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Template: Temperament data

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

mapping generators: ~2, ~3

⟨⟨1 4 10 4 13 12]]

Optimal tuning (CTE): ~3/2 = 696.9521

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness: 0.0137

Catalog of 5-limit rank-2 temperaments

Sintel's TE logflat badness.

Error > 25 ¢, badness < 3

Temperament Complexity Error (¢) Badness Mapping Comma list
2.63 40.1 .392 [⟨1 0 -1], ⟨0 1 2]] 10/9
Antitonic 2.83 41.6 .508 [⟨2 3 0], ⟨0 0 1]] 9/8
University 5.59 25.3 2.39 [⟨1 1 2], ⟨0 3 2]] 144/125

Complexity < 30, error < 25 ¢, badness < 3

Temperament Complexity Error (¢) Badness Mapping Comma list
Father 3.17 20.4 .349 [⟨1 0 4], ⟨0 1 -1]] 16/15
Dicot 3.73 11.0 .306 [⟨1 1 2], ⟨0 2 1]] 25/24
Bug 4.30 17.9 .769 [⟨1 0 0], ⟨0 2 3]] 27/25
Meantone 5.08 2.44 .173 [⟨1 0 -4], ⟨0 1 4]] 81/80
Mavila 5.68 9.36 .928 [⟨1 0 7], ⟨0 1 -3]] 135/128
Augmented 6.40 3.71 .523 [⟨3 0 7], ⟨0 1 0]] 128/125
Porcupine 6.86 4.14 .722 [⟨1 2 3], ⟨0 3 5]] 250/243
Blackwood 7.29 7.14 1.50 [⟨5 8 0], ⟨0 0 1]] 256/243
Diminished 7.54 4.79 1.11 [⟨4 0 3], ⟨0 1 1]] 648/625
Supersharp 7.58 12.3 2.88 [⟨2 0 -5], ⟨0 1 3]] 800/729
Diaschismic 8.76 1.29 .467 [⟨2 0 11], ⟨0 1 -2]] 2048/2025
Magic 9.98 1.71 .919 [⟨1 0 2], ⟨0 5 1]] 3125/3072
Hanson 11.1 .422 .310 [⟨1 0 1], ⟨0 6 5]] 15625/15552
Negri 11.3 2.61 2.04 [⟨1 2 2], ⟨0 4 -3]] 16875/16384
Tetracot 11.5 1.39 1.14 [⟨1 1 1], ⟨0 4 9]] 20000/19683
Helmholtz 12.8 .0881 .0999 [⟨1 0 15], ⟨0 1 -8]] 32805/32768
Sensipent 14.1 .550 .826 [⟨1 6 8], ⟨0 7 9]] 78732/78125
Würschmidt 16.3 .404 .952 [⟨1 7 3], ⟨0 8 1]] 393216/390625
Amity 16.4 .217 .515 [⟨1 3 6], ⟨0 5 13]] 1600000/1594323
Valentine 16.7 1.14 2.88 [⟨1 1 2], ⟨0 9 5]] 1990656/1953125
Compton 17.4 .778 2.22 [⟨12 19 0], ⟨0 0 1]] 531441/524288
Orson 17.5 .332 .957 [⟨1 0 3], ⟨0 7 -3]] 2109375/2097152
Misty 21.3 .476 2.50 [⟨3 0 26], ⟨0 1 -4]] 67108864/66430125
Gravity 21.4 .415 2.19 [⟨1 5 12], ⟨0 6 17]] 129140163/128000000
Parakleismic 24.8 .123 1.02 [⟨1 5 6], ⟨0 13 14]] [8 14 -13⟩
Quartonic 24.9 .331 2.75 [⟨1 2 3], ⟨0 11 18]] [3 -18 11⟩
Escapade 25.8 .213 1.97 [⟨1 2 2], ⟨0 9 -7]] [32 -7 -9⟩
Vishnuzmic 26.5 .0727 .731 [⟨2 4 5], ⟨0 7 3]] [23 6 -14⟩
Vulture 27.3 .0890 .972 [⟨1 0 -6], ⟨0 4 21]] [24 -21 4⟩
Unidecmic 29.3 .143 1.93 [⟨2 5 8], ⟨0 6 11]] [-7 22 -12⟩

