095.692281 generators [1200., 55.27549315] 531441/524288 (3)^12/(2)^19 pythagorean [[12, 19, 28], [0, 0, -1]] comp 13.18334747 rms 1.382394464 bad 3167.444999 generators [100.0000000, 14.66378756]
Message: 5118 - Contents - Hide Contents Date: Tue, 23 Jul 2002 02:46:59 Subject: Re: Modified best 5-limit geometric list From: Gene W Smith On Tue, 23 Jul 2002 02:39:56 -0700 Gene W Smith <genewardsmith@xxxx.xxx> writes:> 256/243 (2)^8/(3)^5 quintal (blackwood?) > [[5, 8, 12], [0, 0, -1]]Which is better? I called the 7-limit version quintal, but they both should have the same name.> 3125/3072 (5)^5/(2)^10/(3) diesic > [[1, 0, 2], [0, 5, 1]]Should be magic; I keep copying myself, and never make the change.
Message: 5119 - Contents - Hide Contents Date: Sat, 27 Jul 2002 03:07:15 Subject: 11-limit spacial temperaments list From: genewardsmith This may be the first time ever anyone has looked at spacial temperaments as such, but even if it isn't I doubt there is much of a market. However, it's useful to have them, and here is a list of everything satisfying complexity < 10, rms error < 20, and badness < 10000. The ordering is by complexity, least to greatest. 33/32 (3)*(11)/(2)^5 [[1, 0, 0, 0, 5], [0, 1, 0, 0, -1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]] comp 2.857236368 rms 18.60142732 bad 3542.232933 generators [1200., 1885.422018, 2764.728986, 3347.241179] 56/55 (2)^3*(7)/(5)/(11) [[1, 0, 0, 0, 3], [0, 1, 0, 0, 0], [0, 0, 1, 0, -1], [0, 0, 0, 1, 1]] comp 2.887939081 rms 10.36903038 bad 2082.947320 generators [1200., 1906.341694, 2801.423426, 3367.851086] 36/35 (2)^2*(3)^2/(5)/(7) [[1, 0, 0, 2, 0], [0, 1, 0, 2, 0], [0, 0, 1, -1, 0], [0, 0, 0, 0, 1]] comp 2.994519510 rms 17.48752667 bad 4210.795253 generators [1200., 1901.955001, 2810.698904, 4161.072022] 25/24 (5)^2/(2)^3/(3) [[1, 1, 2, 0, 0], [0, 2, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]] comp 3.025592935 rms 19.08172863 bad 4838.048328 generators [1200., 347.6988830, 3354.254274, 4136.746308] 45/44 (3)^2*(5)/(2)^2/(11) [[1, 0, 0, 0, -2], [0, 1, 0, 0, 2], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0]] comp 3.252337250 rms 12.45498136 bad 4532.323997 generators [1200., 1891.805670, 2766.860826, 3358.676575] 64/63 (2)^6/(3)^2/(7) [[1, 0, 0, 6, 0], [0, 1, 0, -2, 0], [0, 0, 1, 0, 0], [0, 0, 0, 0, 1]] comp 3.320882425 rms 7.143746136 bad 2885.316848 generators [1200., 1909.101900, 2794.916460, 4159.920689] 100/99 (2)^2*(5)^2/(3)^2/(11) [[1, 0, 0, 0, 2], [0, 1, 0, 0, -2], [0, 0, 1, 0, 2], [0, 0, 0, 1, 0]] comp 3.652996092 rms 4.697903284 bad 3055.977200 generators [1200., 1903.569386, 2781.022118, 3369.453721] 49/48 (7)^2/(2)^4/(3) [[1, 0, 0, 2, 0], [0, 2, 0, 1, 0], [0, 0, 1, 0, 0], [0, 0, 0, 0, 1]] comp 3.733539376 rms 9.638226848 bad 6991.990407 generators [1200., 949.3214630, 2778.953548, 4143.957775] 77/75 (7)*(11)/(3)/(5)^2 [[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 2], [0, 0, 0, 1, -1]] comp 3.811821523 rms 11.65840053 bad 9382.136804 generators [1200., 1905.751787, 2804.032044, 