Superparticular temperaments

Let us call a temperament "superparticular" if its kernel has a basis (not necessarily reduced) consisting of superparticular commas. Because these increase in number as the prime limit increases, and since commas of ever higher complexity appear, it seems likely that however we define "interesting" that sooner or later all of the linear temperaments of interest will be superparticular ones. This raises the question of when this happens for our purposes.

In the 5-limit case, it is clearly not so, unless we think meantone is the last word. In the 7-limit case, we do miss some important systems, but the inclusion of some fairly obvious commas would catch them. This makes it seem plausible that we might not actually miss anything of significance in the 11-limit, and could probably insure that we do not merely by including some obvious additions.

There are 150 7-limit superparticular linear temperaments; temperaments which are not superparticular include the following:

Hexidecimal [36/35, 135/128]

Double Wide [50/49, 875/864]

Porcupine [64/63, 250/243]

Superpythagorean [64/63, 245/243]

Muggles [126/125, 525/512]

Quartaminorthirds [126/125, 1029/1024]

Supermajor Seconds [81/80, 1029/1024]

Schismic [225/224, 3125/3087]

Superkleismic [875/864, 1029/1024]

Semififth [81/80, 6144/6125]

Diaschismic [126/125, 2048/2025]

Quartaminorthirds [126/125, 1029/1024]

Octafiths [245/243, 2401/2400]

Magic [225/224, 245/243]

Orwell [225/224, 1728/1715]

Tritonic [225/224, 50421/50000]

Supersupermajor [245/243, 1029/1024]

Shrutar [245/243, 2048/2025]

Hemiwuerschmidt [2401/2400, 3136/3125]

Hemikleismic [4000/3969, 6144/6125]

Hemithird [1029/1024, 3136/3125]

Wizard [225/224, 118098/117649]

Duodecimal [225/224, 250047/250000]

Slender [225/224, 589824/588245]

Amity [4375/4374, 5120/5103]

Hemififth [2401/2400, 5120/5103]

If we toss out Hexidecimal, Slender, and Wizard, none of which are essential, we find that the TM reduced bases for the remaining temperaments employ superparticulars plus 250/243, 245/243, 525/512, 875/864, 1029/1024, 1728/1715, 2048/2025, 3125/3087, 3136/3125, 5120/5103, 4000/3969, 6144/6125 and 250047/250000, not all of which may be necessary (I didn't check.)

Here are the superparticular 7-limit linear temperaments, in order of increasing geometric badness. Some absurdly funky systems are included.

Ennealimmal
[18, 27, 18, -34, 22, 1] [[9, 15, 22, 26], [0, -2, -3, -2]]

generators [133.3333333, 48.99915090]

comp 58.840778 rms .130449 bad 451.645972

Meantone
[1, 4, 10, 12, -13, 4] [[1, 2, 4, 7], [0, -1, -4, -10]]

generators [1200., 503.3520320]

comp 15.101806 rms 3.665035 bad 835.864548

Miracle
[6, -7, -2, 15, 20, -25] [[1, 1, 3, 3], [0, 6, -7, -2]]

generators [1200., 116.5729472]

comp 24.926629 rms 1.637405 bad 1017.380174

[0, 2, 2, -1, -3, 3] [[2, 3, 5, 6], [0, 0, -1, -1]]

generators [600.0000000, 273.4076908]

comp 4.275602 rms 59.723378 bad 1091.789279

[0, 0, 1, 2, -1, 0] [[1, 1, 2, 3], [0, 0, 0, -1]]

generators [1200., 593.9303320]

comp 1.661977 rms 405.957997 bad 1121.323723

[0, 1, 0, -3, 0, 2] [[1, 2, 3, 3], [0, 0, -1, 0]]

generators [1200., 570.6132559]

comp 2.050856 rms 287.792504 bad 1210.458384

[0, 0, 1, 3, -2, 0] [[1, 2, 3, 3], [0, 0, 0, 1]]

generators [1200., 206.0696683]

comp 1.661977 rms 473.700237 bad 1308.439092

[0, 0, 1, 2, -2, 0] [[1, 2, 2, 3], [0, 0, 0, -1]]

generators [1200., 193.9303319]

comp 1.661977 rms 511.941281 bad 1414.067236

[0, 0, 3, 7, -5, 0] [[3, 5, 7, 9], [0, 0, 0, -1]]

generators [400.0000000, 193.9303319]

comp 4.985930 rms 61.312549 bad 1524.199406

[2, 3, 1, -6, 4, 0] [[1, 2, 3, 3], [0, -2, -3, -1]]

generators [1200., 270.7929767]

comp 6.691597 rms 34.566097 bad 1547.782446

Pajara
[2, -4, -4, 2, 12, -11] [[2, 3, 5, 6], [0, 1, -2, -2]]

generators [600.0000000, 108.8143299]

comp 11.925109 rms 10.903178 bad 1550.521642

Tripletone
[3, 0, -6, -14, 18, -7] [[3, 5, 7, 8], [0, -1, 0, 2]]

generators [400.0000000, 88.72066409]

comp 14.168743 rms 8.100679 bad 1626.237917

[1, -1, -2, -2, 6, -4] [[1, 2, 2, 2], [0, -1, 1, 2]]

generators [1200., 478.6926428]

comp 4.972221 rms 65.953083 bad 1630.556687

Blackwood
[0, 5, 0, -14, 0, 8] [[5, 8, 12, 14], [0, 0, -1, 0]]

generators [240.0000000, 90.61325640]

comp 10.254281 rms 15.815352 bad 1662.988586

[1, -1, 3, 10, -2, -4] [[1, 2, 2, 4], [0, -1, 1, -3]]

generators [1200., 459.5087948]

comp 6.535126 rms 43.659491 bad 1864.603865

Catakleismic
[6, 5, 22, 37, -18, -6] [[1, 0, 1, -3], [0, 6, 5, 22]]

generators [1200., 316.7238784]

comp 34.269866 rms 1.610555 bad 1891.474472

[2, 1, 3, 4, 1, -3] [[1, 1, 2, 2], [0, 2, 1, 3]]

generators [1200., 322.2119006]

comp 6.245166 rms 48.926006 bad 1908.216790

Dominant seventh
[1, 4, -2, -16, 6, 4] [[1, 2, 4, 2], [0, -1, -4, 2]]

