This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 76000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950
6800 - 6825 -
Message: 6825 - Contents - Hide Contents Date: Sun, 18 May 2003 02:01:30 Subject: Re: Efficient MOS From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Those with good memories may recall the notation n+m;s I introduced to > describe MOS on ets. Here the n+m part defines the generator and et; > if we assume n>m and d = gcd(n,m) then if q/r is the next element of > the nth row of the Farey sequence after m/n, so that if u = m/d and v > = n/d then u*r-v*q=1, the generator Gen(n,m) = (q+r)/(u+v), the > period is 1/d,This should be Gen(n,m) = (q+r)/(n+m)
Message: 6826 - Contents - Hide Contents Date: Wed, 21 May 2003 06:55:04 Subject: 7-limit temperaments with fifth and octave as generators From: Gene Ward Smith Given Paul's comments on the special nature of the fifth as generator, I thought I would list these. Meantone [[1, 2, 4, 7], [0, -1, -4, -10]] [1, 4, 10, 4, 13, 12] [1200., 503.3520320] bad 835.864430 comp 15.101806 rms 3.665035 Infraschismic [[1, 2, -1, 19], [0, -1, 8, -39]] [1, -8, 39, -15, 59, 113] [1200., 498.2414296] bad 1159.472962 comp 72.040511 rms .223412 Schismic [[1, 2, -1, -3], [0, -1, 8, 14]] [1, -8, -14, -15, -25, -10] [1200., 497.8598384] bad 1704.355358 comp 24.414474 rms 2.859338 Dominant seventh [[1, 2, 4, 2], [0, -1, -4, 2]] [1, 4, -2, 4, -6, -16] [1200., 497.7740225] bad 1950.956905 comp 9.836560 rms 20.163282 Superpythagorean [[1, 2, 6, 2], [0, -1, -9, 2]] [1, 9, -2, 12, -6, -30] [1200., 489.6151808] bad 2430.162102 comp 19.470320 rms 6.410458 [[1, 2, 1, 1], [0, -1, 3, 4]] [1, -3, -4, -7, -9, -1] [1200., 532.1557550] bad 2695.579145 comp 8.756575 rms 35.154715 Pelogic [[1, 2, 1, 5], [0, -1, 3, -5]] [1, -3, 5, -7, 5, 20] [1200., 526.8909182] bad 2828.823659 comp 12.337509 rms 18.584500 Flattone [[1, 2, 4, -1], [0, -1, -4, 9]] [1, 4, -9, 4, -17, -32] [1200., 506.5439220] bad 2965.536792 comp 19.685796 rms 7.652395 [[1, 2, 0, 4], [0, -1, 6, -3]] [1, -6, 3, -12, 2, 24] [1200., 468.4644943] bad 4593.171154 comp 15.521698 rms 19.064883 [[1, 2, 9, 17], [0, -1, -16, -34]] [1, 16, 34, 23, 51, 34] [1200., 500.9569337] bad 4849.147837 comp 53.592633 rms 1.688322 [[1, 2, 16, 14], [0, -1, -33, -27]] [1, 33, 27, 50, 40, -30] [1200., 497.3975646] bad 4988.312470 comp 65.757540 rms 1.153619 [[1, 2, -1, -8], [0, -1, 8, 26]] [1, -8, -26, -15, -44, -38] [1200., 498.7882616] bad 5312.978320 comp 41.357685 rms 3.106173 [[1, 2, -1, 2], [0, -1, 8, 2]] [1, -8, -2, -15, -6, 18] [1200., 498.4152538] bad 5451.864616 comp 16.648693 rms 19.669112 [[1, 2, -13, -15], [0, -1, 37, 43]] [1, -37, -43, -61, -71, 4] [1200., 496.9546632] bad 5627.373521 comp 85.068723 rms .777617 [[1, 2, 8, 2], [0, -1, -14, 2]] [1, 14, -2, 20, -6, -44] [1200., 486.3802354] bad 5995.726244 comp 29.525064 rms 6.877967 [[1, 2, 0, 2], [0, -1, 6, 2]] [1, -6, -2, -12, -6, 12] [1200., 467.9535131] bad 6466.522095 comp 12.766699 rms 