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Message: 10625 - Contents - Hide Contents
Date: Sun, 14 Mar 2004 01:43:34
Subject: Tuning 72 notes
From: Gene Ward Smith
One has to work at it to come up with a badness figure which will not
lead to the conclusion that ennealimmal is the best 7- or 9-limit
linear temperament for tuning 72 notes to the octave, and miracle the
second-best. Who's on third? This is no longer so clear; the question
requires us to choose our badness measure. Sticking to log-flat
badness measures for temperaments consistent with the 72-et standard
val (and this situation provides an example where log-flatness clearly
seems to be the way to go) we still are left with choices. TOP error
seems a reasonable choice for an error measure, but the results are
highly dependent on complexity measures. If like me you are
particularly interested in complete 7 or 9 limit o/u-tonalities,
Graham complexity in either it's 7 or 9 version is the best choice.
In the 7-limit, perhaps suprisingly, catakleismic/hanson does not come
in third. Instead, we have this:
Wedgie: <24 20 16 -24 -42 -19|
Mapping: [<4 6 9 11|, <0 6 5 4|]
TM basis: {2401/2400, 15625/15552}
TOP generators: [300.07 17.00]
ets: 68, 72, 140, 212
In the 9-limit, catakleismic/hanson does come in third, but
<12 34 20 26 -2 -49|, <12 22 -4 7 -40 -71| and waage make respectible
showings also. I think my complexity cutoff was set too low; surely
tuning 72 notes is not an unreasonable thing to contemplate, and the
commas are by no means beyond the bounds of utility.
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Message: 10626 - Contents - Hide Contents Date: Mon, 15 Mar 2004 06:00:08 Subject: Gene's private reserve -- 7-limit From: Gene Ward Smith I decided to make some top 100 lists based on what I find useful in my own compositional practice; while people may complain no one should be interested in microtemperaments, the fact is I am. Moreover, I like complete o/u-tonalities, which means Graham complexity is the complexity measure of choice. Below I give the top 100 with a log-flat badness figure derived from 7-limit Graham complexity and TOP error, with cutoffs of 20 cents for TOP error and 100 for complexity. The next posting will be the same, for 9-limit; for me the other interesting limits are 11 and 13, so I plan to get to them eventually. 1 [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] bad 26.518833 top err .036377 graham 27 2 [22, -5, 3, -59, -57, 21] [[1, 3, 2, 3], [0, -22, 5, -3]] bad 53.601055 top err .073527 graham 27 3 [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]] bad 78.847244 top err .307998 graham 16 4 [40, 22, 21, -58, -79, -13] [[1, 21, 13, 13], [0, -40, -22, -21]] bad 85.360253 top err .053350 graham 40 5 [4, -32, -15, -60, -35, 55] [[1, 1, 7, 5], [0, 4, -32, -15]] bad 85.691520 top err .066120 graham 36 6 [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]] bad 93.944647 top err 5.871540 graham 4 7 [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] bad 106.641366 top err .631014 graham 13 8 [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] bad 111.836820 top err 3.106578 graham 6 9 [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] bad 114.473365 top err .946061 graham 11 10 [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]] bad 114.743119 top err 3.187309 graham 6 11 [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] bad 117.154200 top err 1.171542 graham 10 12 [37, 46, 75, -13, 15, 45] [[1, 15, 19, 30], [0, -37, -46, -75]] bad 121.726682 top err .021640 graham 75 13 [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]] bad 122.524768 top err 7.657798 graham 4 14 [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] bad 127.584947 top err 14.176105 graham 3 15 [38, -3, 8, -93, -94, 27] [[1, -7, 3, 1], [0, 38, -3, 8]] bad 128.468744 top err .076424 graham 41 16 [14, 59, 33, 61, 13, -89] [[1, -3, -17, -8], [0, 14, 59, 33]] bad 130.053641 top err .037361 graham 59 17 [33, 78, 90, 47, 50, -10] [[3, 9, 17, 20], [0, -11, -26, -30]] bad 135.548384 top err .016734 graham 90 18 [19, 19, 57, -14, 37, 79] [[19, 30, 44, 53], [0, 1, 1, 3]] bad 149.622591 top err .046052 graham 57 19 [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]] bad 151.169908 top err 1.049791 graham 12 20 [2, 1, 3, -3, -1, 4] [[1, 1, 2, 2], [0, 2, 1, 3]] bad 151.541080 top err 16.837898 graham 3 21 [2, 1, -1, -3, -7, -5] [[1, 1, 2, 3], [0, 2, 1, -1]] bad 151.866975 top err 16.874108 graham 3 22 [55, 73, 93, -12, -7, 11] [[1, -19, -25, -32], [0, 55, 73, 93]] bad 153.712613 top err .017772 graham 93 23 [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]] bad 156.178134 top err 3.187309 graham 7 24 [34, 29, 23, -33, -59, -28] [[1, -7, -5, -3], [0, 34, 29, 23]] bad 161.265468 top err .139503 graham 34 25 [58, 49, 39, -57, -101, -47] [[1, -13, -10, -7], [0, 58, 49, 39]] bad 161.751212 top err .048083 graham 58 26 [96, 99, 6, -66, -260, -264] [[3, -2, 0, 8], [0, 32, 33, 2]] bad 163.318824 top err .016663 graham 99 27 [1, -8, 39, -15, 59, 113] [[1, 2, -1, 19], [0, -1, 8, -39]] bad 164.610262 top err .074518 graham 47 28 [24, 20, 