generators [1200., 497.7740225]

comp 9.836560 rms 20.163282 bad 1950.956873

[0, 1, 1, 0, -2, 2] [[1, 2, 3, 3], [0, 0, -1, -1]]

generators [1200., 273.4076908]

comp 2.137801 rms 442.481365 bad 2022.224915

[0, 0, 2, 5, -3, 0] [[2, 3, 5, 6], [0, 0, 0, -1]]

generators [600.0000000, 193.9303319]

comp 3.323954 rms 186.002433 bad 2055.079021

[0, 1, 1, -1, -2, 2] [[1, 2, 2, 3], [0, 0, 1, 1]]

generators [1200., 326.5923086]

comp 2.137801 rms 458.017472 bad 2093.227909

[2, 1, -1, -5, 7, -3] [[1, 1, 2, 3], [0, 2, 1, -1]]

generators [1200., 318.5700997]

comp 6.245166 rms 53.747748 bad 2096.274853

[1, 1, 0, -3, 3, -1] [[1, 2, 3, 3], [0, -1, -1, 0]]

generators [1200., 540.2785958]

comp 3.054488 rms 225.884103 bad 2107.475631

[1, -1, 0, 3, 3, -4] [[1, 2, 2, 3], [0, -1, 1, 0]]

generators [1200., 442.1793558]

comp 3.917133 rms 142.097096 bad 2180.328488

[1, 1, 2, 2, 0, -1] [[1, 1, 2, 2], [0, 1, 1, 2]]

generators [1200., 484.4129532]

comp 3.529341 rms 188.646684 bad 2349.829555

[0, 2, -2, -10, 3, 3] [[2, 3, 5, 5], [0, 0, -1, 1]]

generators [600.0000000, 291.2560966]

comp 6.120838 rms 62.740535 bad 2350.552312

[0, 1, 0, -3, 0, 1] [[1, 1, 2, 3], [0, 0, 1, 0]]

generators [1200., 229.3867439]

comp 2.050856 rms 561.132227 bad 2360.128218

Augmented
[3, 0, 6, 14, -1, -7] [[3, 5, 7, 9], [0, -1, 0, -2]]

generators [400.0000000, 110.2596913]

comp 12.019943 rms 16.598678 bad 2398.160778

[0, 1, 1, 0, -1, 1] [[1, 1, 2, 2], [0, 0, 1, 1]]

generators [1200., 326.5923086]

comp 2.137801 rms 526.643864 bad 2406.863715

[0, 0, 1, 3, -1, 0] [[1, 1, 3, 3], [0, 0, 0, -1]]

generators [1200., 193.9303319]

comp 1.661977 rms 875.848123 bad 2419.238652

[1, 2, 1, -3, 1, 1] [[1, 2, 3, 3], [0, -1, -2, -1]]

generators [1200., 406.8431437]

comp 3.917133 rms 157.889659 bad 2422.648543

[0, 1, 0, -2, 0, 1] [[1, 1, 2, 2], [0, 0, -1, 0]]

generators [1200., 170.6132563]

comp 2.050856 rms 577.847450 bad 2430.432628

Neutral thirds
[2, 1, -4, -12, 12, -3] [[1, 1, 2, 4], [0, 2, 1, -4]]

generators [1200., 358.0334333]

comp 9.849244 rms 25.068246 bad 2431.810368

[0, 1, 1, -1, -1, 1] [[1, 1, 2, 3], [0, 0, -1, -1]]

generators [1200., 273.4076908]

comp 2.137801 rms 539.762916 bad 2466.820304

Beatles
[2, -9, -4, 16, 12, -19] [[1, 1, 5, 4], [0, 2, -9, -4]]

generators [1200., 356.3080304]

comp 19.942653 rms 6.245316 bad 2483.820838

Diminished
[4, 4, 4, -2, 5, -3] [[4, 6, 9, 11], [0, 1, 1, 1]]

generators [300.0000000, 85.69820677]

comp 11.405897 rms 19.136993 bad 2489.617178

[1, 2, 3, 1, -2, 1] [[1, 2, 3, 4], [0, -1, -2, -3]]

generators [1200., 460.9163575]

comp 4.972221 rms 100.967191 bad 2496.209750

Nonkleismic
[10, 9, 7, -9, 17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]

generators [1200., 309.9514712]

comp 27.531739 rms 3.320167 bad 2516.675801

[1, 0, 1, 2, 1, -2] [[1, 2, 2, 3], [0, -1, 0, -1]]

generators [1200., 557.7664030]

comp 3.054488 rms 271.083238 bad 2529.178945

Sharp
[2, 1, 6, 11, -4, -3] [[1, 1, 2, 1], [0, 2, 1, 6]]

generators [1200., 360.2272895]

comp 9.849244 rms 26.099830 bad 2531.881842

[0, 0, 1, 1, -1, 0] [[1, 1, 1, 2], [0, 0, 0, 1]]

generators [1200., 206.0696683]

comp 1.661977 rms 917.874641 bad 2535.322905

[0, 0, 2, 4, -3, 0] [[2, 3, 4, 5], [0, 0, 0, 1]]

generators [600.0000000, 206.0696683]

comp 3.323954 rms 231.177173 bad 2554.199699

[1, 2, 0, -6, 3, 1] [[1, 2, 3, 3], [0, -1, -2, 0]]

generators [1200., 352.0867411]

comp 4.352116 rms 135.480508 bad 2566.124420

Tertiathirds
[4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, -4, 3, -2]]

generators [1200., 125.4687958]

comp 14.729697 rms 12.188571 bad 2644.480844

Decimal
[4, 2, 2, -1, 8, -6] [[2, 4, 5, 6], [0, -2, -1, -1]]

generators [600.0000000, 249.0224992]

comp 10.574200 rms 23.945252 bad 2677.407574

Decitone
[2, 6, 6, -3, -4, 5] [[2, 3, 4, 5], [0, 1, 3, 3]]

generators [600.0000000, 128.5123366]

comp 11.925109 rms 18.863889 bad 2682.600333

[2, -1, 1, 5, 4, -6] [[1, 2, 2, 3], [0, -2, 1, -1]]

generators [1200., 288.4250230]

