User:FloraC/Sandbox: Difference between revisions

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== Scale tree ==
== Accidentals ==
Version 1 (bounded by branch depth = 7)
; 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>


{| class="wikitable center-all"
== Template: Temperament data ==
! colspan="7" | Generator
{{User:FloraC/Temperament data
! Cents
| Subgroup = 2.3.5.7
! L
| Comma list = 81/80, 126/125
! s
| Mapping = {{mapping| 1 0 -4 -13 | 0 1 4 10 }}
! L/s
| Mapping generators = ~2, ~3
! Comments
| 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"
|-
|-
| 4\7 || || || || || || || 685.714 || 1 || 1 || 1.000 ||
! Temperament !! Complexity !! Error (¢) !! Badness !! Mapping !! Comma list
|-
|-
| || || || || || || 27\47 || 689.362 || 7 || 6 || 1.167 ||
| || 2.63 || 40.1 || .392 || {{mapping| 1 0 -1 | 0 1 2 }} || 10/9
|-
|-
| || || || || || 23\40 || || 690.000 || 6 || 5 || 1.200 ||
| Antitonic || 2.83 || 41.6 || .508 || {{mapping| 2 3 0 | 0 0 1 }} || 9/8
|-
|-
| || || || || || || 42\73 || 690.411 || 11 || 9 || 1.222 ||
| 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"
|-
|-
| || || || || 19\33 || || || 690.909 || 5 || 4 || 1.250 ||
! Temperament !! Complexity !! Error (¢) !! Badness !! Mapping !! Comma list
|-
|-
| || || || || || || 53\92 || 691.304 || 14 || 11 || 1.273 ||
| Father || 3.17 || 20.4 || .349 || {{mapping| 1 0 4 | 0 1 -1 }} || 16/15
|-
|-
| || || || || || 34\59 || || 691.525 || 9 || 7 || 1.286 ||
| Dicot || 3.73 || 11.0 || .306 || {{mapping| 1 1 2 | 0 2 1 }} || 25/24
|-
|-
| || || || || || || 49\85 || 691.765 || 13 || 10 || 1.300 ||
| Bug || 4.30 || 17.9 || .769 || {{mapping| 1 0 0 | 0 2 3 }} || 27/25
|-
|-
| || || || 15\26 || || || || 692.308 || 4 || 3 || 1.333 ||
| Meantone || 5.08 || 2.44 || .173 || {{mapping| 1 0 -4 | 0 1 4 }} || 81/80
|-
|-
| || || || || || || 56\97 || 692.784 || 15 || 11 || 1.364 ||
| Mavila || 5.68 || 9.36 || .928 || {{mapping| 1 0 7 | 0 1 -3 }} || 135/128
|-
|-
| || || || || || 41\71 || || 692.958 || 11 || 8 || 1.375 ||
| Augmented || 6.40 || 3.71 || .523 || {{mapping| 3 0 7 | 0 1 0 }} || 128/125
|-
|-
| || || || || || || 67\116 || 693.103 || 18 || 13 || 1.385 ||
| Porcupine || 6.86 || 4.14 || .722 || {{mapping| 1 2 3 | 0 3 5 }} || 250/243
|-
|-
| || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 ||
| Blackwood || 7.29 || 7.14 || 1.50 || {{mapping| 5 8 0 | 0 0 1 }} || 256/243
|-
|-
| || || || || || || 63\109 || 693.578 || 17 || 12 || 1.417 ||
| Diminished || 7.54 || 4.79 || 1.11 || {{mapping| 4 0 3 | 0 1 1 }} || 648/625
|-
|-
| || || || || || 37\64 || || 693.750 || 10 || 7 || 1.429 ||
| Supersharp || 7.58 || 12.3 || 2.88 || {{mapping| 2 0 -5 | 0 1 3 }} || 800/729
|-
|-
| || || || || || || 48\83 || 693.976 || 13 || 9 || 1.444 ||
| Diaschismic || 8.76 || 1.29 || .467 || {{mapping| 2 0 11 | 0 1 -2 }} || 2048/2025
|-
|-
| || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
| Magic || 9.98 || 1.71 || .919 || {{mapping| 1 0 2 | 0 5 1 }} || 3125/3072
|-
|-
| || || || || || || 51\88 || 695.455 || 14 || 9 || 1.556 ||
| Hanson || 11.1 || .422 || .310 || {{mapping| 1 0 1 | 0 6 5 }} || 15625/15552
|-
|-
| || || || || || 40\69 || || 695.652 || 11 || 7 || 1.571 ||
