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== Accidentals ==
; Wiki templates : {{Flat}} {{natural}} {{sharp}}
; Unicode characters : ♭ ♮ ♯
; Bravura font : <span class="bravura">&#xE260;</span> <span class="bravura">&#xE261;</span> <span class="bravura">&#xE262;</span>
; Heji font : <span class="heji">e</span> <span class="heji">n</span> <span class="heji">v</span>


== Template: Temperament data ==
{{User:FloraC/Temperament data
| Subgroup = 2.3.5.7
| Comma list = 81/80, 126/125
| Mapping = {{mapping| 1 0 -4 -13 | 0 1 4 10 }}
| Mapping generators = ~2, ~3
| CTE = ~3/2 = 696.9521
| Wedgie = 1 4 10 4 13 12
| 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{{c}}, badness < 3
{| class="wikitable sortable center-2 center-3 center-4"
|-
! Temperament !! Complexity !! Error (¢) !! Badness !! Mapping !! Comma list
|-
|  || 2.63 || 40.1 || .392 || {{mapping| 1 0 -1 | 0 1 2 }} || 10/9
|-
| Antitonic || 2.83 || 41.6 || .508 || {{mapping| 2 3 0 | 0 0 1 }} || 9/8
|-
| University || 5.59 || 25.3 || 2.39 || {{mapping| 1 1 2 | 0 3 2 }} || 144/125
|}
Complexity < 30, error < 25{{c}}, badness < 3
{| class="wikitable sortable center-2 center-3 center-4"
|-
! Temperament !! Complexity !! Error (¢) !! Badness !! Mapping !! Comma list
|-
| Father || 3.17 || 20.4 || .349 || {{mapping| 1 0 4 | 0 1 -1 }} || 16/15
|-
| Dicot || 3.73 || 11.0 || .306 || {{mapping| 1 1 2 | 0 2 1 }} || 25/24
|-
| Bug || 4.30 || 17.9 || .769 || {{mapping| 1 0 0 | 0 2 3 }} || 27/25
|-
| Meantone || 5.08 || 2.44 || .173 || {{mapping| 1 0 -4 | 0 1 4 }} || 81/80
|-
| Mavila || 5.68 || 9.36 || .928 || {{mapping| 1 0 7 | 0 1 -3 }} || 135/128
|-
| Augmented || 6.40 || 3.71 || .523 || {{mapping| 3 0 7 | 0 1 0 }} || 128/125
|-
| Porcupine || 6.86 || 4.14 || .722 || {{mapping| 1 2 3 | 0 3 5 }} || 250/243
|-
| Blackwood || 7.29 || 7.14 || 1.50 || {{mapping| 5 8 0 | 0 0 1 }} || 256/243
|-
| Diminished || 7.54 || 4.79 || 1.11 || {{mapping| 4 0 3 | 0 1 1 }} || 648/625
|-
| Supersharp || 7.58 || 12.3 || 2.88 || {{mapping| 2 0 -5 | 0 1 3 }} || 800/729
|-
| Diaschismic || 8.76 || 1.29 || .467 || {{mapping| 2 0 11 | 0 1 -2 }} || 2048/2025
|-
| Magic || 9.98 || 1.71 || .919 || {{mapping| 1 0 2 | 0 5 1 }} || 3125/3072
|-
| Hanson || 11.1 || .422 || .310 || {{mapping| 1 0 1 | 0 6 5 }} || 15625/15552
|-
| Negri || 11.3 || 2.61 || 2.04 || {{mapping| 1 2 2 | 0 4 -3 }} || 16875/16384
|-
| Tetracot || 11.5 || 1.39 || 1.14 || {{mapping| 1 1 1 | 0 4 9 }} || 20000/19683
|-