16, -24, -42, -19] [[4, 6, 9, 11], [0, 6, 5, 4]] bad 167.638136 top err .291038 graham 24 29 [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] bad 169.852100 top err 1.698521 graham 10 30 [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] bad 171.757764 top err 4.771049 graham 6 31 [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]] bad 179.406875 top err .620785 graham 17 32 [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] bad 180.990733 top err 7.239629 graham 5 33 [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] bad 183.851136 top err 1.276744 graham 12 34 [78, 72, -12, -67, -238, -230] [[6, 15, 19, 16], [0, -13, -12, 2]] bad 185.389623 top err .022888 graham 90 35 [60, -8, 11, -152, -151, 48] [[1, 19, 0, 6], [0, -60, 8, -11]] bad 185.994225 top err .040224 graham 68 36 [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]] bad 187.279432 top err .299647 graham 25 37 [23, -1, 13, -55, -44, 33] [[1, 9, 2, 7], [0, -23, 1, -13]] bad 189.152064 top err .328389 graham 24 38 [15, 51, 72, 46, 72, 24] [[3, 3, 1, 0], [0, 5, 17, 24]] bad 190.153222 top err .036681 graham 72 39 [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]] bad 190.492116 top err 5.291448 graham 6 40 [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]] bad 191.882395 top err .479706 graham 20 41 [56, 24, 26, -92, -116, -7] [[2, 4, 5, 6], [0, -28, -12, -13]] bad 193.718370 top err .061772 graham 56 42 [4, 4, 0, -3, -11, -11] [[4, 6, 9, 11], [0, 1, 1, 0]] bad 198.154432 top err 12.384652 graham 4 43 [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]] bad 198.959296 top err 5.526647 graham 6 44 [20, -30, -10, -94, -72, 61] [[10, 16, 23, 28], [0, -2, 3, 1]] bad 200.488970 top err .080196 graham 50 45 [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] bad 205.403314 top err .912903 graham 15 46 [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]] bad 209.180089 top err 3.268439 graham 8 47 [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] bad 211.351646 top err 5.870879 graham 6 48 [3, -24, -54, -45, -94, -58] [[3, 5, 5, 4], [0, -1, 8, 18]] bad 211.576339 top err .065120 graham 57 49 [41, 14, 60, -73, -20, 100] [[1, -14, -3, -20], [0, 41, 14, 60]] bad 217.304404 top err .060362 graham 60 50 [39, 42, 6, -24, -100, -104] [[3, 2, 4, 8], [0, 13, 14, 2]] bad 221.669129 top err .125663 graham 42 51 [23, -13, 42, -74, 2, 134] [[1, 11, -3, 20], [0, -23, 13, -42]] bad 224.748425 top err .074297 graham 55 52 [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]] bad 226.094010 top err 14.130876 graham 4 53 [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]] bad 226.094010 top err 14.130876 graham 4 54 [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]] bad 228.058272 top err 14.253642 graham 4 55 [17, 35, -21, 16, -81, -147] [[1, -1, -3, 6], [0, 17, 35, -21]] bad 229.263286 top err .073107 graham 56 56 [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]] bad 229.293222 top err 3.582707 graham 8 57 [9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]] bad 231.854142 top err 1.371918 graham 13 58 [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]] bad 232.504691 top err 14.531543 graham 4 59 [57, 57, 0, -42, -160, -160] [[57, 90, 132, 160], [0, 1, 1, 0]] bad 234.286461 top err .072110 graham 57 60 [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]] bad 234.805888 top err 3.668842 graham 8 61 [23, 40, 1, 10, -63, -110] [[1, 6, 10, 3], [0, -23, -40, -1]] bad 237.044800 top err .148153 graham 40 62 [78, 19, 29, -151, -173, 14] [[1, 29, 9, 13], [0, -78, -19, -29]] bad 237.317411 top err .039007 graham 78 63 [39, 30, -18, -43, -138, -126] [[3, 10, 11, 6], [0, -13, -10, 6]] bad 237.524367 top err .073107 graham 57 64 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]] bad 238.136875 top err 2.939961 graham 9 65 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]] bad 238.136875 top err 2.939961 graham 9 66 [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]] bad 239.419651 top err 3.740932 graham 8 67 [21, 3, -36, -44, -116, -92] [[3, 5, 7, 8], [0, -7, -1, 12]] bad 243.155160 top err .074840 graham 57 68 [62, 17, 24, -117, -136, 8] [[1, 15, 6, 8], [0, -62, -17, -24]] bad 243.206036 top err .063269 graham 62 69 [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]] bad 243.351404 top err 9.734056 graham 5 70 [42, -35, -7, -153, -129, 82] [[7, 9, 18, 20], [0, 6, -5, -1]] bad 245.608557 top err .041425 graham 77 71 [40, 75, -20, 26, -144, -257] [[5, 6, 8, 15], [0, 8, 15, -4]] bad 246.431399 top err .027305 graham 95 72 [21, 15, -12, -25, -78, -70] [[3, 2, 5, 10], [0, 7, 5, -4]] bad 247.770369 top err .227521 graham 33 73 [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]] bad 248.495400 top err .276106 graham 30 74 [18, 15, -6, -18, -60, -56] [[3, 6, 8, 8], [0, -6, -5, 2]] bad 258.439259 top err .448679 graham 24 75 [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]] bad 259.596304 top err .536356 graham 22 76 [12, 10, -9, -12, -48, -49] [[1, 0, 1, 4], [0, 12, 10, -9]] bad 259.647200 top err .588769 graham 21 77 [32, 86, 51, 62, -9, -123] [[1, 13, 33, 21], [0, -32, -86, -51]] bad 260.834732 