3365.661917] 55/54 (5)*(11)/(2)/(3)^3 [[1, 0, 0, 0, 1], [0, 1, 0, 0, 3], [0, 0, 1, 0, -1], [0, 0, 0, 1, 0]] comp 3.814620480 rms 10.55929831 bad 8528.874811 generators [1200., 1906.422187, 2777.131166, 3367.833199] 50/49 (2)*(5)^2/(7)^2 [[2, 0, 0, 1, 0], [0, 1, 0, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 0, 1]] comp 3.891820298 rms 8.095281768 bad 7227.629199 generators [600.0000000, 1901.955001, 2777.569810, 4151.317943] 99/98 (3)^2*(11)/(2)/(7)^2 [[1, 0, 0, 0, 1], [0, 1, 0, 0, -2], [0, 0, 1, 0, 0], [0, 0, 0, 1, 2]] comp 3.966980247 rms 4.745598368 bad 4662.202897 generators [1200., 1900.324226, 2785.679522, 3374.171224] 176/175 (2)^4*(11)/(5)^2/(7) [[1, 0, 0, 0, -4], [0, 1, 0, 0, 0], [0, 0, 1, 0, 2], [0, 0, 0, 1, 1]] comp 4.013858905 rms 2.093551491 bad 2181.193312 generators [1200., 1903.085976, 2790.209292, 3371.684758] 225/224 (3)^2*(5)^2/(2)^5/(7) [[1, 0, 0, -5, 0], [0, 1, 0, 2, 0], [0, 0, 1, 2, 0], [0, 0, 0, 0, 1]] comp 4.026020476 rms 1.577430795 bad 1668.515370 generators [1200., 1900.722983, 2783.324928, 4149.834958] 81/80 (3)^4/(2)^4/(5) [[1, 0, -4, 0, 0], [0, 1, 4, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]] comp 4.132030727 rms 5.635084765 bad 6787.630072 generators [1200., 1896.317430, 3364.336729, 4146.828764] 441/440 (3)^2*(7)^2/(2)^3/(5)/(11) [[1, 0, 0, 0, -3], [0, 1, 0, 0, 2], [0, 0, 1, 0, -1], [0, 0, 0, 1, 2]] comp 4.260379115 rms .8994907037 bad 1262.518976 generators [1200., 1901.430980, 2786.270044, 3367.341179] 385/384 (5)*(7)*(11)/(2)^7/(3) [[1, 0, 0, 0, 7], [0, 1, 0, 0, 1], [0, 0, 1, 0, -1], [0, 0, 0, 1, -1]] comp 4.322037437 rms 1.030496055 bad 1554.135766 generators [1200., 1901.354659, 2784.612746, 3367.124941] 126/125 (2)*(3)^2*(7)/(5)^3 [[1, 0, 0, -1, 0], [0, 1, 0, -2, 0], [0, 0, 1, 3, 0], [0, 0, 0, 0, 1]] comp 4.396397238 rms 2.682546126 bad 4405.871537 generators [1200., 1901.955001, 2790.776728, 4152.129400] 121/120 (11)^2/(2)^3/(3)/(5) [[1, 0, 1, 0, 2], [0, 1, 1, 0, 1], [0, 0, 2, 0, 1], [0, 0, 0, 1, 0]] comp 4.517667160 rms 4.072710005 bad 7663.984940 generators [1200., 1901.955001, -156.9226966, 3367.927957] 896/891 (2)^7*(7)/(3)^4/(11) [[1, 0, 0, 0, 7], [0, 1, 0, 0, -4], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1]] comp 5.006104786 rms 1.936418735 bad 6088.340810 generators [1200., 1903.925434, 2788.284144, 3369.893224] 540/539 (2)^2*(3)^3*(5)/(7)^2/(11) [[1, 0, 0, 0, 2], [0, 1, 0, 0, 3], [0, 0, 1, 0, 1], [0, 0, 0, 1, -2]] comp 5.187443078 rms .7484217537 bad 2811.338392 generators [1200., 1901.732843, 2785.955790, 3369.689857] 3025/3024 (5)^2*(11)^2/(2)^4/(3)^3/(7) [[1, 0, 0, 0, 2], [0, 1, 0, 1, 2], [0, 0, 1, 0, -1], [0, 0, 0, 2, 1]] comp 5.895318737 rms .1081738857 bad 770.2982011 generators [1200., 1901.955001, 2786.170613, 733.4354526] 1375/1372 (5)^3*(11)/(2)^2/(7)^3 [[1, 0, 0, 0, 2], [0, 1, 0, 0, 0], [0, 0, 1, 0, -3], [0, 0, 0, 1, 3]] comp 6.019980961 rms .5552206980 bad 4389.764646 generators [1200., 1901.851244, 2785.575886, 3369.229406] 9801/9800 (3)^4*(11)^2/(2)^3/(5)^2/(7)^2 [[2, 0, 