39.674691 [[1, 2, -1, -30], [0, -1, 8, 79]] [1, -8, -79, -15, -128, -161] [1200., 498.3393014] bad 6536.767371 comp 127.074856 rms .404803 [[1, 2, -6, -13], [0, -1, 20, 38]] [1, -20, -38, -34, -63, -32] [1200., 499.1780246] bad 7239.522633 comp 63.056929 rms 1.820725 [[1, 2, 4, 12], [0, -1, -4, -22]] [1, 4, 22, 4, 32, 40] [1200., 501.4086553] bad 7903.032213 comp 33.983455 rms 6.843191 [[1, 2, 11, -10], [0, -1, -21, 31]] [1, 21, -31, 31, -52, -131] [1200., 495.8167399] bad 8049.514486 comp 77.870556 rms 1.327465 [[1, 2, -5, 2], [0, -1, 18, 2]] [1, -18, -2, -31, -6, 46] [1200., 488.4529600] bad 8056.407803 comp 36.781140 rms 5.955127 [[1, 2, 4, 15], [0, -1, -4, -29]] [1, 4, 29, 4, 43, 56] [1200., 504.6434851] bad 8172.009202 comp 45.418971 rms 3.961451 [[1, 2, 11, 9], [0, -1, -21, -15]] [1, 21, 15, 31, 21, -24] [1200., 495.7080636] bad 8383.389707 comp 40.527877 rms 5.104015 [[1, 2, -8, 9], [0, -1, 25, -15]] [1, -25, 15, -42, 21, 105] [1200., 495.4499387] bad 8498.205838 comp 64.879485 rms 2.018889 [[1, 2, -6, -18], [0, -1, 20, 50]] [1, -20, -50, -34, -82, -60] [1200., 499.3603580] bad 8661.804640 comp 79.600666 rms 1.367020 [[1, 2, 9, 12], [0, -1, -16, -22]] [1, 16, 22, 23, 32, 6] [1200., 501.3431135] bad 8800.446366 comp 38.667369 rms 5.885935 [[1, 2, -18, -3], [0, -1, 49, 14]] [1, -49, -14, -80, -25, 105] [1200., 497.6823538] bad 9506.298851 comp 95.772428 rms 1.036407 [[1, 2, -3, 11], [0, -1, 13, -20]] [1, -13, 20, -23, 29, 83] [1200., 491.4400621] bad 9539.393884 comp 49.549142 rms 3.885514 [[1, 2, 28, 26], [0, -1, -62, -56]] [1, 62, 56, 96, 86, -44] [1200., 496.9820268] bad 11549.825230 comp 127.430034 rms .711266 [[1, 2, 6, -7], [0, -1, -9, 24]] [1, 9, -24, 12, -41, -81] [1200., 490.4322040] bad 12799.334640 comp 49.681423 rms 5.185604 [[1, 2, 14, 27], [0, -1, -28, -58]] [1, 28, 58, 42, 89, 56] [1200., 500.5566262] bad 13078.169160 comp 92.349249 rms 1.533487 [[1, 2, 7, -4], [0, -1, -11, 16]] [1, 11, -16, 15, -28, -68] [1200., 510.4083259] bad 13307.986690 comp 40.474090 rms 8.123780 [[1, 2, -4, -9], [0, -1, 15, 28]] [1, -15, -28, -26, -47, -23] [1200., 505.8354701] bad 13412.109770 comp 46.922728 rms 6.091589 [[1, 2, 13, 18], [0, -1, -26, -37]] [1, 26, 37, 39, 56, 13] [1200., 492.6818432] bad 13887.938240 comp 64.729582 rms 3.314608
Message: 6827 - Contents - Hide Contents Date: Thu, 22 May 2003 22:50:54 Subject: Re: 7-limit temperaments with fifth and octave as generators From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Given Paul's comments on the special nature of the fifth as >generator,pajara and injera both qualify.
Message: 6828 - Contents - Hide Contents Date: Fri, 23 May 2003 05:51:43 Subject: Re: 7-limit temperaments with fifth and octave as generators From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> Given Paul's comments on the special nature of the fifth as >> generator, >> pajara and injera both qualify.If you allow 1/2 octave, it's more like having a half-fourth or half- fifth as generator.
