Catalog of 3.5.7 subgroup rank two temperaments

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Merge into No-twos subgroup temperaments

Table

name comma 3.5.7 |> comma n/d comma c comma name a&b (3) a&b (2) complexity damage mapping TE gen1 TE gen2 per/3 MOS
Izar | -11 -7 12 > 13841287201/13839609375 0.210 b271 & b1071 171p & 676cd 11.675 0.003 [< 271 397 480 ], < 1071 1569 1897 ]> 1901.9584 877.2825 1 2*,3,5,7,9,11*,13*,24,37,50,63,76,...258*,271*
Canopus | 3 4 -5 > 16875/16807 6.990 small BP diesis, mirkwai b13 & b101 8d & 64cd 4.866 0.249 [< 13 19 23 ], < 101 148 179 ]> 1901.7828 583.9055 1 2,3*,4,7,10*,13*,23,36,49,62,75,88,101,114*
*Procyon | -8 -3 7 > 823543/820125 7.200 b13 & b144 8d & 91p 6.810 0.183 [< 13 19 23 ], < 144 211 255 ]> 1902.198 145.4124 1 ...13*,14,27,40,53,66,79,92,105,118,131,144,157*
*Gacrux | 6 8 -10 > 284765625/282475249 13.981
BPS (Bohlen-Pierce-Stearns) | -5 1 2 > 245/243 14.191 minor BP diesis, sensamagic b13 & b17 8d & 11cd 2.749 0.895 [< 13 19 23 ], < 17 25 30 ]> 1903.7424 440.902 1 2,3,4*,5,9,13*,17,30,43,56,69,82*,95*
Sirius | -2 5 -3 > 3125/3087 21.181 major BP diesis, gariboh b13 & b19 8d & 12p 4.021 0.913 [< 13 19 23 ], < 19 28 34 ]> 1902.4482 293.7397 1 2,3,4,5,6*,7,13*,19,32,45,58,71,84,97,110,123,136*
*Betelgeuse | -13 -2 9 > 40353607/39858075 21.391 b13 & b148 8d & 93bccd 8.754 0.423 [< 13 19 23 ], < 148 217 262 ]> 1902.5991 732.4139 1 2*,3*,5*,8,13*,...226*
*Miaplacidus | 1 9 -8 > 5859375/5764801 28.171 b13 & b94d 8d & 59ccdd 7.788 0.627 [< 13 19 23 ], < 94 138 167 ]> 1901.88 585.8431 1 2,3*,4,7,10,13*,16,29,42,55,68,81,94,...224*
Arcturus | -7 6 -1 > 15625/15309 35.371 great BP diesis b13 & b2 8d & 1cdd 4.823 1.271 [< 13 19 23 ], < 2 3 4 ]> 1903.8685 878.9253 1 2*,3,5,7,9,11,13*
*Pollux | -18 -1 11 > 1977326743/1937102445 35.581 b13 & b139 8d & 88dd 10.697 0.576 [< 13 19 23 ], < 139 204 246 ]> 1902.8625 438.6389 1 2,3,4*,5,9*,13
*Castor | -1 14 -11 > 6103515625/5931980229 49.352 b13 & b113d 8d & 71ccddd 11.259 0.760 [< 13 19 23 ], < 113 166 201 ]> 1902.0166 438.5094 1 2,3,4*,5,9*,13
*Muphrid | -12 7 1 > 546875/531441 49.562 b13 & b2d 8d & 1c 6.599 1.302 [< 13 19 23 ], < 2 3 3 ]> 1904.3123 879.0503 1 2*,3,5,7,9,11,13*
*Rigel | -9 11 -4 > 48828125/47258883 56.552 b13 & b21d 8d & 5d 8.844 1.108 [< 13 19 23 ], < 21 31 38 ]> 1903.2571 731.7183 1 2*,3*,5*,8*,13*
*Mirzam | -17 8 3 > 133984375/129140163 63.752 b13 & b32d 8d & 20c 9.347 1.182 [< 13 19 23 ], < 32 47 56 ]> 1904.3391 293.1889 1 2,3,4,5,6*,7,13*
| 15 -9 -1 > 14348907/13671875 83.676 b17 & b32 11cd & 20cd 8.228 1.762 [< 17 25 30 ], < 32 47 57 ]> 1898.8326 891.5018 1 2*,3,5,7,9,11,13,15*,17*,32,49*,66,115,164*
*Capella | 10 -8 1 > 413343/390625 97.866 small link b17 & b19 12p & 11cd 6.452 2.629 [< 17 25 30 ], < 19 28 34 ]> 1897.8856 895.415 1 2*,3,5,7,9,11,13,15,17*,19,36*,53*,89,142*,195*