| Helmholtz || 12.8 || .0881 || .0999 || {{mapping| 1 0 15 | 0 1 -8 }} || 32805/32768
|-
| Sensipent || 14.1 || .550 || .826 || {{mapping| 1 6 8 | 0 7 9 }} || 78732/78125
|-
| Würschmidt || 16.3 || .404 || .952 || {{mapping| 1 7 3 | 0 8 1 }} || 393216/390625
|-
| Amity || 16.4 || .217 || .515 || {{mapping| 1 3 6 | 0 5 13 }} || 1600000/1594323
|-
| Valentine || 16.7 || 1.14 || 2.88 || {{mapping| 1 1 2 | 0 9 5 }} || 1990656/1953125
|-
| Compton || 17.4 || .778 || 2.22 || {{mapping| 12 19 0 | 0 0 1 }} || 531441/524288
|-
| Orson || 17.5 || .332 || .957 || {{mapping| 1 0 3 | 0 7 -3 }} || 2109375/2097152
|-
| Misty || 21.3 || .476 || 2.50 || {{mapping| 3 0 26 | 0 1 -4 }} || 67108864/66430125
|-
| Gravity || 21.4 || .415 || 2.19 || {{mapping| 1 5 12 | 0 6 17 }} || 129140163/128000000
|-
| Parakleismic || 24.8 || .123 || 1.02 || {{mapping| 1 5 6 | 0 13 14 }} || {{monzo| 8 14 -13 }}
|-
| Quartonic || 24.9 || .331 || 2.75 || {{mapping| 1 2 3 | 0 11 18 }} || {{monzo| 3 -18 11 }}
|-
| Escapade || 25.8 || .213 || 1.97 || {{mapping| 1 2 2 | 0 9 -7 }} || {{monzo| 32 -7 -9 }}
|-
| Vishnuzmic || 26.5 || .0727 || .731 || {{mapping| 2 4 5 | 0 7 3 }} || {{monzo| 23 6 -14 }}
|-
| Vulture || 27.3 || .0890 || .972 || {{mapping| 1 0 -6 | 0 4 21 }} || {{monzo| 24 -21 4 }}
|-
| Unidecmic || 29.3 || .143 || 1.93 || {{mapping| 2 5 8 | 0 6 11 }} || {{monzo| -7 22 -12 }}
|}
3 < badness < 6
{| class="wikitable sortable center-2 center-3 center-4"
|-
! Temperament !! Complexity !! Error (¢) !! Badness !! Mapping !! Comma list
|-
| Avila || 7.79 || 16.7 || 4.27 || {{mapping| 1 0 -7 | 0 1 6 }} || 729/640
|-
| Enipucrop || 8.16 || 11.5 || 3.38 || {{mapping| 1 2 2 | 0 3 -2 }} || 1125/1024
|-
| Laconic || 8.87 || 10.1 || 3.80 || {{mapping| 1 1 1 | 0 3 7 }} || 2187/2000
|-
| Lafayette || 9.31 || 10.8 || 4.70 || {{mapping| 1 1 2 | 0 5 3 }} || 3456/3125
|-
| Symbolic || 9.38 || 9.40 || 4.19 || {{mapping| 1 3 2 | 0 4 -1 }} || 2048/1875
|-
| Sixix || 9.81 || 7.06 || 3.59 || {{mapping| 1 3 4 | 0 5 6 }} || 3125/2916
|-
| Whitewood || 10.1 || 6.49 || 3.63 || {{mapping| 7 11 0 | 0 0 1 }} || 2187/2048
|-
| Ripple || 11.2 || 4.35 || 3.26 || {{mapping| 1 2 3 | 0 5 8 }} || 6561/6250
|-
| Progress || 11.9 || 6.38 || 5.77 || {{mapping| 1 0 5 | 0 3 -5 }} || 32768/30375
|-
| Superpyth || 12.2 || 3.26 || 3.17 || {{mapping| 1 0 -12 | 0 1 9 }} || 20480/19683
|-
| Wesley || 13.6 || 4.25 || 5.81 || {{mapping| 1 4 3 | 0 7 2 }} || 78125/73728
|-