0, 0, 3], [0, 1, 0, 0, -2], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]] comp 6.558685862 rms .2935901386e-1 bad 356.3080343 generators [600.0000000, 1901.942581, 2786.331654, 3368.843848] 5632/5625 (2)^9*(11)/(3)^2/(5)^4 [[1, 0, 0, 0, -9], [0, 1, 0, 0, 2], [0, 0, 1, 0, 4], [0, 0, 0, 1, 0]] comp 6.993059136 rms .2430345239 bad 4064.470495 generators [1200., 1902.129577, 2786.808340, 3369.064493] 2401/2400 (7)^4/(2)^5/(3)/(5)^2 [[1, 1, 1, 2, 0], [0, 2, 1, 1, 0], [0, 0, 2, 1, 0], [0, 0, 0, 0, 1]] comp 7.001210638 rms .1009378214 bad 1697.929473 generators [1200., 350.9685108, 617.6855869, 4151.277988] 4000/3993 (2)^5*(5)^3/(3)/(11)^3 [[1, 2, 0, 0, 1], [0, 3, 0, 0, -1], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0]] comp 7.277091769 rms .4609639205 bad 9407.109365 generators [1200., -165.9852708, 2785.912380, 3368.915093] 160083/160000 (3)^3*(7)^2*(11)^2/(2)^8/(5)^4 [[1, 0, 0, 0, 4], [0, 2, 0, 0, -3], [0, 0, 1, 0, 2], [0, 0, 0, 1, -1]] comp 8.009661546 rms .1093194687 bad 3603.863536 generators [1200., 950.9520900, 2786.387123, 3368.712970] 200704/200475 (2)^12*(7)^2/(3)^6/(5)^2/(11) [[1, 0, 0, 0, 12], [0, 1, 0, 0, -6], [0, 0, 1, 0, -2], [0, 0, 0, 1, 2]] comp 8.318381546 rms .2154655877 bad 8581.675756 generators [1200., 1902.164272, 2786.650868, 3368.943828] 4375/4374 (5)^4*(7)/(2)/(3)^7 [[1, 0, 0, 1, 0], [0, 1, 0, 7, 0], [0, 0, 1, -4, 0], [0, 0, 0, 0, 1]] comp 8.514085954 rms .4939105833e-1 bad 2209.726294 generators [1200., 1901.970692, 2786.247899, 4151.309660] 41503/41472 (7)^3*(11)^2/(2)^9/(3)^4 [[1, 0, 0, 1, 3], [0, 1, 0, 0, 2], [0, 0, 1, 0, 0], [0, 0, 0, 2, -3]] comp 8.658539469 rms .1810507469 bad 8810.966005 generators [1200., 1901.922750, 2786.182918, 1084.258868] 131072/130977 (2)^17/(3)^5/(7)^2/(11) [[1, 0, 0, 0, 17], [0, 1, 0, 0, -5], [0, 0, 1, 0, 0], [0, 0, 0, 1, -2]] comp 8.760551242 rms .1254478302 bad 6473.219372 generators [1200., 1902.082652, 2786.470618, 3369.041318] 496125/495616 (3)^4*(5)^3*(7)^2/(2)^12/(11)^2 [[1, 0, 0, 0, -6], [0, 1, 0, 0, 2], [0, 0, 2, 0, 3], [0, 0, 0, 1, 1]] comp 8.950239742 rms .1701889922 bad 9774.730881 generators [1200., 1901.809793, 1393.008192, 3368.566607] 43923/43904 (3)*(11)^4/(2)^7/(7)^3 [[1, 0, 0, 3, 4], [0, 1, 0, 3, 2], [0, 0, 1, 0, 0], [0, 0, 0, 4, 3]] comp 9.113321395 rms .9348367894e-1 bad 5876.506598 generators [1200., 1901.925304, 2786.274940, -1484.226778] 180224/180075 (2)^14*(11)/(3)/(5)^2/(7)^4 [[1, 0, 0, 0, -14], [0, 1, 0, 0, 1], [0, 0, 1, 0, 2], [0, 0, 0, 1, 4]] comp 9.183316130 rms .1339543440 bad 8748.920624 generators [1200., 1902.050698, 2786.516624, 3369.087298] 151263/151250 (3)^2*(7)^5/(2)/(5)^4/(11)^2 [[1, 0, 0, 1, 2], [0, 1, 0, 0, 1], [0, 0, 1, 0, -2], [0, 0, 0, 2, 5]] comp 9.621494742 rms .1438141844e-1 bad 1185.808221 generators [1200., 1901.953231, 2786.325998, 1084.404501]
Message: 5120 - Contents - Hide Contents
Date: Sun, 28 Jul 2002 23:03:48
Subject: 11-limit planar temperament list
From: Gene W Smith
This is limited by rms error < 20, complexity < 40 and badness < 5000.