| 13 -4 -4 > 1594323/1500625 104.857 b4 & b32 3bcdd & 20cd 7.126 2.550 [< 4 6 7 ], < 32 47 57 ]> 474.1446 55.0036 4 ...32*,36*,68*,104*,172*,276*
| 16 0 -9 > 43046721/40353607 111.847 b9 & b36 6bcd & 23d 8.773 2.209 [< 9 13 16 ], < 36 53 64 ]> 210.9385 44.1126 9 18,27,36*,45*,81,126,171*,216*,387*
Toliman | 5 -7 3 > 83349/78125 112.057 minor link b6 & b17 4p & 11cd 5.649 3.437 [< 6 9 11 ], < 17 25 30 ]> 1898.4178 333.298 1 2,3,4,5*,6*,11,17*,23,40,57*,74,131*
Vega | 8 -3 -2 > 6561/6125 119.047 major link b4 & b15 3bcdd & 10bccdd 4.377 4.713 [< 4 6 7 ], < 15 22 27 ]> 1892.0204 494.8658 1 2,3*,4*,7,11,15,19*,23*,42*,65*
*Achernar | 0 -6 5 > 16807/15625 126.247 b7 & b17 4bcd & 11cd 4.847 4.514 [< 7 10 12 ], < 17 25 30 ]> 1901.955 559.351 1 2,3*,4,7*,10,17*
% | 3 -2 0 > 27/25 133.238 large limma, BP small semitone b4 & b6 3bcdd & 4p 1.629 14.176 [< 4 6 7 ], < 6 9 11 ]> 939.6121 549.9897 2 4*,6,10*,14,24*,34,58,82*
| 6 2 -5 > 18225/16807 140.228 minimal BP chroma b9 & b10 6bcd & 6ccdd 4.885 4.974 [< 9 13 16 ], < 10 15 18 ]> 1895.1156 201.762 1 ...9*,10,19*,28*,47*
% | -2 -1 2 > 49/45 147.428 BP minor semitone, swetismic neutral second b4 & b7 3bcdd & 4bcd 1.925 13.274 [< 4 6 7 ], < 7 10 12 ]> 1917.7266 516.1077 1 2,3*,4*,7,11*,15*,26*
| 1 3 -3 > 375/343 154.418 BP major semitone, minor BP chroma b3 & b6 2cd & 4p 2.941 9.098 [< 3 4 5 ], < 6 9 11 ]> 632.9248 234.1367 3 6*,9*,15,24*,33,57*,81*
| -7 0 4 > 2401/2187 161.619 b4 & b40cd 3bcdd & 25p 3.868 7.240 [< 4 6 7 ], < 40 58 70 ]> 478.34 41.8631 4 ...44*,48,92*,136,228*,320*
| -4 4 -1 > 625/567 168.609 BP great semitone, major BP chroma b2 & b7 1cdd & 4bcd 3.195 9.146 [< 2 3 4 ], < 7 10 12 ]> 1914.567 1069.8876 1 2*,3,5,7*,9*,16,25,34*
| -9 5 1 > 21875/19683 182.799 maximal BP chroma b9 & b11 6bcd & 7d 4.970 6.373 [< 9 13 16 ], < 11 16 19 ]> 1913.8948 858.1831 1 2*,3,5,7,9*,11,20*,29*
% | 9 0 -5 > 19683/16807 273.465 b5 & b10 3d & 6ccdd 4.904 9.663 [< 5 7 9 ], < 10 15 18 ]> 377.3036 145.1888 5 10*,15*,25*,40,65*

names: * = proposed

names: % = not shown on PTS plots

MOS: * = maximally even

complexity = ln(n*d)/4

damage = 1200 *log(n/d) / log(n*d)

(complexity and damage are calculated as in the "Middle Path" tables)

Graphs

357plot_cplx_damage.png
Complexity vs. damage plot. z<1 corresponds to the "Middle Path" inclusion criterion.

Projective tuning space diagrams

357ptslines1n.png
Temperaments supported by 13edt, labeled by name
357ptslines1c.png
Temperaments supported by 13edt, labeled by comma
357ptslines2n.png
Temperaments not supported by 13edt, labeled by name
357ptslines2c.png
Temperaments not supported by 13edt, labeled by comma
357ptslines12n.png
Both sets, labeled by name
357ptslines12c.png
Both sets, labeled by comma
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