comp 6.691597 rms 59.930923 bad 2683.555226

[1, -3, -4, -1, 9, -7] [[1, 2, 1, 1], [0, -1, 3, 4]]

generators [1200., 532.1557550]

comp 8.756575 rms 35.154715 bad 2695.579144

[2, -2, 1, 8, 4, -8] [[1, 2, 2, 3], [0, -2, 2, -1]]

generators [1200., 218.7617340]

comp 8.112184 rms 41.524693 bad 2732.637321

[1, 1, 1, 0, 1, -1] [[1, 2, 3, 3], [0, -1, -1, -1]]

generators [1200., 514.3017931]

comp 2.851474 rms 336.706254 bad 2737.726652

[0, 0, 4, 9, -6, 0] [[4, 6, 9, 11], [0, 0, 0, 1]]

generators [300.0000000, 6.069667048]

comp 6.647907 rms 63.402645 bad 2802.058943

[1, 0, -1, -2, 4, -2] [[1, 1, 2, 3], [0, 1, 0, -1]]

generators [1200., 466.5645474]

comp 3.529341 rms 225.688309 bad 2811.229153

[1, 4, 3, -4, -2, 4] [[1, 2, 4, 4], [0, -1, -4, -3]]

generators [1200., 496.0501669]

comp 7.402240 rms 51.362639 bad 2814.321331

Injera
[2, 8, 8, -4, -7, 8] [[2, 3, 4, 5], [0, 1, 4, 4]]

generators [600.0000000, 93.65102578]

comp 15.871133 rms 11.218941 bad 2825.971103

Squares
[4, 16, 9, -24, -3, 16] [[1, 3, 8, 6], [0, -4, -16, -9]]

generators [1200., 425.9591136]

comp 28.922922 rms 3.443812 bad 2880.870808

[1, -1, 1, 5, 1, -4] [[1, 2, 2, 3], [0, -1, 1, -1]]

generators [1200., 374.4750766]

comp 4.352116 rms 154.263172 bad 2921.885210

[1, 0, 0, 0, 3, -2] [[1, 2, 2, 3], [0, -1, 0, 0]]

generators [1200., 549.7582051]

comp 2.851474 rms 360.237866 bad 2929.059955

Flattone
[1, 4, -9, -32, 17, 4] [[1, 2, 4, -1], [0, -1, -4, 9]]

generators [1200., 506.5439220]

comp 19.685796 rms 7.652394 bad 2965.536695

Hemifourth
[2, 8, 1, -20, 4, 8] [[1, 2, 4, 3], [0, -2, -8, -1]]

generators [1200., 252.7423121]

comp 15.298626 rms 12.690078 bad 2970.086942

[2, -2, -2, 1, 9, -8] [[2, 3, 5, 6], [0, 1, -1, -1]]

generators [600.0000000, 167.6687407]

comp 8.311748 rms 43.142169 bad 2980.483611

[1, -1, -5, -9, 11, -4] [[1, 2, 2, 1], [0, -1, 1, 5]]

generators [1200., 441.3069864]

comp 8.956788 rms 37.258834 bad 2989.054558

Semisixths
[7, 9, 13, 5, -1, -2] [[1, -1, -1, -2], [0, 7, 9, 13]]

generators [1200., 443.6203855]

comp 24.364978 rms 5.052931 bad 2999.683372

[0, 0, 2, 5, -4, 0] [[2, 4, 5, 6], [0, 0, 0, 1]]

generators [600.0000000, 6.069667048]

comp 3.323954 rms 288.509783 bad 3187.648638

Kleismic
[6, 5, 3, -7, 12, -6] [[1, 0, 1, 2], [0, 6, 5, 3]]

generators [1200., 316.6640534]

comp 16.383068 rms 12.273810 bad 3294.350652

[0, 2, 0, -6, 0, 3] [[2, 3, 5, 6], [0, 0, -1, 0]]

generators [600.0000000, 170.6132563]

comp 4.101712 rms 197.096298 bad 3315.956645

Pentadecimal
[3, 5, 9, 7, -6, 1] [[1, 2, 3, 4], [0, -3, -5, -9]]

generators [1200., 158.7318223]

comp 14.382059 rms 16.143425 bad 3339.164305

[0, 0, 1, 4, -2, 0] [[1, 2, 4, 4], [0, 0, 0, -1]]

generators [1200., 593.9303320]

comp 1.661977 rms 1211.073885 bad 3345.188138

[1, -3, -2, 4, 6, -7] [[1, 2, 1, 2], [0, -1, 3, 2]]

generators [1200., 513.7543330]

comp 7.402240 rms 61.746458 bad 3383.283558

[0, 1, -1, -6, 2, 2] [[1, 2, 3, 3], [0, 0, -1, 1]]

generators [1200., 291.2560966]

comp 3.060419 rms 364.029464 bad 3409.559323

Heptadecal
[0, 0, 7, 16, -11, 0] [[7, 11, 16, 20], [0, 0, 0, -1]]

generators [171.4285715, 79.64461697]

comp 11.633838 rms 25.354978 bad 3431.699329

[0, 1, 0, -4, 0, 2] [[1, 2, 3, 4], [0, 0, -1, 0]]

generators [1200., 170.6132563]

comp 2.050856 rms 838.919678 bad 3528.505242

[2, 3, 5, 3, -2, 0] [[1, 2, 3, 4], [0, -2, -3, -5]]

generators [1200., 287.3658368]

comp 8.353937 rms 51.525959 bad 3595.906950

[0, 1, 0, -2, 0, 2] [[1, 2, 2, 2], [0, 0, 1, 0]]

generators [1200., 229.3867439]

comp 2.050856 rms 861.314591 bad 3622.698490

[1, 2, 4, 4, -4, 1] [[1, 2, 3, 4], [0, -1, -2, -4]]

generators [1200., 348.5748041]

comp 6.139951 rms 97.801004 bad 3687.000243

[0, 2, 0, -5, 0, 3] [[2, 3, 4, 5], [0, 0, 1, 0]]

generators [600.0000000, 229.3867439]

comp 4.101712 rms 219.922942 bad 3699.993093

[1, 2, -2, -10, 6, 1] [[1, 2, 3, 2], [0, -1, -2, 2]]