3 < badness < 6

Temperament Complexity Error (¢) Badness Mapping Comma list
Avila 7.79 16.7 4.27 [⟨1 0 -7], ⟨0 1 6]] 729/640
Enipucrop 8.16 11.5 3.38 [⟨1 2 2], ⟨0 3 -2]] 1125/1024
Laconic 8.87 10.1 3.80 [⟨1 1 1], ⟨0 3 7]] 2187/2000
Lafayette 9.31 10.8 4.70 [⟨1 1 2], ⟨0 5 3]] 3456/3125
Symbolic 9.38 9.40 4.19 [⟨1 3 2], ⟨0 4 -1]] 2048/1875
Sixix 9.81 7.06 3.59 [⟨1 3 4], ⟨0 5 6]] 3125/2916
Whitewood 10.1 6.49 3.63 [⟨7 11 0], ⟨0 0 1]] 2187/2048
Ripple 11.2 4.35 3.26 [⟨1 2 3], ⟨0 5 8]] 6561/6250
Progress 11.9 6.38 5.77 [⟨1 0 5], ⟨0 3 -5]] 32768/30375
Superpyth 12.2 3.26 3.17 [⟨1 0 -12], ⟨0 1 9]] 20480/19683
Wesley 13.6 4.25 5.81 [⟨1 4 3], ⟨0 7 2]] 78125/73728
Stump 14.4 2.91 4.71 [⟨1 0 6], ⟨0 3 -7]] 273375/262144
Passion 14.5 2.42 3.95 [⟨1 2 2], ⟨0 5 -4]] 262144/253125
Doublewide 14.7 3.09 5.32 [⟨2 1 3], ⟨0 4 3]] 390625/373248
Immunity 17.2 1.59 4.33 [⟨1 0 -8], ⟨0 2 13]] 1638400/1594323
Shibboleth 17.3 1.91 5.34 [⟨1 4 5], ⟨0 9 10]] 1953125/1889568
Unicorn 18.0 1.12 3.53 [⟨1 2 3], ⟨0 8 13]] 1594323/1562500
Mynic 18.6 1.69 5.86 [⟨1 9 9], ⟨0 10 9]] 10077696/9765625
Ampersand 19.9 .917 3.89 [⟨1 1 3], ⟨0 6 -7]] 34171875/33554432
Sycamore 20.5 1.06 4.93 [⟨2 2 3], ⟨0 5 7]] 48828125/47775744
Fifive 20.8 .992 4.83 [⟨2 2 3], ⟨0 5 7]] 9765625/9565938
Rodan 22.2 .669 3.95 [⟨1 1 -1], ⟨0 3 17]] [20 -17 3⟩
Mabila 23.8 .753 5.45 [⟨1 6 1], ⟨0 10 -3]] [28 -3 -10⟩
Ditonic 26.3 .399 3.92 [⟨1 6 3], ⟨0 13 2]] [-27 -2 13⟩
Trisedodge 27.7 .518 5.92 [⟨5 1 7], ⟨0 3 2]] [19 10 -15⟩
Demibuzzard 28.6 .243 3.06 [⟨1 4 -1], ⟨0 8 -11]] [-36 11 8⟩

30 < complexity < 60, badness < 3

Temperament Complexity Error (¢) Badness Mapping Comma list
Luna 33.5 .0239 .483 [⟨1 4 2], ⟨0 15 -2]] [38 -2 -15⟩
Enneadecal 35.8 .0454 1.12 [⟨19 0 14], ⟨0 1 1]] [-14 -19 19⟩
Gammic 37.2 .0740 2.06 [⟨1 1 2], ⟨0 20 11]] [-29 -11 20⟩
Ennealimmal 38.7 .0128 .403 [⟨9 1 1], ⟨0 2 3]] [1 -27 18⟩
Alphatricot 39.3 .0330 1.08 [⟨1 0 -13], ⟨0 3 29]] [39 -29 3⟩
Kwazy 43.2 .00762 .331 [⟨2 1 6], ⟨0 8 -5]] [-53 10 16⟩
Minortone 45.3 .0139 .698 [⟨1 16 32], ⟨0 17 35]] [-16 35 -17⟩
Monzismic 51.7 .00326 .244 [⟨1 0 -27], ⟨0 2 37]] [54 -37 2⟩
Vavoom 54.2 .0269 2.31 [⟨1 0 4], ⟨0 17 -18]] [-68 18 17⟩

5-limit rank-2 temperaments by TE simple badness

Breed's TE simple badness.