top err .035267 graham 86 78 [76, 76, 57, -56, -123, -81] [[19, 31, 45, 54], [0, -4, -4, -3]] bad 261.710560 top err .045310 graham 76 79 [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]] bad 262.233496 top err 1.337926 graham 14 80 [13, 9, 1, -16, -35, -23] [[1, 4, 4, 3], [0, -13, -9, -1]] bad 267.037238 top err 1.580102 graham 13 81 [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]] bad 272.169318 top err 1.610469 graham 13 82 [74, 51, 44, -91, -138, -41] [[1, -25, -16, -13], [0, 74, 51, 44]] bad 273.596225 top err .049963 graham 74 83 [63, 9, 63, -132, -77, 121] [[9, 15, 21, 26], [0, -7, -1, -7]] bad 273.634592 top err .068943 graham 63 84 [59, 41, 78, -72, -42, 66] [[1, 4, 4, 6], [0, -59, -41, -78]] bad 274.339233 top err .045092 graham 78 85 [11, -6, 10, -35, -15, 40] [[1, 4, 1, 5], [0, -11, 6, -10]] bad 274.666756 top err .950404 graham 17 86 [52, 56, 41, -32, -81, -62] [[1, -21, -22, -15], [0, 52, 56, 41]] bad 274.867264 top err .087649 graham 56 87 [60, 45, -30, -68, -216, -196] [[15, 24, 35, 42], [0, -4, -3, 2]] bad 277.368300 top err .034243 graham 90 88 [26, -37, -12, -119, -92, 76] [[1, -1, 6, 4], [0, 26, -37, -12]] bad 278.973072 top err .070288 graham 63 89 [3, 2, -1, -4, -10, -8] [[1, 1, 2, 3], [0, 3, 2, -1]] bad 281.038694 top err 17.564918 graham 4 90 [12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]] bad 284.925888 top err .421488 graham 26 91 [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] bad 287.050049 top err 1.698521 graham 13 92 [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]] bad 289.868296 top err .894655 graham 18 93 [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]] bad 290.869402 top err 2.403879 graham 11 94 [2, -57, -28, -95, -50, 95] [[1, 1, 19, 11], [0, 2, -57, -28]] bad 292.014128 top err .083888 graham 59 95 [45, -18, 45, -133, -55, 155] [[9, 14, 21, 25], [0, 5, -2, 5]] bad 294.245784 top err .074136 graham 63 96 [5, -11, -12, -29, -33, 3] [[1, 4, -3, -3], [0, -5, 11, 12]] bad 295.707979 top err 1.023211 graham 17 97 [11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]] bad 296.963913 top err .916555 graham 18 98 [1, -3, -2, -7, -6, 4] [[1, 2, 1, 2], [0, -1, 3, 2]] bad 298.143022 top err 18.633939 graham 4 99 [5, -40, 24, -75, 24, 168] [[1, 1, 7, 0], [0, 5, -40, 24]] bad 299.445924 top err .073107 graham 64 100 [13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]] bad 300.099731 top err .567296 graham 23
Message: 10627 - Contents - Hide Contents Date: Mon, 15 Mar 2004 19:14:32 Subject: Nonoctave ennealimmal From: Gene Ward Smith The 1 in the wedgie for ennealimmal (<18 27 18 1 -22 -34|) caught my eye, and it occurred to me that a generator of 5/3 and a period of 3 (or 5, but 3 seems more plausible) works for a non-octave ennealimmal. We can take the generators as the TOP tunings of 3 and 5/3, or as 3 and 3^(451/970), etc etc. The mapping is [<9 1 1 12|, <-18 0 1 -22|]. I accidentally shut down my Maple session, so I'm giving DE scales of size 13, 15, 28 and 43 within the "tritave" in terms of the Scala files I created. These are justly tuned versions; if you temper this by your favorite ennealimmal tuning you get tempered versions, but you can simply take them as is, and make use of the inherent approximations. ! ennon13.scl Nonoctave Ennealimmal, [3, 5/3] just tuning 13 ! 27/25 7/6 63/50 10/7 54/35 5/3 9/5 35/18 21/10 50/21 18/7 25/9 3 ! ennon15.scl Nonoctave Ennealimmal, [3, 5/3] just tuning 15 ! 27/25 7/6 63/50 250/189 10/7 54/35 5/3 9/5 35/18 21/10 245/108 50/21 18/7 25/9 3 ! ennon28.scl Nonoctave Ennealimmal, [3, 5/3] just tuning 28 ! 21/20 27/25 245/216 7/6 49/40 63/50 250/189 49/36 10/7 72/49 54/35 100/63 5/3 7/4 9/5 189/100 35/18 49/24 21/10 108/49 245/108 50/21 49/20 18/7 500/189 25/9 20/7 3 ! ennon43.scl Nonoctave Ennealimmal, [3, 5/3] just tuning 43 ! 36/35 21/20 27/25 10/9 245/216 7/6 6/5 49/40 63/50 35/27 250/189 49/36 25/18 10/7 72/49 3/2 54/35 100/63 81/50 5/3 12/7 7/4 9/5 50/27 189/100 35/18 2 49/24 21/10 54/25 108/49 245/108 81/35 50/21 49/20 5/2 18/7 500/189 27/10 25/9 20/7 35/12 3
Message: 10628 - Contents - Hide Contents Date: Mon, 15 Mar 2004 06:03:42 Subject: Gene's private reserve -- 9-limit From: Gene Ward Smith 1 [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]] bad 47.145262 top err .036378 graham 36 2 [4, -32, -15, -60, -35, 55] [[1, 1, 7, 5], [0, 4, -32, -15]] bad 105.792000 top err .066120 graham 40 3 [37, 46, 75, -13, 15, 45] [[1, 15, 19, 30], [0, -37, -46, -75]] bad 121.726682 top err .021640 graham 75 4 [14, 59, 33, 61, 13, -89] [[1, -3, -17, -8], [0, 14, 59, 33]] bad 130.053641 top err .037361 graham 59 5 [33, 78, 90, 47, 50, -10] [[3, 9, 17, 20], [0, -11, -26, -30]] bad 135.548384 top err .016734 graham 90 6 [19, 19, 57, -14, 37, 79] [[19, 30, 44, 53], [0, 1, 1, 3]] bad 149.622591 top err .046052 graham 57 7 [1, -8, 39, -15, 59, 113] [[1, 2, -1, 19], [0, -1, 8, -39]] bad 164.610262 top err .074518 graham 47 8 [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]] bad 169.852100 top err 1.698521 graham 10 9 [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]] bad 171.757764 top err 4.771049 graham 6 10 [22, -5, 