generators [1200., 454.5534905]

comp 6.535126 rms 87.356149 bad 3730.795007

[1, 0, 2, 4, 0, -2] [[1, 1, 2, 2], [0, 1, 0, 2]]

generators [1200., 398.8312176]

comp 4.006648 rms 238.412445 bad 3827.288461

[2, 5, 3, -7, 1, 3] [[1, 1, 1, 2], [0, 2, 5, 3]]

generators [1200., 315.3245950]

comp 9.333509 rms 44.754977 bad 3898.802316

[0, 0, 5, 12, -8, 0] [[5, 8, 12, 14], [0, 0, 0, 1]]

generators [240.0000000, 46.06966903]

comp 8.309884 rms 57.395986 bad 3963.432270

[2, 4, 1, -9, 4, 2] [[1, 2, 3, 3], [0, -2, -4, -1]]

generators [1200., 203.6485512]

comp 8.112184 rms 60.827594 bad 4002.913457

[0, 4, 0, -11, 0, 6] [[4, 6, 9, 11], [0, 0, 1, 0]]

generators [300.0000000, 29.38674378]

comp 8.203424 rms 60.054354 bad 4041.428204

[2, 4, 4, -2, -1, 2] [[2, 3, 4, 5], [0, 1, 2, 2]]

generators [600.0000000, 196.6785324]

comp 8.311748 rms 61.297826 bad 4234.770049

[0, 2, 4, 4, -6, 3] [[2, 3, 5, 6], [0, 0, -1, -2]]

generators [600.0000000, 152.3110691]

comp 6.467985 rms 104.261749 bad 4361.772496

[1, 4, 5, 0, -5, 4] [[1, 2, 4, 5], [0, -1, -4, -5]]

generators [1200., 520.6704216]

comp 8.756575 rms 57.900863 bad 4439.699141

[1, 1, 1, -1, 2, -1] [[1, 1, 2, 3], [0, 1, 1, 1]]

generators [1200., 285.6982066]

comp 2.851474 rms 548.057280 bad 4456.201822

[0, 1, -1, -4, 1, 1] [[1, 1, 2, 2], [0, 0, -1, 1]]

generators [1200., 291.2560966]

comp 3.060419 rms 485.134310 bad 4543.847061

[2, 1, 0, -3, 6, -3] [[1, 1, 2, 3], [0, 2, 1, 0]]

generators [1200., 417.2373637]

comp 5.542165 rms 148.387948 bad 4557.824204

[0, 0, 1, 0, -1, 0] [[1, 1, 0, 2], [0, 0, 0, -1]]

generators [1200., 193.9303319]

comp 1.661977 rms 1673.375576 bad 4622.142544

[2, 1, 2, 2, 2, -3] [[1, 1, 2, 2], [0, 2, 1, 2]]

generators [1200., 420.5480918]

comp 5.542165 rms 155.619746 bad 4779.953169

[2, 0, 0, 0, 6, -5] [[2, 3, 5, 6], [0, 1, 0, 0]]

generators [600.0000000, 250.2417952]

comp 5.702948 rms 148.630126 bad 4833.989876

[0, 3, 0, -9, 0, 5] [[3, 5, 7, 9], [0, 0, 1, 0]]

generators [400.0000000, 96.05341086]

comp 6.152568 rms 133.979799 bad 5071.684361

[1, 1, 3, 5, -2, -1] [[1, 2, 3, 4], [0, -1, -1, -3]]

generators [1200., 458.2361773]

comp 4.722914 rms 231.090354 bad 5154.683175

[3, 5, 1, -12, 7, 1] [[1, 2, 3, 3], [0, -3, -5, -1]]

generators [1200., 157.4375200]

comp 10.922722 rms 43.469957 bad 5186.220813

[0, 2, 2, -1, -4, 4] [[2, 4, 5, 6], [0, 0, 1, 1]]

generators [600.0000000, 26.59230899]

comp 4.275602 rms 287.723620 bad 5259.809056

[0, 1, 0, -4, 0, 1] [[1, 1, 3, 4], [0, 0, -1, 0]]

generators [1200., 570.6132559]

comp 2.050856 rms 1255.321105 bad 5279.894151

[0, 1, 0, -1, 0, 1] [[1, 1, 2, 1], [0, 0, -1, 0]]

generators [1200., 570.6132559]

comp 2.050856 rms 1277.867746 bad 5374.725567

[6, 7, 5, -8, 9, -3] [[1, 3, 4, 4], [0, -6, -7, -5]]

generators [1200., 286.0989745]

comp 17.627870 rms 17.388827 bad 5403.435759

[0, 2, 6, 8, -9, 3] [[2, 3, 5, 7], [0, 0, -1, -3]]

generators [600.0000000, 294.0505316]

comp 9.353407 rms 62.423802 bad 5461.222835

[0, 1, 2, 1, -4, 2] [[1, 2, 2, 3], [0, 0, 1, 2]]

generators [1200., 65.87074902]

comp 3.233992 rms 522.430484 bad 5463.947575

[2, -3, -1, 6, 7, -9] [[1, 1, 3, 3], [0, 2, -3, -1]]

generators [1200., 300.3607896]

comp 9.333509 rms 63.538867 bad 5535.149429

[1, 2, 6, 8, -7, 1] [[1, 2, 3, 5], [0, -1, -2, -6]]

generators [1200., 435.7072302]

comp 8.956788 rms 70.186382 bad 5630.635755

[1, 1, -2, -8, 6, -1] [[1, 2, 3, 2], [0, -1, -1, 2]]

generators [1200., 541.5638500]

comp 5.319858 rms 199.354602 bad 5641.911563

[4, 9, -8, -44, 24, 5] [[1, 1, 1, 4], [0, 4, 9, -8]]

generators [1200., 177.4647360]

comp 27.604056 rms 7.456594 bad 5681.804332

[2, 11, 6, -17, -4, 13] [[1, 1, -1, 1], [0, 2, 11, 6]]

generators [1200., 363.1818634]

comp 19.942653 rms 14.296447 bad 5685.831463

[0, 3, 0, -8, 0, 5] [[3, 5, 7, 8], [0, 0, -1, 0]]

generators [400.0000000, 37.27992253]