There are a lot of 11-limit commas and therefore a lot of reasonable
systems, and the badness is set so as to limit the number, but it's
pretty arbitrary otherwise. Of course if we were to look at, for example,
spacial temperaments in the 19-limit, it would be even more out of hand
with exploding numbers of systems.
[1, 1, -2, -2, 2, -2, -2, 5, 1, -6] {36/35, 56/55}
[[1, 0, 0, 2, 5], [0, 1, 0, 2, 2], [0, 0, 1, -1, -2]]
comp 8.612269456 rms 18.93178096 bad 4120.837552
generators [1200., 1905.077357, 2819.545579]
[1, -2, 1, -2, 2, -2, 5, -2, -1, -6] {56/55, 45/44}
[[1, 0, 0, -5, -2], [0, 1, 0, 2, 2], [0, 0, 1, 2, 1]]
comp 9.294368925 rms 18.94576168 bad 4989.555266
generators [1200., 1894.806413, 2782.197962]
[1, 0, -1, 2, -2, 2, -6, 9, -6, 6] {56/55, 64/63}
[[1, 0, 0, 6, 9], [0, 1, 0, -2, -2], [0, 0, 1, 0, -1]]
comp 9.571162783 rms 11.60092948 bad 3287.793582
generators [1200., 1911.208074, 2806.289813]
[1, -3, 2, 2, -2, 2, 1, 2, -8, 6] {56/55, 100/99}
[[1, 0, 0, -1, 2], [0, 1, 0, -2, -2], [0, 0, 1, 3, 2]]
comp 10.48363973 rms 12.23674678 bad 4354.579196
generators [1200., 1909.065474, 2795.586754]
[2, 0, -2, -1, 1, -1, -4, 10, -4, -3] {56/55, 49/48}
[[1, 0, 0, 2, 5], [0, 2, 0, 1, 1], [0, 0, 1, 0, -1]]
comp 10.63096771 rms 13.11941875 bad 4834.444644
generators [1200., 951.4862286, 2793.240998]
[1, 0, 2, 2, -2, -4, -6, 2, 12, -8] {64/63, 100/99}
[[1, 0, 0, 6, 2], [0, 1, 0, -2, -2], [0, 0, 1, 0, 2]]
comp 11.40622127 rms 7.842944503 bad 3446.145599
generators [1200., 1909.527660, 2790.324320]
[0, 3, -3, 0, 0, 0, -7, 7, 2, 0] {56/55, 128/125}
[[3, 0, 7, 0, 2], [0, 1, 0, 0, 0], [0, 0, 0, 1, 1]]
comp 11.57779068 rms 10.41216358 bad 4749.033317
generators [400.0000000, 1905.928440, 3366.473576]
[0, 0, 2, 0, 4, -4, 0, -11, 12, 2] {64/63, 50/49}
[[2, 0, 11, 12, 0], [0, 1, -2, -2, 0], [0, 0, 0, 0, 1]]
comp 11.92510945 rms 9.546325800 bad 4688.049473
generators [600.0000000, 1907.232522, 4157.670515]
[2, -2, 4, 0, -4, 4, -1, 4, -2, -2] {50/49, 99/98}
[[2, 0, 0, 1, 4], [0, 1, 0, 0, -2], [0, 0, 1, 1, 2]]
comp 12.70642452 rms 8.095335006 bad 4659.005552
generators [600.0000000, 1901.942581, 2777.587752]
[1, -2, 4, -2, 2, 4, 5, -9, -2, 8] {176/175, 99/98}
[[1, 0, 0, -5, -9], [0, 1, 0, 2, 2], [0, 0, 1, 2, 4]]
comp 13.87057757 rms 4.770738943 bad 3418.393201
generators [1200., 1900.667338, 2786.672948]
[1, -2, 2, -2, -2, 8, 5, 2, -14, 6] {100/99, 225/224}
[[1, 0, 0, -5, 2], [0, 1, 0, 2, -2], [0, 0, 1, 2, 2]]
comp 14.53451469 rms 4.698274549 bad 3783.900689