Message: 6829 - Contents - Hide Contents Date: Fri, 23 May 2003 21:28:22 Subject: 7-limit temperaments with half-octave and fifth as generators From: Gene Ward Smith Pajara [[2, 3, 5, 6], [0, 1, -2, -2]] [2, -4, -4, -11, -12, 2] [600.0000000, 108.8143299] bad 1550.521632 comp 11.925109 rms 10.903177 Diaschismic [[2, 3, 5, 7], [0, 1, -2, -8]] [2, -4, -16, -11, -31, -26] [600.0000000, 103.7370914] bad 2780.393711 comp 26.972971 rms 3.821631 Injera [[2, 3, 4, 5], [0, 1, 4, 4]] [2, 8, 8, 8, 7, -4] [600.0000000, 93.65102578] bad 2825.971089 comp 15.871133 rms 11.218941 [[2, 3, 6, 9], [0, 1, -8, -20]] [2, -16, -40, -30, -69, -48] [600.0000000, 101.5749456] bad 4064.299560 comp 64.925320 rms .964179 [[2, 3, -2, 8], [0, 1, 39, -14]] [2, 78, -28, 119, -50, -284] [600.0000000, 102.2165991] bad 4935.947684 comp 180.673948 rms .151210 [[2, 3, 5, 4], [0, 1, -2, 9]] [2, -4, 18, -11, 23, 53] [600.0000000, 107.6367356] bad 5144.994437 comp 33.692710 rms 4.532241 [[2, 3, 7, 11], [0, 1, -14, -32]] [2, -28, -64, -49, -107, -70] [600.0000000, 100.9911836] bad 6429.978171 comp 103.514062 rms .600082 [[2, 3, 5, 3], [0, 1, -2, 15]] [2, -4, 30, -11, 42, 81] [600.0000000, 104.7617956] bad 6895.573550 comp 52.950863 rms 2.459372 [[2, 3, 5, 6], [0, 1, -2, -3]] [2, -4, -6, -11, -15, -3] [600.0000000, 87.99444750] bad 7787.634784 comp 13.583828 rms 42.204739 [[2, 3, 3, 2], [0, 1, 10, 22]] [2, 20, 44, 27, 64, 46] [600.0000000, 98.50753458] bad 9472.021305 comp 68.515807 rms 2.017721
Message: 6830 - Contents - Hide Contents Date: Fri, 23 May 2003 21:30:17 Subject: 7-limit temperaments with octave and half-fifth or half-fourth as generators From: Gene Ward Smith Hemififth [[1, 1, -5, -1], [0, 2, 25, 13]] [2, 25, 13, 35, 15, -40] [1200., 351.4712147] bad 1262.620150 comp 46.451565 rms .585156 [[1, 2, 3, 3], [0, -2, -3, -1]] [2, 3, 1, 0, -4, -6] [1200., 270.7929767] bad 1547.782448 comp 6.691597 rms 34.566097 [[1, 1, 2, 4], [0, 2, 1, -4]] [2, 1, -4, -3, -12, -12] [1200., 358.0334333] bad 2431.810370 comp 9.849244 rms 25.068246 Beatles [[1, 1, 5, 4], [0, 2, -9, -4]] [2, -9, -4, -19, -12, 16] [1200., 356.3080304] bad 2483.820841 comp 19.942653 rms 6.245316 [[1, 1, 19, 11], [0, 2, -57, -28]] [2, -57, -28, -95, -50, 95] [1200., 351.1142096] bad 2515.083696 comp 111.653682 rms .201747 Hemifourth [[1, 2, 4, 3], [0, -2, -8, -1]] [2, 8, 1, 8, -4, -20] [1200., 252.7423121] bad 2970.086939 comp 15.298626 rms 12.690078 Semififth [[1, 1, 0, 6], [0, 2, 8, -11]] [2, 8, -11, 8, -23, -48] [1200., 348.3528922] bad 3109.919809 comp 28.865737 rms 3.732363 [[1, 2, -1, 8], [0, -2, 16, -25]] [2, -16, 25, -30, 34, 103] [1200., 249.2114324] bad 3750.376628 comp 61.748496 rms .983608 [[1, 1, 1, 2], [0, 2, 5, 3]] [2, 5, 3, 3, -1, -7] [1200., 315.3245950] bad 3898.802315 comp 9.333509 rms 44.754977 [[1, 1, 7, 30], [0, 2, -16, -93]] [2, -16, -93, -30, -153, -171] [1200., 350.8743312] bad 3928.712657 comp 147.709229 rms .180067 [[1, 2, -7, -9], [0, -2, 45, 57]] [2, -45, -57, -76, -96, -6] [1200., 248.5874435] bad 6992.965578 comp 108.728913 rms .591523 [[1, 1, 7, -1], [0, 2, -16, 13]] [2, -16, 13, -30, 15, 75] [1200., 351.0948128] bad 7594.526402 comp 46.156599 rms 3.564783 [[1, 2, 5, 3], [0, -2, -13, -1]] [2, 13, 1, 16, -4, -34] [1200., 247.5778574] bad 7614.478971 comp 25.171790 rms 12.017441 [[1, 2, 0, -1], [0, -2, 11, 18]] [2, -11, -18, -22, -34, -11] [1200., 253.4171263] bad 7788.113101 comp 32.924467 rms 7.184470