comp 6.152568 rms 155.873944 bad 5900.467447

[2, 2, 2, -1, 2, -1] [[2, 3, 4, 5], [0, 1, 1, 1]]

generators [600.0000000, 285.6982066]

comp 5.702948 rms 184.020398 bad 5985.009686

[0, 2, 1, -3, -1, 2] [[1, 1, 3, 3], [0, 0, -2, -1]]

generators [1200., 582.3155032]

comp 3.845808 rms 405.273928 bad 5994.098393

[1, 0, -2, -4, 5, -2] [[1, 1, 2, 3], [0, 1, 0, -2]]

generators [1200., 290.2267039]

comp 4.722914 rms 272.846672 bad 6086.096297

[1, 0, -2, -4, 6, -2] [[1, 2, 2, 2], [0, -1, 0, 2]]

generators [1200., 467.6680326]

comp 4.722914 rms 274.917379 bad 6132.285324

[0, 0, 3, 8, -5, 0] [[3, 5, 8, 9], [0, 0, 0, -1]]

generators [400.0000000, 60.59699869]

comp 4.985930 rms 249.612405 bad 6205.240006

[2, 2, 1, -3, 4, -2] [[1, 2, 3, 3], [0, -2, -2, -1]]

generators [1200., 336.7290694]

comp 5.670929 rms 194.291227 bad 6248.296967

[1, 0, 2, 4, -1, -2] [[1, 2, 2, 3], [0, -1, 0, -2]]

generators [1200., 255.7142370]

comp 4.006648 rms 401.256919 bad 6441.467346

[0, 1, 3, 5, -6, 2] [[1, 2, 3, 4], [0, 0, -1, -3]]

generators [1200., 394.0505317]

comp 4.676704 rms 311.829091 bad 6820.187515

[0, 2, 0, -5, 0, 2] [[2, 2, 4, 5], [0, 0, 1, 0]]

generators [600.0000000, 29.38674378]

comp 4.101712 rms 405.448549 bad 6821.283937

[1, 1, 4, 7, -3, -1] [[1, 1, 2, 1], [0, 1, 1, 4]]

generators [1200., 541.9922676]

comp 6.138398 rms 182.241625 bad 6866.851562

[1, 0, 3, 6, -2, -2] [[1, 2, 2, 4], [0, -1, 0, -3]]

generators [1200., 541.4436503]

comp 5.319858 rms 246.900497 bad 6987.502433

[2, 2, 2, -1, 3, -2] [[2, 3, 5, 6], [0, -1, -1, -1]]

generators [600.0000000, 114.3017931]

comp 5.702948 rms 216.492360 bad 7041.115475

[2, 0, 1, 2, 4, -4] [[1, 2, 2, 3], [0, -2, 0, -1]]

generators [1200., 352.7582023]

comp 5.670929 rms 229.676703 bad 7386.274048

[0, 2, 2, -1, -2, 2] [[2, 2, 4, 5], [0, 0, 1, 1]]

generators [600.0000000, 26.59230899]

comp 4.275602 rms 405.399657 bad 7411.017522

[0, 1, 2, 3, -2, 1] [[1, 1, 3, 3], [0, 0, -1, -2]]

generators [1200., 370.4928875]

comp 3.233992 rms 730.932254 bad 7644.606586

[2, 1, 10, 20, -10, -3] [[1, 1, 2, 0], [0, 2, 1, 10]]

generators [1200., 334.5486096]

comp 15.864707 rms 30.441426 bad 7661.770385

[1, 2, -1, -9, 5, 1] [[1, 2, 3, 3], [0, -1, -2, 1]]

generators [1200., 224.5581578]

comp 5.297379 rms 278.453554 bad 7814.026023

[2, 0, 0, 0, 5, -4] [[2, 3, 4, 5], [0, -1, 0, 0]]

generators [600.0000000, 149.7582050]

comp 5.702948 rms 251.915620 bad 8193.208145

[2, -1, 0, 3, 6, -7] [[1, 1, 3, 3], [0, 2, -1, 0]]

generators [1200., 484.9100284]

comp 6.537864 rms 203.252537 bad 8687.758969

[4, 4, 12, 17, -8, -3] [[4, 6, 9, 10], [0, 1, 1, 3]]

generators [300.0000000, 125.9743488]

comp 18.891658 rms 24.610643 bad 8783.408917

[2, 3, 2, -3, 2, 0] [[1, 2, 3, 3], [0, -2, -3, -2]]

generators [1200., 252.5079222]

comp 6.537864 rms 209.085647 bad 8937.087487

[4, -4, -1, 9, 13, -16] [[1, 1, 3, 3], [0, 4, -4, -1]]

generators [1200., 189.5576458]

comp 15.650108 rms 37.374885 bad 9154.076279

[2, -1, 2, 7, 2, -6] [[1, 2, 2, 3], [0, -2, 1, -2]]

generators [1200., 235.1959794]

comp 7.234335 rms 178.776933 bad 9356.399761

[1, 4, 0, -12, 3, 4] [[1, 2, 4, 3], [0, -1, -4, 0]]

generators [1200., 476.6658998]

comp 7.962349 rms 148.388409 bad 9407.677587

[0, 1, 3, 3, -3, 1] [[1, 1, 2, 3], [0, 0, -1, -3]]

generators [1200., 194.0505316]

comp 4.676704 rms 436.527296 bad 9547.531318

[3, 2, 0, -6, 9, -4] [[1, 1, 2, 3], [0, 3, 2, 0]]

generators [1200., 267.7220970]

comp 8.482929 rms 136.530107 bad 9824.717681

[8, 13, 4, -27, 16, 2] [[1, 2, 3, 3], [0, -8, -13, -4]]

generators [1200., 63.00613990]

comp 28.070300 rms 12.646377 bad 9964.608569

[2, 3, 2, -4, 2, 1] [[1, 1, 1, 2], [0, 2, 3, 2]]

generators [1200., 505.3868143]

comp 6.537864 rms 234.327566 bad 10016.019720

[1, 4, 1, -8, 1, 4] [[1, 2, 4, 3], [0, -1, -4, -1]]

generators [1200., 518.8451304]

comp 7.416607 rms 187.506165 bad 10313.974820

[1, 1, 3, 6, -3, -1] [[1, 2, 3, 3], [0, -1, -1, -3]]