| Negri || 11.3 || 2.61 || 2.04 || {{mapping| 1 2 2 | 0 4 -3 }} || 16875/16384
|-
|-
| || || || || || || 69\119 || 695.798 || 19 || 12 || 1.583 ||
| Tetracot || 11.5 || 1.39 || 1.14 || {{mapping| 1 1 1 | 0 4 9 }} || 20000/19683
|-
|-
| || || || || 29\50 || || || 696.000 || 8 || 5 || 1.600 ||
| Helmholtz || 12.8 || .0881 || .0999 || {{mapping| 1 0 15 | 0 1 -8 }} || 32805/32768
|-
|-
| || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || Golden meantone
| Sensipent || 14.1 || .550 || .826 || {{mapping| 1 6 8 | 0 7 9 }} || 78732/78125
|-
|-
| || || || || || 47\81 || || 696.296 || 13 || 8 || 1.625 ||
| Würschmidt || 16.3 || .404 || .952 || {{mapping| 1 7 3 | 0 8 1 }} || 393216/390625
|-
|-
| || || || || || || 65\112 || 696.429 || 18 || 11 || 1.636 ||  
| Amity || 16.4 || .217 || .515 || {{mapping| 1 3 6 | 0 5 13 }} || 1600000/1594323
|-
|-
| || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region
| Valentine || 16.7 || 1.14 || 2.88 || {{mapping| 1 1 2 | 0 9 5 }} || 1990656/1953125
|-
|-
| || || || || || || 61\105 || 697.143 || 17 || 10 || 1.700 ||
| Compton || 17.4 || .778 || 2.22 || {{mapping| 12 19 0 | 0 0 1 }} || 531441/524288
|-
|-
| || || || || || 43\74 || || 697.297 || 12 || 7 || 1.714 ||
| Orson || 17.5 || .332 || .957 || {{mapping| 1 0 3 | 0 7 -3 }} || 2109375/2097152
|-
|-
| || || || || || || 68\117 || 697.436 || 19 || 11 || 1.727 ||
| Misty || 21.3 || .476 || 2.50 || {{mapping| 3 0 26 | 0 1 -4 }} || 67108864/66430125
|-
|-
| || || || || 25\43 || || || 697.674 || 7 || 4 || 1.750 ||
| Gravity || 21.4 || .415 || 2.19 || {{mapping| 1 5 12 | 0 6 17 }} || 129140163/128000000
|-
|-
| || || || || || || 57\98 || 697.959 || 16 || 9 || 1.778 ||
| Parakleismic || 24.8 || .123 || 1.02 || {{mapping| 1 5 6 | 0 13 14 }} || {{monzo| 8 14 -13 }}
|-
|-
| || || || || ||32\55 || || 698.182 || 9 || 5 || 1.800 ||
| Quartonic || 24.9 || .331 || 2.75 || {{mapping| 1 2 3 | 0 11 18 }} || {{monzo| 3 -18 11 }}
|-
|-
| || || || || || || 39\67 || 698.507 || 11 || 6 || 1.833 ||
| Escapade || 25.8 || .213 || 1.97 || {{mapping| 1 2 2 | 0 9 -7 }} || {{monzo| 32 -7 -9 }}
|-
|-
| || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)
| Vishnuzmic || 26.5 || .0727 || .731 || {{mapping| 2 4 5 | 0 7 3 }} || {{monzo| 23 6 -14 }}
|-
|-
| || || || || || || 38\65 || 701.539 || 11 || 5 || 2.200 ||
| Vulture || 27.3 || .0890 || .972 || {{mapping| 1 0 -6 | 0 4 21 }} || {{monzo| 24 -21 4 }}
|-
|-
| || || || || || 31\53 || || 701.887 || 9 || 4 || 2.250 ||
| 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"
|-
|-
| || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 ||
! Temperament !! Complexity !! Error (¢) !! Badness !! Mapping !! Comma list
|-
|-
| || || || || 24\41 || || || 702.409 || 7 || 3 || 2.333 ||
| Avila || 7.79 || 16.7 || 4.27 || {{mapping| 1 0 -7 | 0 1 6 }} || 729/640
|-
|-
| || || || || || || 65\111 || 702.703 || 19 || 8 || 2.375 ||
| Enipucrop || 8.16 || 11.5 || 3.38 || {{mapping| 1 2 2 | 0 3 -2 }} || 1125/1024
|-
|-
| || || || || || 41\70 || || 702.857 || 12 || 5 || 2.400 ||
| Laconic || 8.87 || 10.1 || 3.80 || {{mapping| 1 1 1 | 0 3 7 }} || 2187/2000
|-
|-
| || || || || || || 58\99 || 703.030 || 17 || 7 || 2.428 ||
| Lafayette || 9.31 || 10.8 || 4.70 || {{mapping| 1 1 2 | 0 5 3 }} || 3456/3125