Message: 6831 - Contents - Hide Contents Date: Sat, 24 May 2003 00:13:29 Subject: Re: 7-limit temperaments with fifth and octave as generators From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <gwsmith@s...> >> wrote:>>> Given Paul's comments on the special nature of the fifth as >>> generator, >>>> pajara and injera both qualify. >> If you allow 1/2 octave, it's more like having a half-fourth or half- > fifth as generator.nonsense. neither the half-fourth nor the half-fifth are anywhere near any of the possible generators for pajara or injera.
Message: 6832 - Contents - Hide Contents
Date: Sat, 24 May 2003 21:29:18
Subject: {2,3,5,13} temperaments
From: Gene Ward Smith
The rationale for this subgroup is that while the 11-th partial is a
little awkwardly placed, between 10 and 12, the 13-th partial can
substitute for the 12; so that we have 8:10:13 as a variant on 8:10:12.
Clearly the standout system for this sort of thing is kleismic, but
orwell and (various forms of) meantone are possibilities, and
especially so if 7 is added to the mix.
[[1, 0, 1, 0], [0, 6, 5, 14]] [6, 5, 14, -6, 0, 14]
[1200., 317.1782480]
bad 1544.881453 comp 28.167578 rms .935507
[[1, 3, -9, -18], [0, -3, 24, 46]] [3, -24, -46, -45, -84, -18]
[1200., 566.0950734]
bad 2739.918388 comp 91.512154 rms .157192
[[1, 0, 3, 8], [0, 7, -3, -19]] [7, -3, -19, -21, -56, -33]
[1200., 271.5856780]
bad 2754.217033 comp 44.981181 rms .654015
[[1, 2, 2, 3], [0, -1, 1, 2]] [1, -1, -2, -4, -7, -1]
[1200., 440.2031692]
bad 3009.058093 comp 6.220665 rms 37.360090
[[1, 1, 2, 4], [0, 2, 1, -1]] [2, 1, -1, -3, -9, -6]
[1200., 357.0595734]
bad 3392.073388 comp 8.035192 rms 25.241994
[[1, 0, -6, -7], [0, 4, 21, 27]] [4, 21, 27, 24, 28, -15]
[1200., 475.5692196]
bad 3690.343087 comp 59.467771 rms .501365
[[1, 2, 4, 2], [0, -1, -4, 4]] [1, 4, -4, 4, -10, -24]
[1200., 506.7816758]
bad 3722.944279 comp 14.138269 rms 8.948391
[[1, 2, 4, 10], [0, -1, -4, -15]] [1, 4, 15, 4, 20, 20]
[1200., 504.1468810]
bad 4152.345940 comp 23.774699 rms 3.529510
[[1, 2, 1, 5], [0, -1, 3, -3]] [1, -3, 3, -7, 1, 18]
[1200., 523.6054531]
bad 4497.303591 comp 12.018087 rms 14.960028
[[1, 2, -1, 1], [0, -2, 16, 13]] [2, -16, -13, -30, -28, 29]
[1200., 249.1776838]
bad 4773.426380 comp 47.050279 rms 1.035993
[[1, 1, 1, 4], [0, 4, 9, -2]] [4, 9, -2, 5, -18, -38]
[1200., 176.6699375]
bad 5379.129474 comp 26.393924 rms 3.709838
[[1, 3, 6, 2], [0, -5, -13, 6]] [5, 13, -6, 9, -28, -62]
[1200., 339.6788753]
bad 5779.802130 comp 39.791279 rms 1.753832
[[1, 2, 4, 6], [0, -2, -8, -11]] [2, 8, 11, 8, 10, -4]
[1200., 251.2502482]
bad 6102.317289 comp 23.089213 rms 5.499554