generators [1200., 1903.626101, 2781.116797]
[0, 0, 1, 0, -4, 10, 0, 4, -13, 12] {81/80, 126/125}
[[1, 0, -4, -13, 0], [0, 1, 4, 10, 0], [0, 0, 0, 0, 1]]
comp 15.10180563 rms 5.640679273 bad 4999.235518
generators [1200., 1896.439145, 4147.010107]
[2, -3, 4, 4, -4, -2, -5, 4, 4, -2] {100/99, 245/242}
[[1, 0, 1, 4, 4], [0, 1, 0, -2, -2], [0, 0, 2, 3, 4]]
comp 15.28049091 rms 5.288631285 bad 4827.102898
generators [1200., 1902.349260, 790.0166976]
[1, -3, 5, 2, -2, -4, 1, -5, 10, -8] {176/175, 126/125}
[[1, 0, 0, -1, -5], [0, 1, 0, -2, -2], [0, 0, 1, 3, 5]]
comp 15.38428372 rms 2.743964593 bad 2547.250784
generators [1200., 1902.348538, 2791.393074]
[1, 3, 2, -3, -2, 0, -5, 2, 16, -16] {100/99, 385/384}
[[1, 0, 0, 5, 2], [0, 1, 0, 3, -2], [0, 0, 1, -3, 2]]
comp 15.78362763 rms 4.716489415 bad 4668.054634
generators [1200., 1903.304566, 2780.387947]
[2, 3, 1, -1, 1, 2, -11, 3, 10, 4] {176/175, 121/120}
[[1, 0, 1, 4, 2], [0, 1, 1, -1, 1], [0, 0, 2, -3, 1]]
comp 15.78752896 rms 4.078252712 bad 4038.866834
generators [1200., 1902.084929, -156.7887088]
[1, -2, -3, -2, -1, -4, 5, 12, -9, -19] {225/224, 385/384}
[[1, 0, 0, -5, 12], [0, 1, 0, 2, -1], [0, 0, 1, 2, -3]]
comp 16.93706091 rms 1.584465965 bad 1870.587544
generators [1200., 1900.698768, 2783.222881]
[1, -2, 3, -2, 6, -6, 5, -13, 11, -4] {225/224, 441/440}
[[1, 0, 0, -5, -13], [0, 1, 0, 2, 6], [0, 0, 1, 2, 3]]
comp 17.12658309 rms 1.882869462 bad 2285.583154
generators [1200., 1900.058168, 2783.119618]
[1, -3, -4, 2, 3, -1, 1, 8, -20, 13] {126/125, 385/384}
[[1, 0, 0, -1, 8], [0, 1, 0, -2, 3], [0, 0, 1, 3, -4]]
comp 17.97763590 rms 3.571482050 bad 4894.179827
generators [1200., 1900.444933, 2788.289556]
[3, 0, -3, 1, 4, 1, -10, 11, -10, 17] {441/440, 385/384}
[[1, 1, 0, 3, 5], [0, 3, 0, -1, 4], [0, 0, 1, 0, -1]]
comp 18.38423633 rms 1.217389384 bad 1764.182444
generators [1200., 233.6891156, 2784.876116]
[0, 2, -2, 4, -4, -8, -11, 11, 14, -16] {176/175, 896/891}
[[2, 0, 11, 0, 14], [0, 1, -2, 0, -4], [0, 0, 0, 1, 1]]
comp 20.07262936 rms 2.568435319 bad 4636.382920
generators [600.0000000, 1904.449829, 3372.039360]
[3, 1, 5, -3, 3, 6, -6, -6, 8, 12] {176/175, 540/539}
[[1, 0, 0, 2, -2], [0, 1, 0, 1, 1], [0, 0, 3, -1, 5]]
comp 20.79255579 rms 2.292930734 bad 4520.230946
generators [1200., 1902.851684, 929.9710573]
[1, -1, 1, 6, -10, 4, -10, 17, -7, 2] {441/440, 896/891}
[[1, 0, 0, 10, 17], [0, 1, 0, -6, -10], [0, 0, 1, 1, 1]]
comp 21.23275110 rms 2.274972896 bad 4725.980212
generators [1200., 1903.401073, 2788.404311]