| Stump || 14.4 || 2.91 || 4.71 || {{mapping| 1 0 6 | 0 3 -7 }} || 273375/262144
|-
| Passion || 14.5 || 2.42 || 3.95 || {{mapping| 1 2 2 | 0 5 -4 }} || 262144/253125
|-
| Doublewide || 14.7 || 3.09 || 5.32 || {{mapping| 2 1 3 | 0 4 3 }} || 390625/373248
|-
| Immunity || 17.2 || 1.59 || 4.33 || {{mapping| 1 0 -8 | 0 2 13 }} || 1638400/1594323
|-
| Shibboleth || 17.3 || 1.91 || 5.34 || {{mapping| 1 4 5 | 0 9 10 }} || 1953125/1889568
|-
| Unicorn || 18.0 || 1.12 || 3.53 || {{mapping| 1 2 3 | 0 8 13 }} || 1594323/1562500
|-
| Mynic || 18.6 || 1.69 || 5.86 || {{mapping| 1 9 9 | 0 10 9 }} || 10077696/9765625
|-
| Ampersand || 19.9 || .917 || 3.89 || {{mapping| 1 1 3 | 0 6 -7 }} || 34171875/33554432
|-
| Sycamore || 20.5 || 1.06 || 4.93 || {{mapping| 2 2 3 | 0 5 7 }} || 48828125/47775744
|-
| Fifive || 20.8 || .992 || 4.83 || {{mapping| 2 2 3 | 0 5 7 }} || 9765625/9565938
|-
| Rodan || 22.2 || .669 || 3.95 || {{mapping| 1 1 -1 | 0 3 17 }} || {{monzo| 20 -17 3 }}
|-
| Mabila || 23.8 || .753 || 5.45 || {{mapping| 1 6 1 | 0 10 -3 }} || {{monzo| 28 -3 -10 }}
|-
| Ditonic || 26.3 || .399 || 3.92 || {{mapping| 1 6 3 | 0 13 2 }} || {{monzo| -27 -2 13 }}
|-
| Trisedodge || 27.7 || .518 || 5.92 || {{mapping| 5 1 7 | 0 3 2 }} || {{monzo| 19 10 -15 }}
|-
| Demibuzzard || 28.6 || .243 || 3.06 || {{mapping| 1 4 -1 | 0 8 -11 }} || {{monzo| -36 11 8 }}
|}
30 < complexity < 60, badness < 3
{| class="wikitable sortable center-2 center-3 center-4"
|-
! Temperament !! Complexity !! Error (¢) !! Badness !! Mapping !! Comma list
|-
| Luna || 33.5 || .0239 || .483 || {{mapping| 1 4 2 | 0 15 -2 }} || {{monzo| 38 -2 -15 }}
|-
| Enneadecal || 35.8 || .0454 || 1.12 || {{mapping| 19 0 14 | 0 1 1 }} || {{monzo| -14 -19 19 }}
|-
| Gammic || 37.2 || .0740 || 2.06 || {{mapping| 1 1 2 | 0 20 11 }} || {{monzo| -29 -11 20 }}
|-
| Ennealimmal || 38.7 || .0128 || .403 || {{mapping| 9 1 1 | 0 2 3 }} || {{monzo| 1 -27 18 }}
|-
| Alphatricot || 39.3 || .0330 || 1.08 || {{mapping| 1 0 -13 | 0 3 29 }} || {{monzo| 39 -29 3 }}
|-
| Kwazy || 43.2 || .00762 || .331 || {{mapping| 2 1 6 | 0 8 -5 }} || {{monzo| -53 10 16 }}
|-
| Minortone || 45.3 || .0139 || .698 || {{mapping| 1 16 32 | 0 17 35 }} || {{monzo| -16 35 -17 }}
|-
| Monzismic || 51.7 || .00326 || .244 || {{mapping| 1 0 -27 | 0 2 37 }} || {{monzo| 54 -37 2 }}
|-
| Vavoom || 54.2 || .0269 || 2.31 || {{mapping| 1 0 4 | 0 17 -18 }} || {{monzo| -68 18 17 }}
|}
== 5-limit rank-2 temperaments by TE simple badness ==
Breed's TE simple badness.