3, -59, -57, 21] [[1, 3, 2, 3], [0, -22, 5, -3]] bad 176.537906 top err .073527 graham 49 11 [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] bad 180.990733 top err 7.239629 graham 5 12 [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]] bad 183.851136 top err 1.276744 graham 12 13 [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]] bad 187.279432 top err .299647 graham 25 14 [15, 51, 72, 46, 72, 24] [[3, 3, 1, 0], [0, 5, 17, 24]] bad 190.153222 top err .036681 graham 72 15 [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]] bad 198.821014 top err 3.106578 graham 8 16 [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]] bad 211.351646 top err 5.870879 graham 6 17 [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]] bad 226.094010 top err 14.130876 graham 4 18 [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]] bad 226.817684 top err 14.176105 graham 4 19 [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] bad 227.796054 top err .631014 graham 19 20 [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]] bad 228.058272 top err 14.253642 graham 4 21 [17, 35, -21, 16, -81, -147] [[1, -1, -3, 6], [0, 17, 35, -21]] bad 229.263286 top err .073107 graham 56 22 [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]] bad 229.293222 top err 3.582707 graham 8 23 [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]] bad 233.703326 top err .912904 graham 16 24 [3, -24, -54, -45, -94, -58] [[3, 5, 5, 4], [0, -1, 8, 18]] bad 234.433616 top err .065120 graham 60 25 [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]] bad 234.805888 top err 3.668842 graham 8 26 [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]] bad 248.495400 top err .276106 graham 30 27 [23, -13, 42, -74, 2, 134] [[1, 11, -3, 20], [0, -23, 13, -42]] bad 258.627857 top err .074297 graham 59 28 [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]] bad 259.596304 top err .536356 graham 22 29 [32, 86, 51, 62, -9, -123] [[1, 13, 33, 21], [0, -32, -86, -51]] bad 260.834732 top err .035267 graham 86 30 [2, 1, 3, -3, -1, 4] [[1, 1, 2, 2], [0, 2, 1, 3]] bad 269.406365 top err 16.837898 graham 4 31 [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]] bad 273.411591 top err .946061 graham 17 32 [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] bad 287.050049 top err 1.698521 graham 13 33 [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]] bad 289.868296 top err .894655 graham 18 34 [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]] bad 290.869402 top err 2.403879 graham 11 35 [5, -40, 24, -75, 24, 168] [[1, 1, 7, 0], [0, 5, -40, 24]] bad 299.445924 top err .073107 graham 64 36 [2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]] bad 302.412996 top err 8.400361 graham 6 37 [2, -57, -28, -95, -50, 95] [[1, 1, 19, 11], [0, 2, -57, -28]] bad 312.146463 top err .083888 graham 61 38 [23, 40, 1, 10, -63, -110] [[1, 6, 10, 3], [0, -23, -40, -1]] bad 313.490954 top err .148153 graham 46 39 [8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]] bad 313.872084 top err .968741 graham 18 40 [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]] bad 315.388978 top err .307997 graham 32 41 [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]] bad 315.651990 top err 1.610469 graham 14 42 [13, 14, 35, -8, 19, 42] [[1, 5, 6, 12], [0, -13, -14, -35]] bad 320.869279 top err .261934 graham 35 43 [12, 34, 20, 26, -2, -49] [[2, 4, 7, 7], [0, -6, -17, -10]] bad 329.419540 top err .284965 graham 34 44 [12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]] bad 330.446592 top err .421488 graham 28 45 [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]] bad 338.267387 top err 9.396316 graham 6 46 [40, 22, 21, -58, -79, -13] [[1, 21, 13, 13], [0, -40, -22, -21]] bad 341.441012 top err .053350 graham 80 47 [1, 33, 27, 50, 40, -30] [[1, 2, 16, 14], [0, -1, -33, -27]] bad 347.940945 top err .319505 graham 33 48 [22, 48, -38, 25, -122, -223] [[2, 7, 13, -1], [0, -11, -24, 19]] bad 349.810020 top err .047297 graham 86 49 [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]] bad 350.426022 top err 9.734056 graham 6 50 [0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]] bad 355.421548 top err .617051 graham 24 51 [11, -11, 22, -43, 4, 82] [[11, 17, 26, 30], [0, 1, -1, 2]] bad 367.883802 top err .337818 graham 33 52 [11, 83, 87, 106, 107, -31] [[1, 3, 13, 14], [0, -11, -83, -87]] bad 368.663283 top err .048707 graham 87 53 [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]] bad 375.778589 top err 5.871540 graham 8 54 [3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]] bad 377.962898 top err 2.624742 graham 12 55 [35, 62, -3, 17, -103, -181] [[1, 11, 19, 2], [0, -35, -62, 3]] bad 384.893321 top err .072226 graham 73 56 [13, 67, -6, 76, -46, -202] [[1, -1, -11, 4], [0, 13, 67, -6]] bad 385.382622 top err .072318 graham 73 57 [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]] bad 385.664389 top err 3.187309 graham 11 58 [20, -30, -10, -94, -72, 61] [[10, 16, 23, 28], [0, -2, 3, 1]] bad 392.958382 top err .080196 graham 70 59 [41, 14, 60, -73, -20, 100] [[1, -14, -3, -20], [0, 41, 14, 60]] bad 405.876336 top err .060362 