[4, -2, 0, -1, 10, -5, -5, -2, 1, 13] {441/440, 243/242}
[[1, 1, 1, 2, 2], [0, 2, 1, 1, 5], [0, 0, 2, 1, 0]]
comp 22.01175542 rms 1.462086952 bad 3323.608780
generators [1200., 350.4014207, 617.6277206]
[2, -4, 0, -4, 5, -10, 10, -1, 2, -23] {225/224, 243/242}
[[1, 1, 0, -3, 2], [0, 2, 0, 4, 5], [0, 0, 1, 2, 0]]
comp 23.21703942 rms 1.809515516 bad 4699.800337
generators [1200., 350.1386568, 2783.638546]
[4, -2, -6, -1, 3, -3, -5, 23, -19, -2] {385/384, 1375/1372}
[[1, 1, 1, 2, 5], [0, 2, 1, 1, 0], [0, 0, 2, 1, -3]]
comp 25.85062167 rms 1.077522792 bad 3661.035538
generators [1200., 350.6699179, 616.8206966]
[2, -4, 6, 8, -12, 0, -9, 12, 3, -6] {441/440, 8019/8000}
[[2, 0, 0, 9, 12], [0, 1, 0, -4, -6], [0, 0, 1, 2, 3]]
comp 27.40496903 rms .9167078854 bad 3604.161607
generators [600.0000000, 1901.338331, 2786.378136]
[5, -4, -3, -3, 9, -9, 0, 10, -8, -6] {540/539, 1375/1372}
[[1, 0, 0, 0, 2], [0, 1, 3, 3, 0], [0, 0, 5, 4, -3]]
comp 27.81436519 rms .7561992066 bad 3085.383114
generators [1200., 1901.732844, -583.8771683]
[4, 4, 0, -6, -2, -2, -11, 17, 17, -31] {385/384, 9801/9800}
[[2, 1, 0, 7, 8], [0, 2, 0, 3, -1], [0, 0, 1, -1, 0]]
comp 28.20690243 rms 1.049393382 bad 4434.317767
generators [600.0000000, 650.6253937, 2784.681994]
[2, -5, 8, 0, 4, -10, 6, -18, 21, -12] {441/440, 3136/3125}
[[1, 0, 0, -3, -9], [0, 1, 0, 0, 2], [0, 0, 2, 5, 8]]
comp 29.06943484 rms .9765850405 bad 4449.400901
generators [1200., 1901.674054, 1393.524953]
[6, 0, 6, -10, -2, 10, -1, 10, 1, -17] {540/539, 4000/3993}
[[2, 1, 0, 2, 3], [0, 3, 0, 5, -1], [0, 0, 1, 0, 1]]
comp 32.94779996 rms .7610570219 bad 4742.236934
generators [600.0000000, 433.9278373, 2785.854450]
[4, -8, 12, 4, -12, 12, 1, 5, -13, 8] {1375/1372, 6250/6237}
[[4, 0, 0, -1, 5], [0, 1, 0, -1, -3], [0, 0, 1, 2, 3]]
comp 37.88323708 rms .5572687387 bad 4922.462152
generators [300.0000000, 1901.870393, 2785.537590]
[2, 8, -6, -14, 10, -2, -2, 5, 14, -25] {3025/3024, 4375/4374}
[[2, 0, 0, 2, 5], [0, 1, 0, 7, 5], [0, 0, 1, -4, -3]]
comp 38.57526083 rms .1120888160 bad 1035.939660
generators [600.0000000, 1901.942581, 2786.188555]
Message: 5122 - Contents - Hide Contents
Date: Mon, 29 Jul 2002 04:09:57
Subject: A {385/384, 441/440} temperament scale
From: Gene W Smith
In case anyone thinks these temperaments are useless, here is a scale in
the 190-et
version of this 11-limit planar temperament. It has 34 intervals and 46
triads:
! [13, 24, 13, 24, 18, 24, 13, 24, 13, 24]
! suzz.scl
!