generators [1200., 16.13091449]

comp 4.722914 rms 494.772960 bad 11036.366530

[10, 14, 14, -7, 6, -1] [[2, 2, 3, 4], [0, 5, 7, 7]]

generators [600.0000000, 139.6810044]

comp 32.695891 rms 10.427607 bad 11147.334260

[3, 3, 3, -1, 3, -2] [[3, 5, 7, 8], [0, 1, 1, 1]]

generators [400.0000000, 19.03154112]

comp 8.554423 rms 157.513437 bad 11526.541520

[0, 8, 4, -13, -6, 12] [[4, 6, 9, 11], [0, 0, 2, 1]]

generators [300.0000000, 17.68449710]

comp 15.383232 rms 58.863881 bad 13929.774210

[1, -1, 11, 29, -15, -4] [[1, 2, 2, 7], [0, -1, 1, -11]]

generators [1200., 453.7389106]

comp 18.940672 rms 39.612986 bad 14211.121100

[5, 8, 3, -15, 9, 1] [[1, 2, 3, 3], [0, -5, -8, -3]]

generators [1200., 103.5769428]

comp 17.205721 rms 49.518880 bad 14659.412800

[4, 8, 5, -11, 3, 4] [[1, 1, 1, 2], [0, 4, 8, 5]]

generators [1200., 198.4673924]

comp 15.650108 rms 61.043960 bad 14951.244890

[0, 4, 2, -7, -4, 8] [[2, 4, 5, 6], [0, 0, 2, 1]]

generators [600.0000000, 17.68449710]

comp 7.691616 rms 287.546442 bad 17011.522780

[5, 6, 11, 8, -3, -2] [[1, 3, 4, 6], [0, -5, -6, -11]]

generators [1200., 348.0209534]

comp 18.827083 rms 52.250677 bad 18520.725480

[0, 2, -8, -23, 12, 3] [[2, 3, 4, 7], [0, 0, 1, -4]]

generators [600.0000000, 234.1580374]

comp 15.230577 rms 90.719918 bad 21044.343580

[4, 4, 3, -4, 7, -4] [[1, 3, 4, 4], [0, -4, -4, -3]]

generators [1200., 464.8797794]

comp 11.251842 rms 184.323319 bad 23336.061130

[0, 1, -4, -15, 8, 2] [[1, 2, 3, 3], [0, 0, -1, 4]]

generators [1200., 70.92670834]

comp 7.615289 rms 442.895836 bad 25684.690840

[4, 0, 1, 2, 9, -8] [[1, 3, 2, 3], [0, -4, 0, -1]]

generators [1200., 473.9276617]

comp 11.251842 rms 225.636070 bad 28566.418700

[1, 0, 7, 14, -7, -2] [[1, 1, 2, 0], [0, 1, 0, 7]]

generators [1200., 452.0455031]

comp 11.512379 rms 226.287912 bad 29991.037240

[4, 7, 0, -21, 12, 2] [[1, 2, 3, 3], [0, -4, -7, 0]]

generators [1200., 99.13652417]

comp 15.874272 rms 133.944397 bad 33752.984350

[0, 1, -4, -8, 4, 1] [[1, 1, 1, 4], [0, 0, 1, -4]]

generators [1200., 539.2427832]

comp 7.615289 rms 624.335655 bad 36206.861660

>In the 5-limit case, it is clearly not so, unless we
>think meantone is the last word. In the 7-limit case,
>we do miss some important systems, but the inclusion
>of some fairly obvious commas would catch them.
//
>There are 150 7-limit superparticular linear temperaments;

I dunno Gene; looks like a bust to me. I wonder what
other easy stuff can be done to fractions to study commas.
There are jacks, but they're a subset of the above...
While we would expect superparticulars to be the smallest
intervals of a given complexity, there must be a cleaner
way of doing this... has anything been done on 'badness for
commas'?

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma
<clumma@y...>" <clumma@y...> wrote:
> >In the 5-limit case, it is clearly not so, unless we
> >think meantone is the last word. In the 7-limit case,
> >we do miss some important systems, but the inclusion
> >of some fairly obvious commas would catch them.
> //
> >There are 150 7-limit superparticular linear temperaments;
>
> I dunno Gene; looks like a bust to me. I wonder what
> other easy stuff can be done to fractions to study commas.
> There are jacks, but they're a subset of the above...
> While we would expect superparticulars to be the smallest
> intervals of a given complexity, there must be a cleaner
> way of doing this... has anything been done on 'badness for
> commas'?

the heurstic. note that |n-d| is right in there, so if you define
badness function that falls off instead of being log-flat, you can
easily arrange for the superparticulars to be the "least bad".

however, once a temperament is defined by more than one
comma, this approach will miss, since straightness is being
ignored.

speaking of which, i was wondering -- in the ultimate 5-limit list,
are there any commas which would make the list if one used
heuristic log-flat badness? when i get back to work, i can
calculate heuristic log-flat badness for each of them, and then
i'm hoping gene can run a search to determine if anything's
missing . . .

> > I dunno Gene; looks like a bust to me. I wonder what
> > other easy stuff can be done to fractions to study commas.
> > There are jacks, but they're a subset of the above...
> > While we would expect superparticulars to be the smallest
> > intervals of a given complexity, there must be a cleaner
> > way of doing this... has anything been done on 'badness for
> > commas'?
>
> the heurstic. note that |n-d| is right in there, so if you
> define badness function that falls off instead of being
> log-flat, you can easily arrange for the superparticulars
> to be the "least bad".

Can you explain how the complexity heuristic differs with
Gene's epimericity?

Still don't know how log(d) [the error heuristic] is
reckoned to tell RMS error.

> however, once a temperament is defined by more than one
> comma, this approach will miss, since straightness is being
> ignored.

No problem! I think linear temperaments are the most
interesting. Two sets of accidentals would be too confusing
fer the likes of me! I'd just use JI. If I'm going to
temper, give me error, low complexity, and simple notation!