graham 82 60 [6, 29, -2, 32, -20, -86] [[1, 4, 14, 2], [0, -6, -29, 2]] bad 405.886038 top err .422358 graham 31 61 [4, 21, -3, 24, -16, -66] [[1, 0, -6, 4], [0, 4, 21, -3]] bad 417.917376 top err .725551 graham 24 62 [2, 1, -1, -3, -7, -5] [[1, 1, 2, 3], [0, 2, 1, -1]] bad 421.852709 top err 16.874108 graham 5 63 [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] bad 423.354444 top err 2.939961 graham 12 64 [10, 91, 48, 121, 48, -144] [[1, 1, -3, 0], [0, 10, 91, 48]] bad 423.970638 top err .051198 graham 91 65 [1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]] bad 428.654980 top err 2.536420 graham 13 66 [20, 52, 31, 36, -7, -74] [[1, 3, 6, 5], [0, -20, -52, -31]] bad 429.432255 top err .158814 graham 52 67 [1, 6, 5, 7, 5, -5] [[1, 2, 5, 5], [0, -1, -6, -5]] bad 430.921548 top err 11.970043 graham 6 68 [12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]] bad 432.350324 top err .639571 graham 26 69 [3, 5, 1, 1, -7, -12] [[1, 2, 3, 3], [0, -3, -5, -1]] bad 434.386260 top err 12.066285 graham 6 70 [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]] bad 434.821376 top err 1.698521 graham 16 71 [7, 26, 25, 25, 20, -15] [[1, 5, 15, 15], [0, -7, -26, -25]] bad 437.476180 top err .647154 graham 26 72 [3, -24, -1, -45, -10, 65] [[1, 1, 7, 3], [0, 3, -24, -1]] bad 437.698051 top err .486331 graham 30 73 [7, 38, -4, 44, -26, -116] [[1, 3, 10, 2], [0, -7, -38, 4]] bad 437.858567 top err .248219 graham 42 74 [3, -12, -30, -26, -56, -36] [[3, 5, 6, 6], [0, -1, 4, 10]] bad 438.741401 top err .338535 graham 36 75 [14, 6, 74, -23, 78, 155] [[2, 4, 5, 10], [0, -7, -3, -37]] bad 441.009660 top err .080535 graham 74 76 [11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]] bad 443.612760 top err .916555 graham 22 77 [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]] bad 445.541726 top err 3.094040 graham 12 78 [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]] bad 447.347281 top err 3.106578 graham 12 79 [7, -56, -69, -105, -129, -3] [[1, 4, -17, -21], [0, -7, 56, 69]] bad 448.614773 top err .065120 graham 83 80 [4, 21, -56, 24, -100, -189] [[1, 0, -6, 25], [0, 4, 21, -56]] bad 451.117259 top err .076087 graham 77 81 [21, 3, -36, -44, -116, -92] [[3, 5, 7, 8], [0, -7, -1, 12]] bad 455.326560 top err .074840 graham 78 82 [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]] bad 458.972474 top err 3.187309 graham 12 83 [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]] bad 462.139139 top err 9.431411 graham 7 84 [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]] bad 462.957790 top err 1.049791 graham 21 85 [0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]] bad 463.097566 top err 3.215955 graham 12 86 [1, -3, -2, -7, -6, 4] [[1, 2, 1, 2], [0, -1, 3, 2]] bad 465.848472 top err 18.633939 graham 5 87 [3, 7, -1, 4, -10, -22] [[1, 1, 1, 3], [0, 3, 7, -1]] bad 465.860116 top err 7.279064 graham 8 88 [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]] bad 468.616800 top err 1.171542 graham 20 89 [38, -3, 8, -93, -94, 27] [[1, -7, 3, 1], [0, 38, -3, 8]] bad 476.962184 top err .076424 graham 79 90 [0, 5, 5, 8, 8, -2] [[5, 8, 12, 14], [0, 0, -1, -1]] bad 484.223081 top err 19.368923 graham 5 91 [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]] bad 489.399495 top err 2.895855 graham 13 92 [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]] bad 490.099072 top err 7.657798 graham 8 93 [8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]] bad 494.179835 top err 1.525246 graham 18 94 [5, -11, -12, -29, -33, 3] [[1, 4, -3, -3], [0, -5, 11, 12]] bad 495.234124 top err 1.023211 graham 22 95 [0, 0, 5, 0, 8, 12] [[5, 8, 12, 14], [0, 0, 0, 1]] bad 496.017125 top err 19.840685 graham 5 96 [1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]] bad 497.431297 top err 4.974313 graham 10 97 [1, 4, 5, 4, 5, 0] [[1, 2, 4, 5], [0, -1, -4, -5]] bad 499.434900 top err 19.977396 graham 5 98 [6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]] bad 501.183396 top err 3.480440 graham 12 99 [2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]] bad 507.038958 top err 1.267597 graham 20 100 [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]] bad 508.711523 top err 14.130876 graham 6
Message: 10629 - Contents - Hide Contents Date: Mon, 15 Mar 2004 06:52:36 Subject: Re: Gene's private reserve -- 7-limit From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> Rats--two temperaments with identical TOP errors *and* Graham complexities. I was aware this could happen, but guessed, apparently incorrectly, that it was quite unlikely and then forgot to check. The missing one gets 64th place since it is better in the 9-limit. 64 [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] bad 238.136875 top err 2.939961 graham 9> 64 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]] > bad 238.136875 top err 2.939961 graham 9 > > 65 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]] > bad 238.136875 top err 2.939961 graham 9