{385/384, 441/440} suzz in 190-et version
10
!
82.10526316
233.6842105
315.7894737
467.3684211
581.0526316
732.6315789
814.7368421
966.3157895
1048.421053
2/1
The 72-et version is isomoprhic, with pattern [5,9,5,9,7,9,5,9,5,9]. Of
all the 126 scale patterns, this one comes up best for in terms of both
intervals and triads.
Message: 5123 - Contents - Hide Contents Date: Mon, 29 Jul 2002 21:56:35 Subject: Re: A 5-limit, "geometric" temperament list From: wallyesterpaulrus --- In tuning-math@y..., "hs" <straub@d...> wrote:>> 81/80 (3)^4/(2)^4/(5) meantone >> [[1, 0, -4], [0, 1, 4]] >> Newbie question: what is the meaning of this vector notation? Fromwhat I read so> far, all I can imagine are unison vectors determining a periodicity block. Something > like this? > > > Hans StraubLuckily there are no wedgies here, so i think i can explain this one. Meantone temperament takes the just lattice and modifies it by "tempering out" the interval 81/80. that is, all the consonant intervals, the "rungs" of the just lattice, are detuned slightly so that 81/80 comes out to a unison. (3)^4/(2)^4/(5) is simply the prime factorization of 81/80. 81 = (3) ^4; 1/80 = 1/(2)^4/(5), get it? Then the next line tells you the mapping of the primes in terms of the generators. You left out this line, which was included with the ones you report below: generators [1200., 1896.164845] (these are in cents) So . . . [1, 0, -4], [0, 1, 4] would be better arranged like this: [1] [0] [0] [1] [-4] [4] the first row tells you that the 2/1 is represented by [1]*1200 + [0] *1896; the second row tells you that the 3/1 is represented by [0] *1200 + [1]*1896; the third row tells you that the 5/1 is represented by [-4]*1200 + [4]*1896. Making sense? p.s. i tried to reply directly to the freelists list, but i got the following error message: The original message was received at Mon, 29 Jul 2002 21:21:32 GMT from user10.acadian-asset.com [208.253.47.10] (may be forged) ----- The following addresses had permanent fatal errors ----- <tuning-math@xxxxxxxxx.xxx> ----- Transcript of session follows ----- ... while talking to turing.freelists.org. [206.53.239.180]:>>> RCPT To:<tuning-math@xxxxxxxxx.xxx><<< 554 Service unavailable; [199.171.54.106] blocked using relays.osirusoft.com, reason: emarketing * [with cont.] (Wayb.) (services (appending, also look for 'opt-out')) 554 <tuning- math@xxxxxxxxx.xxx>... Service unavailable anyone know what's up?
Message: 5124 - Contents - Hide Contents
Date: Tue, 30 Jul 2002 01:22:06
Subject: Three {126/125, 176/175} planar temperament scales
From: Gene W Smith
These two commas go well together, and one might consider the result an
11-limit extension of starling. I checked over 4000 scales to find the
three best with these particular steps:
49 intervals 83 triads
! starra.scl (inverse of starrb.scl)
! [22, 34, 29, 22, 22, 41, 22, 22, 29, 22, 34, 29]
12 note {126/125, 176/175} scale, 328-et version
12
!
80.48780488
204.8780488
310.9756098
391.4634146
471.9512195
621.9512195
702.4390244
782.9268293
889.0243902
969.5121951
1093.902439
2/1
49 intervals 83 triads
! starrb.scl (inverse of starra.scl)
! [22, 22, 29, 34, 22, 29, 34, 22, 29, 22, 22, 41]
12 note {126/125, 176/175} scale, 328-et version
12
!
80.48780488
160.9756098
267.0731707
391.4634146
471.9512195
578.0487805
702.4390244
782.9268293
889.0243902
969.5121951
1050.000000
2/1
49 intervals 82 triads
! starrc.scl
! [22, 22, 41, 22, 22, 29, 34, 22, 29, 22, 34, 29]
12 note {126/125, 176/175} scale, 328-et version
12
!
80.48780488
160.9756098
310.9756098
391.4634146
471.9512195
578.0487805
702.4390244
782.9268293
889.0243902
969.5121951
1093.902439
2/1
5000 5050 5100 5150 5200 5250 5300 5350 5400 5450 5500 5550 5600 5650 5700 5750 5800 5850 5900 5950
5100 - 5125 -