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma
<clumma@y...>" <clumma@y...> wrote:
> > > I dunno Gene; looks like a bust to me. I wonder what
> > > other easy stuff can be done to fractions to study commas.
> > > There are jacks, but they're a subset of the above...
> > > While we would expect superparticulars to be the smallest
> > > intervals of a given complexity, there must be a cleaner
> > > way of doing this... has anything been done on 'badness for
> > > commas'?
> >
> > the heurstic. note that |n-d| is right in there, so if you
> > define badness function that falls off instead of being
> > log-flat, you can easily arrange for the superparticulars
> > to be the "least bad".
>
> Can you explain how the complexity heuristic differs with
> Gene's epimericity?

heuristic complexity is merely log(d) . . . epimericity can be
expressed as a function of *both* heuristic complexity and
heuristic eror -- thus it's a kind of badness measure, though not
log-flat.

> Still don't know how log(d) [the error heuristic] is
> reckoned to tell RMS error.

log(d) is the complexity heuristic, not the error heuristic. the error
heuristic is proportional to |n-d|/(d*log(d)).

> > however, once a temperament is defined by more than one
> > comma, this approach will miss, since straightness is being
> > ignored.
>
> No problem! I think linear temperaments are the most
> interesting.

a 7-liimit linear temperament is defined by a *pair* of unison
vectors, so straighness comes into play there already.

> Two sets of accidentals would be too confusing
> fer the likes of me! I'd just use JI.

5-limit JI has two sets of accidentals; other planar
temperaments are no more complex, and can reach effectively
into the 11-limit or whatever . . .

> If I'm going to
> temper, give me error, low complexity, and simple notation!

it should be a book of temperaments. for each, a notation, a
generalized keyboard, a halbestadt-imitation keyboard, a
horogram . . .

>
> -Carl

> heuristic complexity is merely log(d) . . . epimericity can be
> expressed as a function of *both* heuristic complexity and
> heuristic eror -- thus it's a kind of badness measure, though
> not log-flat.
//
> log(d) is the complexity heuristic, not the error heuristic.
> the error heuristic is proportional to |n-d|/(d*log(d)).

I swear you had that backwards when you defined "step" and
"cent" for me recently. Anyway, all is clear now!!

> > > however, once a temperament is defined by more than one
> > > comma, this approach will miss, since straightness is being
> > > ignored.
> >
> > No problem! I think linear temperaments are the most
> > interesting.
>
> a 7-liimit linear temperament is defined by a *pair* of unison
> vectors, so straighness comes into play there already.

When you say "temerament is defined by", you mean all the uvs,
only the commatic ones, ...?

And don't you mean that straigtness would _not_ come into play,
by what you say in the first paragraph (more than one comma =
ignore straightness)?

> > Two sets of accidentals would be too confusing
> > fer the likes of me! I'd just use JI.
>
> 5-limit JI has two sets of accidentals;

Adaptive JI. It should go without saying with me. I didn't
even know about strict JI until joining this list, and it
took 2 years of confusion before we figured it out!!

>>If I'm going to temper, give me error, low complexity, and
>>simple notation!
>
> it should be a book of temperaments. for each, a notation, a
> generalized keyboard, a halbestadt-imitation keyboard, a
> horogram . . .

Sounds good, but what does this have to do with my comment?

-Carl

Carl Lumma wrote:

> When you say "temerament is defined by", you mean all the uvs,
> only the commatic ones, ...?

You only need the commatic UVs to define the temperament. From those, all you need is the wedge product. It's useful to have a chromatic UV for doing the calculations, but they're interchangeble. Choosing each prime axis in turn will eventually give you a valid chromatic UV. The choice is only important when you look at the periodicity block, or corresponding MOS.

If the badness can be calculated solely from the wedge product of the commatic UVs, then it's naturally independent of the specific UVs chosen. I'm not sure if that's what Gene's proposing.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> If the badness can be calculated solely from the wedge product of the
> commatic UVs, then it's naturally independent of the specific UVs
> chosen. I'm not sure if that's what Gene's proposing.

That's the idea. Badness can be calculated from the wedgie, or equivalently, from the mapping.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> it should be a book of temperaments. for each, a notation, a
> generalized keyboard, a halbestadt-imitation keyboard, a
> horogram . . .

A set of poptimal generators, a MOS list, a mapping, a wedgie, a TM reduced basis for the kernel, and a chord indexing function.

I can never figure out these Erv Wilson things which consist of nothing but pictures, so if you can explain horograms that would be nice.

--- In tuning-math@yahoogroups.com, "Carl Lumma
<clumma@y...>" <clumma@y...> wrote:
> > heuristic complexity is merely log(d) . . . epimericity can be
> > expressed as a function of *both* heuristic complexity and
> > heuristic eror -- thus it's a kind of badness measure, though
> > not log-flat.
> //
> > log(d) is the complexity heuristic, not the error heuristic.
> > the error heuristic is proportional to |n-d|/(d*log(d)).
>
> I swear you had that backwards when you defined "step" and
> "cent" for me recently.

where?

> Anyway, all is clear now!!

phew!

> > > > however, once a temperament is defined by more than
one
> > > > comma, this approach will miss, since straightness is
being
> > > > ignored.
> > >
> > > No problem! I think linear temperaments are the most
> > > interesting.
> >
> > a 7-liimit linear temperament is defined by a *pair* of unison
> > vectors, so straighness comes into play there already.
>
> When you say "temerament is defined by", you mean all the
uvs,
> only the commatic ones, ...?

only the commatic ones, of course. the chromatic ones only
come into defining your finite "diatonic block".

> And don't you mean that straigtness would _not_ come into
play,
> by what you say in the first paragraph (more than one comma =
> ignore straightness)?

in what "first paragraph" do i say this?? i've been saying the exact
opposite!!!

> > > Two sets of accidentals would be too confusing
> > > fer the likes of me! I'd just use JI.
> >
> > 5-limit JI has two sets of accidentals;
>
> Adaptive JI. It should go without saying with me. I didn't
> even know about strict JI until joining this list, and it
> took 2 years of confusion before we figured it out!!

you're kidding me, right? well anyway, adaptive JI in general can
be even more complex to specify . . . do you specifically mean
adaptive JI on a meantone basis?