Message: 10630 - Contents - Hide Contents Date: Mon, 15 Mar 2004 07:55:21 Subject: Re: Gene's private reserve -- 7-limit From: Graham Breed Gene Ward Smith wrote:> Rats--two temperaments with identical TOP errors *and* Graham > complexities. I was aware this could happen, but guessed, apparently > incorrectly, that it was quite unlikely and then forgot to check. The > missing one gets 64th place since it is better in the 9-limit.I've verified that, contorsion aside, each temperament I look at does have a unique badness. I need this to test my temperament finder (now in three languages!). I can't prove it but it's always worked so far.> 64 [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]] > bad 238.136875 top err 2.939961 graham 9 > >>> 64 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]] >> bad 238.136875 top err 2.939961 graham 9Oh yes, that'll be because it's a kind of minimax. The 7 mapping presumable doesn't contribute to the TOP error, and the complexities happen to be the same. They should be distinct if you switch to some kind of RMS. Graham ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 10632 - Contents - Hide Contents Date: Tue, 16 Mar 2004 10:26:25 Subject: Minimal filled scale From: Gene Ward Smith Suppose we have a linear temperament with octave period. A chord-type in this temperament is a set of generators. A question we might ask is what is the what is the cardinality of the smallest set of contiguous generators which arise from contiguous generator translates of the chord--the minimal filled scale for the chord. This doesn't depend on the temperament, but only on the chord, considered as a set or list. Maple code for it is this: mfs := proc(l) # minimal filled scale from chord l local i, s, u; u := sort(convert(l,list)); s := -1; for i to nops(u)-1 do s := max(s,u[i+1]-u[i]) od; s+u[nops(u)]-u[1] end: This program sorts the scale (converting it first to a list if it starts out as a set or array Maple data type.) It then finds the maximum chord interval in terms of generator steps, and adds that to the span of the scale--the difference between the least and greatest element in terms of generator steps. Extending it to non-octave periods involves deciding what definition is then best. The minimal filled scale for septimal miracle is Miracle[19], and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit meantone we get the diatonic scale (Meantone[7]), and in the 7-limit, Meantone[16]. And so on and so forth... ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 10633 - Contents - Hide Contents Date: Tue, 16 Mar 2004 18:07:10 Subject: Re: Minimal filled scale From: Gene Ward Smith For some reason this didn't get posted; I'll try again. --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The minimal filled scale for septimal miracle is Miracle[19], and for > 11-limit miracle is Canasta (Miracle[31].) For 5-limit meantone we get > the diatonic scale (Meantone[7]), and in the 7-limit, Meantone[16].This was less than clear, because I forgot to say I was using the major p-limit chord as my defining chord for this. Probably I should have used odd limit for 11-limit meantone, including 9, which would have given me Miracle[28]. One definition for fractional-octave periods is simply to multiply each generator by the number of periods in an octave, so the 7-limit minimal filled scale for pajara would be what you get by putting {0,2,-4,-4} into the program I gave, which would be Paul's favorite Pajara[10]; going to the 9-limit would give Pajara[12].
Message: 10634 - Contents - Hide Contents Date: Tue, 16 Mar 2004 18:15:03 Subject: Re: Minimal filled scale From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "hstraub64" <hstraub64@t...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> Suppose we have a linear temperament with octave period. A chord-type >> in this temperament is a set of generators. A question we might ask is >> what is the what is the cardinality of the smallest set of contiguous >> generators which arise from contiguous generator translates of the >> chord--the minimal filled scale for the chord. This doesn't depend on >> the temperament, but only on the chord, considered as a set or list. >> >> Something like a transitive hull?I don't see a connection.
Message: 10635 - Contents - Hide Contents Date: Tue, 16 Mar 2004 22:12:10 Subject: Chords of Blackjack From: Gene Ward Smith The complete 7-limit o/u-tonalities, or tetrads, as well as the complete 9-limit o/u-tonalities, form a rectangular lattice in a natural way. We can reduce by a comma set in this lattice to minimize the distance (Euclidean or what have you) of the chord from [0,0,0], the major tonic. If we do this to the 16 tetrads of Blackjack, we find we have two pairs of tetrads--{[1,1,-2], [-1,-2,1]} and {[-1,-1,2], [1,2,-1]}--with identical Euclidean distances from [0,0,0], which represent the same chord of Blackjack. Choosing +-[1,2,-1] as our representative gives us this as the chords of Blackjack in reduced form: {[1, 2, 0], [0, 0, 1], [-1, -1, 0], [1, 1, -1], [1, 2, 1], [0, 0, 0], [0, 0, 2], [1, 2, -1], [1, 1, 0], [-1, -1, 1], [0, 1, -2], [-1, -2, 1], [0, 0, -1], [-1, -1, -1], [1, 1, 1], [0, 0, -2]} Should anyone feel inspired to draw a 3D diagram of this, it would be interesting to see what it looked like.