[[1, 2, 4, 5], [0, -1, -4, -3]] [1, 4, 3, 4, 1, -8]
[1200., 508.8799915]
bad 7120.742538 comp 10.122731 rms 33.387264
[[1, 6, 3, 2], [0, -13, -2, 5]] [13, 2, -5, -27, -56, -19]
[1200., 407.6681316]
bad 7410.380474 comp 51.533064 rms 1.340663
[[1, 2, 2, 4], [0, -18, 14, -13]] [18, -14, 13, -64, -46, 82]
[1200., 27.63474175]
bad 7479.765169 comp 92.776267 rms .417508
[[2, 4, 3, 7], [0, -2, 4, 1]] [4, -8, -2, -22, -18, 25]
[600.0000000, 247.2146790]
bad 7497.983252 comp 30.330961 rms 3.915823
[[5, 8, 12, 19], [0, 0, -1, -1]] [0, 5, 5, 8, 8, -7]
[240.0000000, 97.55681404]
bad 7507.779571 comp 14.089424 rms 18.170879
[[1, -2, 8, 4], [0, 12, -19, -1]] [12, -19, -1, -58, -46, 68]
[1200., 358.5725813]
bad 8255.528931 comp 79.858910 rms .621941
[[1, -1, 2, 5], [0, 8, 1, -4]] [8, 1, -4, -17, -36, -13]
[1200., 388.3457874]
bad 8421.189302 comp 32.305788 rms 3.876713
[[1, 0, 2, 4], [0, 5, 1, -1]] [5, 1, -1, -10, -20, -6]
[1200., 378.0513539]
bad 8474.812895 comp 19.307163 rms 10.923055
[[1, 2, 5, 7], [0, -2, -13, -16]] [2, 13, 16, 16, 18, -11]
[1200., 247.2780982]
bad 8599.283217 comp 36.528785 rms 3.096298
[[1, 2, 3, 5], [0, -2, -3, -6]] [2, 3, 6, 0, 2, 3]
[1200., 261.2358895]
bad 8643.523519 comp 11.353199 rms 32.218493
[[1, 1, 0, 4], [0, 2, 8, -1]] [2, 8, -1, 8, -9, -32]
[1200., 349.0071836]
bad 8752.986475 comp 21.665053 rms 8.959581
[[1, 6, 1, 9], [0, -10, 3, -12]] [10, -3, 12, -28, -18, 39]
[1200., 529.7378924]
bad 8815.473407 comp 45.742838 rms 2.024186
[[2, 3, 4, 7], [0, 1, 3, 2]] [2, 6, 4, 5, -1, -13]
[600.0000000, 129.7513798]
bad 9013.794934 comp 15.225397 rms 18.681914
[[1, 2, 2, 4], [0, -4, 3, -3]] [4, -3, 3, -14, -10, 18]
[1200., 125.2778176]
bad 9082.656432 comp 20.450477 rms 10.434146
[[1, 4, -1, 4], [0, -8, 11, -1]] [8, -11, 1, -36, -28, 43]
[1200., 362.3182973]
bad 9114.550166 comp 49.778173 rms 1.767292
[[1, 2, 4, 3], [0, -3, -12, 5]] [3, 12, -5, 12, -19, -56]
[1200., 167.8627430]
bad 9124.252538 comp 35.157654 rms 3.546569
[[1, -1, -4, 6], [0, 9, 22, -8]] [9, 22, -8, 14, -46, -100]
[1200., 344.8448843]
bad 9271.450153 comp 66.030899 rms 1.021655
[[1, 2, 3, 3], [0, -2, -3, 3]] [2, 3, -3, 0, -12, -18]
[1200., 272.5300013]
bad 9320.064406 comp 12.132575 rms 30.420327
[[3, 5, 7, 11], [0, -1, 0, 0]] [3, 0, 0, -7, -11, 0]
[400.0000000, 106.9921230]
bad 9369.193334 comp 11.759093 rms 32.554085
[[1, 10, 5, 19], [0, -22, -7, -40]] [22, 7, 40, -40, -18, 67]
[1200., 458.9800321]
bad 9445.710500 comp 95.904963 rms .493405
[[1, 2, 3, 4], [0, -3, -5, -2]] [3, 5, 2, 1, -8, -14]
[1200., 161.8078498]
bad 9831.730783 comp 14.481501 rms 22.524428
Message: 6833 - Contents - Hide Contents Date: Sat, 24 May 2003 05:23:12 Subject: Re: 7-limit temperaments with fifth and octave as generators From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> If you allow 1/2 octave, it's more like having a half-fourth or > half->> fifth as generator. >> nonsense. neither the half-fourth nor the half-fifth are anywhere near > any of the possible generators for pajara or injera.You missed my point, which was about complexity.