=== Junk temperaments ===
=== High-badness temperaments ===
{| class="wikitable sortable center-2 center-3 center-4 mw-collapsible mw-collapsed"
|-
! Temperament !! Complexity !! Error (¢) !! Badness (moct) !! Mapping !! Comma list
|-
| Yo || .367 || 26.0 || 7.95 || {{mapping| 1 0 -1 | 0 1 -2 }} || 10/9
|-
| Antitonic || .395 || 27.0 || 8.89 || {{mapping| 2 3 0 | 0 0 1 }} || 9/8
|-
| Father || .443 || 13.2 || 4.87 || {{mapping| 1 0 4 | 0 1 -1 }} || 16/15
|-
| Bug || .602 || 11.6 || 5.81 || {{mapping| 1 0 0 | 0 2 3 }} || 27/25
|-
| Supersharp || 1.06 || 7.94 || 7.01 || {{mapping| 2 0 -5 | 0 1 3 }} || 800/729
|-
| Laconic || 1.24 || 6.52 || 6.74 || {{mapping| 1 1 1 | 0 3 7 }} || 2187/2000
|-
| Lafayette || 1.30 || 7.00 || 7.60 || {{mapping| 1 1 2 | 0 5 3 }} || 3456/3125
|-
| Symbolic || 1.31 || 6.09 || 6.66 || {{mapping| 1 3 4 | 0 5 6 }} || 2048/1875
|-
| Sixix || 1.37 || 4.57 || 5.22 || {{mapping| 1 3 2 | 0 4 -1 }} || 3125/2916
|-
| Uncle || 1.40 || 7.53 || 8.80 || {{mapping| 1 0 12 | 0 1 -6 }} || 4096/3645
|-
| Whitewood || 1.42 || 4.20 || 4.95 || {{mapping| 7 11 0 | 0 0 1 }} || 2187/2048
|-
| Progress || 1.66 || 4.13 || 5.72 || {{mapping| 1 0 5 | 0 3 -5 }} || 32768/30375
|}
=== Main sequence ===
{| class="wikitable sortable center-2 center-3 center-4 mw-collapsible mw-collapsed"
|-
! Temperament !! Complexity !! Error (¢) !! Badness (moct) !! Mapping !! Comma list
|-
| Dicot || .521 || 7.09 || 3.08 || {{mapping| 1 1 2 | 0 2 1 }} || 25/24
|-
| Meantone || .711 || 1.58 || .937 || {{mapping| 1 0 -4 | 0 1 4 }} || 81/80
|-
| Mavila || .795 || 6.06 || 4.02 || {{mapping| 1 0 7 | 0 1 -3 }} || 135/128
|-
| Augmented || .894 || 2.40 || 1.79 || {{mapping| 3 0 7 | 0 1 0 }} || 128/125
|-
| Porcupine || .960 || 2.68 || 2.14 || {{mapping| 1 2 3 | 0 3 5 }} || 250/243
|-
| Blackwood || 1.02 || 4.63 || 3.93 || {{mapping| 5 8 0 | 0 0 1 }} || 256/243
|-
| Diminished || 1.05 || 3.10 || 2.73 || {{mapping| 4 0 3 | 0 1 1 }} || 648/625
|-
| Diaschismic || 1.22 || .835 || .852 || {{mapping| 2 0 11 | 0 1 -2 }} || 2048/2025
|-
| Magic || 1.40 || 1.11 || 1.29 || {{mapping| 1 0 2 | 0 5 1 }} || 3125/3072
|-
| Hanson || 1.55 || .274 || .353 || {{mapping| 1 0 1 | 0 6 5 }} || 15625/15552
|-
| Ripple || 1.56 || 2.82 || 3.66 || {{mapping| 1 2 3 | 0 5 8 }} || 6561/6250
|-
| Negri || 1.58 || 1.69 || 2.23 || {{mapping| 1 2 2 | 0 4 -3 }} || 16875/16384