Junk temperaments

High-badness temperaments

Temperament Complexity Error (¢) Badness (moct) Mapping Comma list
Yo .367 26.0 7.95 [⟨1 0 -1], ⟨0 1 -2]] 10/9
Antitonic .395 27.0 8.89 [⟨2 3 0], ⟨0 0 1]] 9/8
Father .443 13.2 4.87 [⟨1 0 4], ⟨0 1 -1]] 16/15
Bug .602 11.6 5.81 [⟨1 0 0], ⟨0 2 3]] 27/25
Supersharp 1.06 7.94 7.01 [⟨2 0 -5], ⟨0 1 3]] 800/729
Laconic 1.24 6.52 6.74 [⟨1 1 1], ⟨0 3 7]] 2187/2000
Lafayette 1.30 7.00 7.60 [⟨1 1 2], ⟨0 5 3]] 3456/3125
Symbolic 1.31 6.09 6.66 [⟨1 3 4], ⟨0 5 6]] 2048/1875
Sixix 1.37 4.57 5.22 [⟨1 3 2], ⟨0 4 -1]] 3125/2916
Uncle 1.40 7.53 8.80 [⟨1 0 12], ⟨0 1 -6]] 4096/3645
Whitewood 1.42 4.20 4.95 [⟨7 11 0], ⟨0 0 1]] 2187/2048
Progress 1.66 4.13 5.72 [⟨1 0 5], ⟨0 3 -5]] 32768/30375

Main sequence

Temperament Complexity Error (¢) Badness (moct) Mapping Comma list
Dicot .521 7.09 3.08 [⟨1 1 2], ⟨0 2 1]] 25/24
Meantone .711 1.58 .937 [⟨1 0 -4], ⟨0 1 4]] 81/80
Mavila .795 6.06 4.02 [⟨1 0 7], ⟨0 1 -3]] 135/128
Augmented .894 2.40 1.79 [⟨3 0 7], ⟨0 1 0]] 128/125
Porcupine .960 2.68 2.14 [⟨1 2 3], ⟨0 3 5]] 250/243
Blackwood 1.02 4.63 3.93 [⟨5 8 0], ⟨0 0 1]] 256/243
Diminished 1.05 3.10 2.73 [⟨4 0 3], ⟨0 1 1]] 648/625
Diaschismic 1.22 .835 .852 [⟨2 0 11], ⟨0 1 -2]] 2048/2025
Magic 1.40 1.11 1.29 [⟨1 0 2], ⟨0 5 1]] 3125/3072
Hanson 1.55 .274 .353 [⟨1 0 1], ⟨0 6 5]] 15625/15552
Ripple 1.56 2.82 3.66 [⟨1 2 3], ⟨0 5 8]] 6561/6250
Negri 1.58 1.69 2.23 [⟨1 2 2], ⟨0 4 -3]] 16875/16384
Tetracot 1.61 .900 1.21 [⟨1 1 1], ⟨0 4 9]] 20000/19683
Superpyth 1.70 2.11 2.99 [⟨1 0 -12], ⟨0 1 9]] 20480/19683
Helmholtz 1.79 .0570 .0851 [⟨1 0 15], ⟨0 1 -8]] 32805/32768
Wesley 1.91 2.75 4.37 [⟨1 4 3], ⟨0 7 2]] 78125/73728
Sensipent 1.97 .356 .584 [⟨1 6 8], ⟨0 7 9]] 78732/78125
Stump 2.02 1.88 3.16 [⟨1 0 6], ⟨0 3 -7]] 273375/262144
Passion 2.02 1.57 2.64 [⟨1 2 2], ⟨0 5 -4]] 262144/253125
Doublewide 2.06 2.00 3.43 [⟨2 1 3], ⟨0 4 3]] 390625/373248
Würschmidt 2.29 .262 .499 [⟨1 7 3], ⟨0 8 1]] 393216/390625
Amity 2.29 .140 .268 [⟨1 3 6], ⟨0 5 13]] 1600000/1594323
Valentine 2.34 .736 1.44 [⟨1 1 2], ⟨0 9 5]] 1990656/1953125
Immunity 2.40 1.03 2.06 [⟨1 0 -8], ⟨0 2 13]] 1638400/1594323
Shibboleth 2.42 1.24 2.50 [⟨1 4 5], ⟨0 9 10]] 1953125/1889568
Compton 2.44 .504 1.02 [⟨12 19 0], ⟨0 0 1]] 531441/524288
Orson 2.44 .215 .438 [⟨1 0 3], ⟨0 7 -3]] 2109375/2097152
Unicorn 2.52 .725 1.52 [⟨1 2 3], ⟨0 8 13]] 1594323/1562500
Mynic 2.60 1.10 2.37 [⟨1 9 9], ⟨0 10 9]] 10077696/9765625
Ampersand 2.78 .594 1.38 [⟨1 1 3], ⟨0 6 -7]] 34171875/33554432
Fifive 2.91 .643 1.56 [⟨2 2 3], ⟨0 5 7]] 9765625/9565938
Misty 2.99 .308 .767 [⟨3 0 26], ⟨0 1 -4]] 67108864/66430125
Gravity 2.99 .269 .669 [⟨1 5 12], ⟨0 6 17]] 129140163/128000000
Rodan 3.10 .433 1.12 [⟨1 1 -1], ⟨0 3 17]] [20 -17 3⟩
3.11 .757 1.96 [⟨1 0 -23], ⟨0 1 16]] [-23 16 -1⟩
Mabila 3.32 .488 1.35 [⟨1 6 1], ⟨0 10 -3]] [28 -3 -10⟩
Parakleismic 3.47 .0798 .231 [⟨1 5 6], ⟨0 13 14]] [8 14 -13⟩
Quartonic 3.48 .214 .621 [⟨1 2 3], ⟨0 11 18]] [3 -18 11⟩
Escapade 3.60 .138 .414 [⟨1 2 2], ⟨0 9 -7]] [32 -7 -9⟩
Ditonic 3.68 .258 .792 [⟨1 6 3], ⟨0 13 2]] [-27 -2 13⟩
Vishnuzmic 3.71 .0471 .145 [⟨2 4 5], ⟨0 7 3]] [23 6 -14⟩
Vulture 3.81 .0576 .183 [⟨1 0 -6], ⟨0 4 21]] [24 -21 4⟩
Trisedodge 3.87 .336 1.08 [⟨5 1 7], ⟨0 3 2]] [19 10 -15⟩
4.00 .157 .524 [⟨1 4 -1], ⟨0 8 -11]] [-36 11 8⟩