Message: 6834 - Contents - Hide Contents Date: Sun, 25 May 2003 07:44:05 Subject: Re: 7-limit temperaments with fifth and octave as generators From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote: >>>> If you allow 1/2 octave, it's more like having a half-fourth or >> half->>> fifth as generator. >>>> nonsense. neither the half-fourth nor the half-fifth are anywhere near >> any of the possible generators for pajara or injera. >> You missed my point, which was about complexity.ah, ok, now that i've seen what you posted after that, this makes perfect sense! sorry for taking "more like" too literally . . .
Message: 6835 - Contents - Hide Contents Date: Thu, 29 May 2003 07:33:49 Subject: Tetrahedrons of temperaments From: Gene Ward Smith Seven-limit linear temperaments can be arranged into tetrahedrons of relationships; I use the example of the meantone-pajara-tripletone-magic-semisiths-superpythagorean tetrahedron below. We can use either the four equal temperaments--12, 19, 22, and 27; or the four commas--64/63, 126/125, 225/224, 245/243--as verticies. The six temperaments are the six edges. Given any edge, it will connect to four other edges with a common vertex (implying a common comma.) One other edge will be skew to the given edge; this gives a complementary temperament. The two temperaments together exactly specify 7-limit JI. So, for instance, knowing octave and fifth in meantone and also in superpythagorean means we know the precise 7-limit interval in question. Meantone [12, 19] [126/125, 225/224] [1, 4, 10, 4, 13, 12] Pajara [12, 22] [64/63, 225/224] [2, -4, -4, -11, -12, 2] Tripletone [12, 27] [64/63, 126/125] [3, 0, -6, -7, -18, -14] Magic [19, 22] [225/224, 245/243] [5, 1, 12, -10, 5, 25] Semisixths [19, 27] [126/125, 245/243] [7, 9, 13, -2, 1, 5] Superpythagorean [22, 27] [64/63, 245/243] [1, 9, -2, 12, -6, -30]
Message: 6836 - Contents - Hide Contents Date: Fri, 30 May 2003 22:54:14 Subject: ...continued... From: Carl Lumma Gene wrote...>If T is a linear temperament, and T[n] a scale (within an octave) of >n notes, then if the number of generator steps for an interval q times >the number of periods in an octave is +-n, q is a chroma for T[n]. In >terms of the programs I sent you, a7d(T,q)[1] = +-n. >>> Thanks. I believe that is the chromatic uv. There should only be >> one for a given T[n]. >>Any one of these, times a comma of T[n], will be another one; hence >they are infinite in number.Yeah, but this is true for any interval in the temperament. What isn't clear to me is: 1. The choice of a chroma q and pi(p)-1 commas (where p is the harmonic limit) specifies a linear temperament, right? It does not specify an n for T[n] or a tuning for T, but it does specify a family of maps (and if we're lucky, a canonic map), right? 2. Yet above it appears that changing n in T[n] changes the chroma (but obviously not the temperament, T). Therefore, we have a problem, unless changing n can only change the chroma among the family of comma-transposed chroma for that T... ? Finally, note that I'm still confused about prime- vs. odd-limit as regards pi(p)-1. Obviously I'm assuming prime-limit here, but should pi(p) be changed to ceiling(p/2)? That is, how many commas does a 9-limit linear temperament require? Paul? -Carl