|-
| Tetracot || 1.61 || .900 || 1.21 || {{mapping| 1 1 1 | 0 4 9 }} || 20000/19683
|-
| Superpyth || 1.70 || 2.11 || 2.99 || {{mapping| 1 0 -12 | 0 1 9 }} || 20480/19683
|-
| Helmholtz || 1.79 || .0570 || .0851 || {{mapping| 1 0 15 | 0 1 -8 }} || 32805/32768
|-
| Wesley || 1.91 || 2.75 || 4.37 || {{mapping| 1 4 3 | 0 7 2 }} || 78125/73728
|-
| Sensipent || 1.97 || .356 || .584 || {{mapping| 1 6 8 | 0 7 9 }} || 78732/78125
|-
| Stump || 2.02 || 1.88 || 3.16 || {{mapping| 1 0 6 | 0 3 -7 }} || 273375/262144
|-
| Passion || 2.02 || 1.57 || 2.64 || {{mapping| 1 2 2 | 0 5 -4 }} || 262144/253125
|-
| Doublewide || 2.06 || 2.00 || 3.43 || {{mapping| 2 1 3 | 0 4 3 }} || 390625/373248
|-
| Würschmidt || 2.29 || .262 || .499 || {{mapping| 1 7 3 | 0 8 1 }} || 393216/390625
|-
| Amity || 2.29 || .140 || .268 || {{mapping| 1 3 6 | 0 5 13 }} || 1600000/1594323
|-
| Valentine || 2.34 || .736 || 1.44 || {{mapping| 1 1 2 | 0 9 5 }} || 1990656/1953125
|-
| Immunity || 2.40 || 1.03 || 2.06 || {{mapping| 1 0 -8 | 0 2 13 }} || 1638400/1594323
|-
| Shibboleth || 2.42 || 1.24 || 2.50 || {{mapping| 1 4 5 | 0 9 10 }} || 1953125/1889568
|-
| Compton || 2.44 || .504 || 1.02 || {{mapping| 12 19 0 | 0 0 1 }} || 531441/524288
|-
| Orson || 2.44 || .215 || .438 || {{mapping| 1 0 3 | 0 7 -3 }} || 2109375/2097152
|-
| Unicorn || 2.52 || .725 || 1.52 || {{mapping| 1 2 3 | 0 8 13 }} || 1594323/1562500
|-
| Mynic || 2.60 || 1.10 || 2.37 || {{mapping| 1 9 9 | 0 10 9 }} || 10077696/9765625
|-
| Ampersand || 2.78 || .594 || 1.38 || {{mapping| 1 1 3 | 0 6 -7 }} || 34171875/33554432
|-
| Fifive || 2.91 || .643 || 1.56 || {{mapping| 2 2 3 | 0 5 7 }} || 9765625/9565938
|-
| Misty || 2.99 || .308 || .767 || {{mapping| 3 0 26 | 0 1 -4 }} || 67108864/66430125
|-
| Gravity || 2.99 || .269 || .669 || {{mapping| 1 5 12 | 0 6 17 }} || 129140163/128000000
|-
| Rodan || 3.10 || .433 || 1.12 || {{mapping| 1 1 -1 | 0 3 17 }} || {{monzo| 20 -17 3 }}
|-
|  || 3.11 || .757 || 1.96 || {{mapping| 1 0 -23 | 0 1 16 }} || {{monzo| -23 16 -1 }}
|-
| Mabila || 3.32 || .488 || 1.35 || {{mapping| 1 6 1 | 0 10 -3 }} || {{monzo| 28 -3 -10 }}
|-
| Parakleismic || 3.47 || .0798 || .231 || {{mapping| 1 5 6 | 0 13 14 }} || {{monzo| 8 14 -13 }}
|-
| Quartonic || 3.48 || .214 || .621 || {{mapping| 1 2 3 | 0 11 18 }} || {{monzo| 3 -18 11 }}