> >>If I'm going to temper, give me error, low complexity, and
> >>simple notation!
> >
> > it should be a book of temperaments. for each, a notation, a
> > generalized keyboard, a halbestadt-imitation keyboard, a
> > horogram . . .
>
> Sounds good, but what does this have to do with my
>comment?

notation.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:

> I can never figure out these Erv Wilson things which consist of
>nothing but pictures, so if you can explain horograms that
>would be nice.

they're simply concentric "clock diagrams" of all the MOSs of a
given generator. search the tuning list archives . . .

>A set of poptimal generators,

I thought something had cast doubt on the poptimal stuff.
And didn't Paul say the entire p range tends to minimax?

>chord indexing function.

What's a chord indexing function?

-Carl

> > > log(d) is the complexity heuristic, not the error heuristic.
> > > the error heuristic is proportional to |n-d|/(d*log(d)).
> >
> > I swear you had that backwards when you defined "step" and
> > "cent" for me recently.
>
> where?

D'oh! You had it right.

>>>however, once a temperament is defined by more than
>>>one comma, this approach will miss, since straightness
>>>is being ignored.
//
>>>a 7-liimit linear temperament is defined by a *pair* of unison
>>>vectors, so straighness comes into play there already.
//
>>And don't you mean that straigtness would _not_ come into
>>play, by what you say in the first paragraph (more than one
>>comma = ignore straightness)?
>
>in what "first paragraph" do i say this?? i've been saying
>the exact opposite!!!

Look ye above.

> > Adaptive JI. It should go without saying with me. I didn't
> > even know about strict JI until joining this list, and it
> > took 2 years of confusion before we figured it out!!
>
> you're kidding me, right?

No, why would you say that? Certainly you remember all those
heated arguments about whether something was 'possible in JI'.
It was quite ironic, actually.

> well anyway, adaptive JI in general can be even more complex
> to specify . . . do you specifically mean adaptive JI on a
> meantone basis?

No. I'm primarily interested in three types of adaptive JI.
One is painstakingly described here...

http://lumma.org/stuff/adaptive.txt

...and completely automatic. Another, writing intervals and
roots on scores as ratios, is complicated to perform (unless
you're a computer), but easy to notate, since I already know
ratios. I was thinking I should scan in the score of Retrofit
to show this technique.

The third is painstakingly described here...

http://lumma.org/stuff/xmw.txt

> > >>If I'm going to temper, give me error, low complexity, and
> > >>simple notation!
> > >
> > > it should be a book of temperaments. for each, a notation,
> > > a generalized keyboard, a halbestadt-imitation keyboard, a
> > > horogram . . .
> >
> > Sounds good, but what does this have to do with my
> >comment?
>
> notation.

But the book contains notation??? Anyway, all I was saying was,
I'm not interested in the more accurate temperaments. There
are already linear temperaments that are too accurate through
the 7-limit. Since we don't really know what the 11- and 13-
limits look like yet, I won't rule out my interest in planar
temperaments there... yet. But a lot of accuracy is required
there. And it's already so weird that tempering might be asking
for trouble. :)

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >A set of poptimal generators,
>
> I thought something had cast doubt on the poptimal stuff.
> And didn't Paul say the entire p range tends to minimax?

didn't i say what?

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:

> >>>however, once a temperament is defined by more than
> >>>one comma, this approach will miss, since straightness
> >>>is being ignored.
> //
> >>>a 7-liimit linear temperament is defined by a *pair* of unison
> >>>vectors, so straighness comes into play there already.
> //
> >>And don't you mean that straigtness would _not_ come into
> >>play, by what you say in the first paragraph (more than one
> >>comma = ignore straightness)?
> >
> >in what "first paragraph" do i say this?? i've been saying
> >the exact opposite!!!
>
> Look ye above.

yes, it's the exact opposite of what you thought -- if there's only
one comma, you're not missing or ignoring anything, because
straightness doesn't even exist (let alone come into play).

> > > Adaptive JI. It should go without saying with me. I didn't
> > > even know about strict JI until joining this list, and it
> > > took 2 years of confusion before we figured it out!!
> >
> > you're kidding me, right?
>
> No, why would you say that?

you didn't refer to pitches as ratios?

> Anyway, all I was saying was,
> I'm not interested in the more accurate temperaments. There
> are already linear temperaments that are too accurate through
> the 7-limit. Since we don't really know what the 11- and 13-
> limits look like yet, I won't rule out my interest in planar
> temperaments there... yet. But a lot of accuracy is required
> there. And it's already so weird that tempering might be asking
> for trouble. :)

i'm also not interested in the more accurate temperaments. they're
mathematical curiosities. what interests me are temperaments where
a "small-numbered" MOS already contains a good deal of the desired
harmonies, and you can develop a grammar around the harmonic meaning
of scalar alterations, etc.

> > I thought something had cast doubt on the poptimal stuff.
> > And didn't Paul say the entire p range tends to minimax?
>
> didn't i say what?

That as p goes to infinity you get minimax.

-Carl

> > Look ye above.
>
> yes, it's the exact opposite of what you thought -- if there's
> only one comma, you're not missing or ignoring anything, because
> straightness doesn't even exist (let alone come into play).

OK.

>>>>Adaptive JI. It should go without saying with me. I
>>>>didn't even know about strict JI until joining this list,
>>>>and it took 2 years of confusion before we figured it out!!
>>>
>>you're kidding me, right?
>>
>>No, why would you say that?
>
>you didn't refer to pitches as ratios?

Nope. Well, I had seed scales in ratios, but I never even
thought of the idea of a global pitch set.

>i'm also not interested in the more accurate temperaments.
>they're mathematical curiosities. what interests me are
>temperaments where a "small-numbered" MOS already contains
>a good deal of the desired harmonies, and you can develop
>a grammar around the harmonic meaning of scalar alterations,
>etc.

Exactly!

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:

> > > I thought something had cast doubt on the poptimal stuff.
> > > And didn't Paul say the entire p range tends to minimax?
> >
> > didn't i say what?
>
> That as p goes to infinity you get minimax.

well, yes, that's pretty well-known, so i don't think i even bothered
saying it explicitly.

but

"as p goes to infinity you get minimax"

sure sounds very different to me than

"the entire p range tends to minimax"

!

> "as p goes to infinity you get minimax"
>
> sure sounds very different to me than
>
> "the entire p range tends to minimax"
>
> !

Yeah, sorry 'bout that. Just in a hurry,
or half asleep.

-Carl