Message: 6837 - Contents - Hide Contents Date: Fri, 30 May 2003 23:02:02 Subject: Re: ...continued... From: Carl Lumma Just a note; for the purposes of the below message, and from now on, I intend "chroma" = "chromatic unison vector" and "comma" = "commatic unison vector". -Carl>Gene wrote... >>> If T is a linear temperament, and T[n] a scale (within an octave) of >> n notes, then if the number of generator steps for an interval q times >> the number of periods in an octave is +-n, q is a chroma for T[n]. In >> terms of the programs I sent you, a7d(T,q)[1] = +-n. >>>>> Thanks. I believe that is the chromatic uv. There should only be >>> one for a given T[n]. >>>> Any one of these, times a comma of T[n], will be another one; hence >> they are infinite in number. >>Yeah, but this is true for any interval in the temperament. > >What isn't clear to me is: > >1. >The choice of a chroma q and pi(p)-1 commas (where p is the harmonic >limit) specifies a linear temperament, right? It does not specify >an n for T[n] or a tuning for T, but it does specify a family of maps >(and if we're lucky, a canonic map), right? > >2. >Yet above it appears that changing n in T[n] changes the chroma (but >obviously not the temperament, T). Therefore, we have a problem, >unless changing n can only change the chroma among the family of >comma-transposed chroma for that T... ? > > >Finally, note that I'm still confused about prime- vs. odd-limit as >regards pi(p)-1. Obviously I'm assuming prime-limit here, but should >pi(p) be changed to ceiling(p/2)? That is, how many commas does a >9-limit linear temperament require? Paul? > >-Carl
Message: 6838 - Contents - Hide Contents Date: Sat, 31 May 2003 11:40:30 Subject: Re: ...continued... From: Carl Lumma>No, the chroma has nothing to do with defining the temperament; it >defines the scale, given the temperament. >Gene says pi(p)-2 commas, which will be correct if you're counting 2. >It does specify the n, but not the tuning for T. If you don't want n, >you don't need the chroma.Thanks. Got it.>This should be pi(p)-2 commas, and no chroma, or 2 vals. In general >pi(p)-n commas, or n vals, specifies an (n-1)-temperament.Can someone give the vals for 5-limit meantone?>> Finally, note that I'm still confused about prime- vs. odd-limit as >> regards pi(p)-1. Obviously I'm assuming prime-limit here, but should >> pi(p) be changed to ceiling(p/2)? That is, how many commas does a >> 9-limit linear temperament require? Paul? >>Exactly as many as a 7-limit linear temperament. >... linear independence. So the 5-limit (2-3-5) requires one comma. >So does the 2-3-7 limit. And so would a system composed of octaves, >fifths, and 7:5 tritones, although it uses 4 prime numbers.Ok, but what about stuff like (2-3-5-9) where we don't have linear independence but wish to consider 9 as consonant as 3 or 5? How does visualization in terms of blocks work on a lattice with a 9-axis? -Carl
Message: 6839 - Contents - Hide Contents Date: Sat, 31 May 2003 20:26:39 Subject: Re: ...continued... From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Can someone give the vals for 5-limit meantone?Various choices are possible: (1) Eytan, consisting of h12 = [12, 19, 28] and h7 = [7, 11, 16] (2) Octave and fifth, consisting of oct = [1, 1, 0] and fif = [0, 1, 4] (3) Nominals and accidentals, consisting of h7 = [7, 11, 16] and h5 = [5, 8, 12] (4) Very meantone, h31 = [31, 49, 72] and h19 = [19, 30, 44] And so forth. I suggest you try sticking each of these pairs into my Maple routine "a5val" and seeing what comes forth.> Ok, but what about stuff like (2-3-5-9) where we don't have linear > independence but wish to consider 9 as consonant as 3 or 5? How does > visualization in terms of blocks work on a lattice with a 9-axis?If we have linearity, then if 9 is as consonant as 5, 3 will have to be twice as consonant.