Message: 10637 - Contents - Hide Contents Date: Wed, 17 Mar 2004 22:46:12 Subject: Symmetric 7-limit comma badness and 2401/2400 From: Gene Ward Smith If you take, for a 7-limit interval q and a symmetric lattice distance dist, the function cents(q) dist(q)^4, you get a log-flat symmetric badness measure for 7-limit commas. Euclidean and Hahn are not very different; below I use Hahn distance, and order some commas with badness less than 3000 from best to worst. The list is completely dominated by superparticulars, and it looks to me as if 2401/2400 is likely to be an absolute minimum in badness. At any rate looking at this makes the Erlich phenomenon--the great importance of 2401/2400 for 7-limit micro ets--more understandable. 2401/2400 184.626652 8/7 231.174094 7/6 266.870906 6/5 315.641287 5/4 386.313714 4/3 498.044999 50/49 559.609830 49/48 571.148984 7/5 582.512193 10/7 617.487808 3/2 701.955001 36/35 780.326114 8/5 813.686286 5/3 884.358713 12/7 933.129094 4375/4374 950.211153 7/4 968.825906 126/125 1117.376095 25/24 1130.758839 21/20 1351.475096 16/15 1787.700573 15/14 1911.084922 225/224 1974.150012 250047/250000 2135.221095 1029/1024 2158.776169 64/63 2208.391452 35/32 2482.233925 |-92 -17 21 25> 2860.311932 10/9 2918.459394
Message: 10638 - Contents - Hide Contents Date: Wed, 17 Mar 2004 23:53:40 Subject: Re: Minimal filled scale From: Carl Lumma Greetings from beautiful Portland!>Suppose we have a linear temperament with octave period. A >chord-type in this temperament is a set of generators. A >question we might ask is what is the what is the cardinality >of the smallest set of contiguous generators which arise from >contiguous generator translates of the chord--the minimal >filled scale for the chord."Continuous generator translates"??>This doesn't depend on the temperament, but only on the chord, >considered as a set or list.Huh? It must depend on the mapping.>The minimal filled scale for septimal miracle is Miracle[19], >and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit >meantone we get the diatonic scale (Meantone[7]), and in the >7-limit, Meantone[16]. And so on and so forth...How is this different from Graham complexity? -Carl
Message: 10639 - Contents - Hide Contents Date: Thu, 18 Mar 2004 18:28:31 Subject: Re: Minimal filled scale From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:> Greetings from beautiful Portland! >>> Suppose we have a linear temperament with octave period. A >> chord-type in this temperament is a set of generators. A >> question we might ask is what is the what is the cardinality >> of the smallest set of contiguous generators which arise from >> contiguous generator translates of the chord--the minimal >> filled scale for the chord. >> "Continuous generator translates"??Contiguous. I mean if we have for example a chord [0 1 4 10], we take [1 2 5 11] [1 3 6 12] etc. until we've filled all the holes, and every note is harmonizable by at least one such chord.>> This doesn't depend on the temperament, but only on the chord, >> considered as a set or list. >> Huh? It must depend on the mapping.See above--it depends on what is mapped to (say [0 1 4 10] for 7-limit meantone) but not at all on where it is mapped from.>> The minimal filled scale for septimal miracle is Miracle[19], >> and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit >> meantone we get the diatonic scale (Meantone[7]), and in the >> 7-limit, Meantone[16]. And so on and so forth... >> How is this different from Graham complexity?How is it the same?
Message: 10640 - Contents - Hide Contents Date: Thu, 18 Mar 2004 18:30:47 Subject: Re: Symmetric 7-limit comma badness and 2401/2400 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:>> If you take, for a 7-limit interval q and a symmetric >> lattice distance dist, the function cents(q) dist(q)^4, >> Why 4? It used to be pi(lim)-1, which would be 3 in the > 7-limit.The exponent should be rank(Group)/rank(Kernel); the rank of the group for p-limit will be pi(p), and the rank of the kernel for codimension one temperaments is of course one.
Message: 10641 - Contents - Hide Contents Date: Thu, 18 Mar 2004 22:03:33 Subject: Re: Symmetric 7-limit comma badness and 2401/2400 From: Carl Lumma>>> >f you take, for a 7-limit interval q and a symmetric >>> lattice distance dist, the function cents(q) dist(q)^4, >>>> Why 4? It used to be pi(lim)-1, which would be 3 in the >> 7-limit. >>The exponent should be rank(Group)/rank(Kernel); the rank of >the group for p-limit will be pi(p), and the rank of the >kernel for codimension one temperaments is of course one.You were among the people to review my code, which used pi(lim)-1. Is the above due to that we're no longer assuming octave equivalence or something? I also asked:>...I'm only returned the 10 best results, and I only >search q with d <= 3000 and cents(q) <= 600. What bounds >does your method require? -Carl
Message: 10642 - Contents - Hide Contents Date: Thu, 18 Mar 2004 22:01:06 Subject: Re: Minimal filled scale From: Carl Lumma>> >reetings from beautiful Portland! >>>>> Suppose we have a linear temperament with octave period. A >>> chord-type in this temperament is a set of generators. A >>> question we might ask is what is the what is the cardinality >>> of the smallest set of contiguous generators which arise from >>> contiguous generator translates of the chord--the minimal >>> filled scale for the chord. >>>> "Continuous generator translates"?? > >Contiguous.Weird; high-level typo; I read it correctly.>I mean if we have for example a chord [0 1 4 10], we take >[1 2 5 11] [1 3 6 12] etc.You mean [2 3 6 12]?>until we've filled all the holes,I still don't get it. You're harmonizing every note of the original chord?>and every >note is harmonizable by at least one such chord.The original chord has this property...>>> The minimal filled scale for septimal miracle is Miracle[19], >>> and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit >>> meantone we get the diatonic scale (Meantone[7]), and in the >>> 7-limit, Meantone[16]. And so on and so forth... >>>> How is this different from Graham complexity? >>How is it the same?I was hoping an explanation of the difference would help us understand it. -Carl