Tunings

5-limit norm-based tunings
Manhattan Euclidean Chebyshevian
Constrained Constrained & skewed
Equilateral CEOP: ~3/2 = 697.654 ¢
(1/5-comma) 		CEE: ~3/2 = 696.895 ¢
(4/17-comma) 		CSEE: ~3/2 = 696.453 ¢
(11/43-comma) 		CEC: ~3/2 = 696.578 ¢
(1/4-comma)
Tenney CTOP: ~3/2 = 698.020 ¢ CTE: ~3/2 = 697.214 ¢ CWE: ~3/2 = 696.651 ¢ CTC: ~3/2 = 696.578 ¢
(1/4-comma)
Benedetti,
Wilson 		CBOP: ~3/2 = 698.160 ¢
(3/17-comma) 		CBE: ~3/2 = 697.374 ¢
(36/169-comma) 		CSBE: ~3/2 = 696.787 ¢
(31/129-comma) 		CBC: ~3/2 = 696.578 ¢
(1/4-comma)
7-limit norm-based tunings
Manhattan Euclidean Chebyshevian
Constrained Constrained & skewed
Equilateral CEOP: ~3/2 = 697.344 ¢ CEE: ~3/2 = 696.884 ¢ CSEE: ~3/2 = 696.725 ¢ CEC: ~3/2 = 696.883 ¢
Tenney CTOP: ~3/2 = 696.646 ¢ CTE: ~3/2 = 696.952 ¢ CWE: ~3/2 = 696.656 ¢ CTC: ~3/2 = 696.883 ¢
Benedetti,
Wilson 		CBOP: ~3/2 = 697.842 ¢ CBE: ~3/2 = 697.015 ¢ CSBE: ~3/2 = 696.631 ¢ CBC: ~3/2 = 696.883 ¢
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