Message: 6840 - Contents - Hide Contents Date: Sat, 31 May 2003 08:21:04 Subject: Re: ...continued... From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> The choice of a chroma q and pi(p)-1 commas (where p is the harmonic > limit) specifies a linear temperament, right? It does not specify > an n for T[n] or a tuning for T, but it does specify a family of maps > (and if we're lucky, a canonic map), right?No, the chroma has nothing to do with defining the temperament; it defines the scale, given the temperament.> Finally, note that I'm still confused about prime- vs. odd-limit as > regards pi(p)-1. Obviously I'm assuming prime-limit here, but should > pi(p) be changed to ceiling(p/2)? That is, how many commas does a > 9-limit linear temperament require? Paul?Exactly as many as a 7-limit linear temperament.
Message: 6841 - Contents - Hide Contents Date: Sat, 31 May 2003 08:24:49 Subject: Re: ...continued... From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> The choice of a chroma q and pi(p)-1 commas (where p is the harmonic > limit) specifies a linear temperament, right?This should be pi(p)-2 commas, and no chroma, or 2 vals. In general pi(p)-n commas, or n vals, specifies an (n-1)-temperament.
Message: 6842 - Contents - Hide Contents Date: Sat, 31 May 2003 12:05:27 Subject: Re: ...continued... From: Graham Breed Carl Lumma wrote:> 1. > The choice of a chroma q and pi(p)-1 commas (where p is the harmonic > limit) specifies a linear temperament, right? It does not specify > an n for T[n] or a tuning for T, but it does specify a family of maps > (and if we're lucky, a canonic map), right?Gene says pi(p)-2 commas, which will be correct if you're counting 2. It does specify the n, but not the tuning for T. If you don't want n, you don't need the chroma.> 2. > Yet above it appears that changing n in T[n] changes the chroma (but > obviously not the temperament, T). Therefore, we have a problem, > unless changing n can only change the chroma among the family of > comma-transposed chroma for that T... ?No, there's no problem.> Finally, note that I'm still confused about prime- vs. odd-limit as > regards pi(p)-1. Obviously I'm assuming prime-limit here, but should > pi(p) be changed to ceiling(p/2)? That is, how many commas does a > 9-limit linear temperament require? Paul?It generally goes by prime numbers, or more generally by prime intervals -- that is a set of intervals none of which can be arrived at by adding and subtracting the other ones. This is like linear independence. So the 5-limit (2-3-5) requires one comma. So does the 2-3-7 limit. And so would a system composed of octaves, fifths, and 7:5 tritones, although it uses 4 prime numbers. Graham
Message: 6844 - Contents - Hide Contents Date: Mon, 02 Jun 2003 19:12:53 Subject: Re: ...continued... From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> How does > visualization in terms of blocks work on a lattice with a 9-axis?not very well, since every pitch will appear in an infinite number of positions.
Message: 6845 - Contents - Hide Contents Date: Mon, 02 Jun 2003 19:23:56 Subject: Re: Interval Database Experiences From: wallyesterpaulrus hi alex, most of your questions have been addressed at length in the old discussions on the tuning list. perhaps you could take some time to look over the tuning list archives, particularly posts by dave keenan. my main response right now is, why would you restrict yourself to rational numbers? while it is true that simple integer ratios are audibly different from their neighbors, and that one can therefore tune successive 3:2s and 4:3s to obtain (theoretically) a just 2187:2048, there are irrational numbers which are also very useful for various tuning systems where simple integer ratios are approximated "close enough" with various musically useful relationships that would be impossible in just intonation.
Message: 6847 - Contents - Hide Contents Date: Mon, 02 Jun 2003 20:00:28 Subject: Re: Interval Database Experiences From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Porres" <decuritiba@y...> wrote:> I still don't know what's the normal tuning error range that we can > actually notice, maybe around 2 cents?as others have mentioned to you on other lists, it totally depends on the context. it's quite rare for anyone to notice a melodic interval difference of 5 cents or less; meanwhile, it's rather easy to tell if a harmonic interval is in just intonation or 1 cent off from it, at least if you hold the interval long enough. certain effects, such as the combinational tones that jacques dudon exploits, can make tiny differences easier to hear.
Message: 6849 - Contents - Hide Contents Date: Tue, 03 Jun 2003 11:54:14 Subject: Re: Interval Database Experiences From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Porres" <decuritiba@y...> wrote:> Hy Wally, > > well, as I said, I was first concerned on creating a tool ( a table) > where you could convert cents to a integer ratioThere's no difficulty in doing that; you don't need a table.
6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950
6800 - 6825 -