|-
| Escapade || 3.60 || .138 || .414 || {{mapping| 1 2 2 | 0 9 -7 }} || {{monzo| 32 -7 -9 }}
|-
| Ditonic || 3.68 || .258 || .792 || {{mapping| 1 6 3 | 0 13 2 }} || {{monzo| -27 -2 13 }}
|-
| Vishnuzmic || 3.71 || .0471 || .145 || {{mapping| 2 4 5 | 0 7 3 }} || {{monzo| 23 6 -14 }}
|-
| Vulture || 3.81 || .0576 || .183 || {{mapping| 1 0 -6 | 0 4 21 }} || {{monzo| 24 -21 4 }}
|-
| Trisedodge || 3.87 || .336 || 1.08 || {{mapping| 5 1 7 | 0 3 2 }} || {{monzo| 19 10 -15 }}
|-
|  || 4.00 || .157 || .524 || {{mapping| 1 4 -1 | 0 8 -11 }} || {{monzo| -36 11 8 }}
|}
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+style=white-space:nowrap | 5-limit norm-based tunings
|-
! rowspan="2" |
! rowspan="2" | Manhattan
! colspan="2" | Euclidean
! rowspan="2" | Chebyshevian
|-
! Constrained
! Constrained & skewed
|-
! Equilateral
| CEOP: ~3/2 = 697.654{{c}}<br>(1/5-comma)
| CEE: ~3/2 = 696.895{{c}}<br>(4/17-comma)
| CSEE: ~3/2 = 696.453{{c}}<br>(11/43-comma)
| CEC: ~3/2 = 696.578{{c}}<br>(1/4-comma)
|-
! Tenney
| CTOP: ~3/2 = 698.020{{c}}
| CTE: ~3/2 = 697.214{{c}}
| CWE: ~3/2 = 696.651{{c}}
| CTC: ~3/2 = 696.578{{c}}<br>(1/4-comma)
|-
! Benedetti, <br>Wilson
| CBOP: ~3/2 = 698.160{{c}}<br>(3/17-comma)
| CBE: ~3/2 = 697.374{{c}}<br>(36/169-comma)
| CSBE: ~3/2 = 696.787{{c}}<br>(31/129-comma)
| CBC: ~3/2 = 696.578{{c}}<br>(1/4-comma)
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+style=white-space:nowrap | 7-limit norm-based tunings
|-
! rowspan="2" |
! rowspan="2" | Manhattan
! colspan="2" | Euclidean
! rowspan="2" | Chebyshevian
|-
! Constrained
! Constrained & skewed
|-
! Equilateral
| CEOP: ~3/2 = 697.344{{c}}
| CEE: ~3/2 = 696.884{{c}}
| CSEE: ~3/2 = 696.725{{c}}
| CEC: ~3/2 = 696.883{{c}}
|-
! Tenney
| CTOP: ~3/2 = 696.646{{c}}
| CTE: ~3/2 = 696.952{{c}}
| CWE: ~3/2 = 696.656{{c}}
| CTC: ~3/2 = 696.883{{c}}
|-
! Benedetti, <br>Wilson
| CBOP: ~3/2 = 697.842{{c}}
| CBE: ~3/2 = 697.015{{c}}
| CSBE: ~3/2 = 696.631{{c}}
| CBC: ~3/2 = 696.883{{c}}
|}

Latest revision as of 12:07, 19 May 2026

Accidentals

Wiki templates
Unicode characters
♭ ♮ ♯
Bravura font
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e n v

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

Wedgie: ⟨⟨ 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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