Message: 10644 - Contents - Hide Contents Date: Thu, 18 Mar 2004 00:36:36 Subject: Re: Symmetric 7-limit comma badness and 2401/2400 From: Carl Lumma>If you take, for a 7-limit interval q and a symmetric >lattice distance dist, the function cents(q) dist(q)^4,Why 4? It used to be pi(lim)-1, which would be 3 in the 7-limit.>you get a log-flat symmetric >badness measure for 7-limit commas. Euclidean and Hahn are >not very different; below I use Hahn distance, and order >some commas with badness less than 3000 from best to worst. >The list is completely dominated by superparticulars, and >it looks to me as if 2401/2400 is likely to be an absolute >minimum in badness. At any rate looking at this makes the >Erlich phenomenon--the great importance of 2401/2400 >for 7-limit micro ets--more understandable. > >2401/2400 184.626652 >8/7 231.174094 >7/6 266.870906 >6/5 315.641287 >5/4 386.313714 >4/3 498.044999 >50/49 559.609830 >49/48 571.148984 >7/5 582.512193 >10/7 617.487808If q = n/d, then using (n-d)/d instead of cents(q) and log(d)^3 instead of dist(q)^4, I get... ((0.19645692845300844 7 2401/2400) (0.4419896533813025 3 4/3) (0.6660493039778589 5 5/4) (0.7075223009389495 7 225/224) (0.8337823128571303 5 6/5) (0.9004848978857011 7 126/125) (0.9587113625980545 7 7/6) (1.0518045661034596 5 81/80) (1.0526168298843461 7 8/7) (1.1239582004626365 3 9/8)) ...I'm only returned the 10 best results, and I only search q with d <= 3000 and cents(q) <= 600. What bounds does your method require? -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 10645 - Contents - Hide Contents Date: Thu, 18 Mar 2004 17:44:12 Subject: Re: Minimal filled scale From: Carl Lumma>>> > mean if we have for example a chord [0 1 4 10], we take >>> [1 2 5 11] [1 3 6 12] etc. >>>> You mean [2 3 6 12]? > >Right. >>>> until we've filled all the holes, >>>> I still don't get it. You're harmonizing every note of the >> original chord? >>No, I'm harmonizing everything with translates of the chord in a >minimal contiguous-generator scale containing the chord. > >>> and every>>> note is harmonizable by at least one such chord. >>>> The original chord has this property... >>No, the numbers from 0 to 10 only find harmonies for 0, 1, 4 and 10. >2, 3, 5, 6, 7, 8 and 9 have no major tetrad. If, however, I take the >numbers from 0 to 15, every one of them has a major tetrad to >harmonize it. The union of the sets {i,i+1,i+4,i+10} as i ranges from >0 to 5 is {0..15}; no smaller value than 5 will work.Got it. Cool. -Carl
Message: 10647 - Contents - Hide Contents Date: Fri, 19 Mar 2004 21:15:33 Subject: Yantras From: Gene Ward Smith Yantras are an idea of Ernest McClain which he modestly attributes to anyone over the age of 2000. We can define them in general as follows--first, define a function for the nth smallest p-limit odd integer. If we order the p-limit odd integers in ascending order of size, we may call the nth number in the sequence pl_p(n). For instance the 5-limit sequence begins 1, 3, 5, 9, 15, 25 ... so pl_5(4)=9. We then may define the n-note p-limit yantra, yantra_p(n), as the reduction to an octave of pl_p(1) through pl_p(n). Yantras have interesting properties. Yantra_p(n) is a scale of n notes, and if n<m then yantra_p(n) is strictly contained in yantra_p(m). Any note which is octave equivalent to pl_p(n), where n<=floor(pl_p(m)/p), will have a complete p-limit otonality for which it is the root.
Message: 10648 - Contents - Hide Contents Date: Fri, 19 Mar 2004 21:57:28 Subject: Yantra commas From: Gene Ward Smith For yantra_p(n) for any p and n, there will be a smallest scale-step interval. As n increases, these form a non-increasing sequence, which only the more talented p-limit commas will form a part of. For 7-limit yantras, for example, the sequence (ignoring repetitions) goes 2, 4/3, 6/5, 8/7, 10/9, 16/15, 21/20, 25/24, 28/27, 36/35, 50/49, 64/63, 81/80, 126/125, 225/224, 2401/2400, 4375/4374... If I take two successive commas of this I get a sequence of linear temperaments. Starting from 25/24 and 28/27, this sequence goes dicot, father, diminished, pajara, dominant seventh, meantone, meantone, miracle, ennealimmal ... . Each member of the sequence has a relationship to those before and after, from a non-empty intersection of their kernels, and the temperaments we get are clearly strong ones.
Message: 10649 - Contents - Hide Contents Date: Fri, 19 Mar 2004 23:33:58 Subject: 5-limit yantra commas From: Gene Ward Smith These start out as 2, 4/3, 6/5, 10/9 and then 16/15 father 25/24 dicot 81/80 meantone 2048/2025 diaschismic 15625/15552 kleismic 32805/32768 schismic semithirds |38 -2 -15> ennealimmal |1 -27 18> kwazy |-53 10 16> monzismic |54 -37 2> The maximal yantra for any given comma, tempered by that comma, might be worth looking at. An example would be yantra_5(22), which is the largest yantra for which 81/80 is the smallest scale step. Tempering it gives a 17-note scale which is Meantone[18] minus a note--in terms of generators of a fifth, the numbers from 0 to 17, except for 15.
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