|-
|-
| || || || 17\29 || || || || 703.448 || 5 || 2 || 2.500 ||
| Symbolic || 9.38 || 9.40 || 4.19 || {{mapping| 1 3 2 | 0 4 -1 }} || 2048/1875
|-
|-
| || || || || || || 61\104 || 703.846 || 18 || 7 || 2.571 ||
| Sixix || 9.81 || 7.06 || 3.59 || {{mapping| 1 3 4 | 0 5 6 }} || 3125/2916
|-
|-
| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
| Whitewood || 10.1 || 6.49 || 3.63 || {{mapping| 7 11 0 | 0 0 1 }} || 2187/2048
|-
|-
| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden parapyth
| Ripple || 11.2 || 4.35 || 3.26 || {{mapping| 1 2 3 | 0 5 8 }} || 6561/6250
|-
|-
| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 ||
| Progress || 11.9 || 6.38 || 5.77 || {{mapping| 1 0 5 | 0 3 -5 }} || 32768/30375
|-
|-
| || || || || || || 64\109 || 704.587 || 19 || 7 || 2.714 ||
| Superpyth || 12.2 || 3.26 || 3.17 || {{mapping| 1 0 -12 | 0 1 9 }} || 20480/19683
|-
|-
| || || || || || 37\63 || || 704.762 || 11 || 4 || 2.750 ||
| Wesley || 13.6 || 4.25 || 5.81 || {{mapping| 1 4 3 | 0 7 2 }} || 78125/73728
|-
|-
| || || || || || || 47\80 || 705.000 || 14 || 5 || 2.800 ||
| Stump || 14.4 || 2.91 || 4.71 || {{mapping| 1 0 6 | 0 3 -7 }} || 273375/262144
|-
|-
| || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
| Passion || 14.5 || 2.42 || 3.95 || {{mapping| 1 2 2 | 0 5 -4 }} || 262144/253125
|-
|-
| || || || || || || 43\73 || 706.849 || 13 || 4 || 3.250 ||
| Doublewide || 14.7 || 3.09 || 5.32 || {{mapping| 2 1 3 | 0 4 3 }} || 390625/373248
|-
|-
| || || || || || 33\56 || || 707.143 || 10 || 3 || 3.333 ||
| Immunity || 17.2 || 1.59 || 4.33 || {{mapping| 1 0 -8 | 0 2 13 }} || 1638400/1594323
|-
|-
| || || || || || || 56\95 || 707.368 || 17 || 5 || 3.400 ||
| Shibboleth || 17.3 || 1.91 || 5.34 || {{mapping| 1 4 5 | 0 9 10 }} || 1953125/1889568
|-
|-
| || || || || 23\39 || || || 707.692 || 7 || 2 || 3.500 ||
| Unicorn || 18.0 || 1.12 || 3.53 || {{mapping| 1 2 3 | 0 8 13 }} || 1594323/1562500
|-
|-
| || || || || || || 59\100 || 708.000 || 18 || 5 || 3.600 ||
| Mynic || 18.6 || 1.69 || 5.86 || {{mapping| 1 9 9 | 0 10 9 }} || 10077696/9765625
|-
|-
| || || || || || 36\61 || || 708.197 || 11 || 3 || 3.667 ||
| Ampersand || 19.9 || .917 || 3.89 || {{mapping| 1 1 3 | 0 6 -7 }} || 34171875/33554432
|-
|-
| || || || || || || 49\83 || 708.434 || 15 || 4 || 3.750 ||
| Sycamore || 20.5 || 1.06 || 4.93 || {{mapping| 2 2 3 | 0 5 7 }} || 48828125/47775744
|-
|-
| || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region
| Fifive || 20.8 || .992 || 4.83 || {{mapping| 2 2 3 | 0 5 7 }} || 9765625/9565938
|-
|-
| || || || || || || 42\71 || 709.859 || 13 || 3 || 4.333 ||
| Rodan || 22.2 || .669 || 3.95 || {{mapping| 1 1 -1 | 0 3 17 }} || {{monzo| 20 -17 3 }}
|-
|-
| || || || || || 29\49 || || 710.204 || 9 || 2 || 4.500 ||
| Mabila || 23.8 || .753 || 5.45 || {{mapping| 1 6 1 | 0 10 -3 }} || {{monzo| 28 -3 -10 }}
|-
|-
| || || || || || || 45\76 || 710.526 || 14 || 3 || 4.667 ||
| Ditonic || 26.3 || .399 || 3.92 || {{mapping| 1 6 3 | 0 13 2 }} || {{monzo| -27 -2 13 }}
|-
|-
| || || || || 16\27 || || || 711.111 || 5 || 1 || 5.000 ||
| Trisedodge || 27.7 || .518 || 5.92 || {{mapping| 5 1 7 | 0 3 2 }} || {{monzo| 19 10 -15 }}
|-
|-
| || || || || || || 35\59 || 711.864 || 11 || 2 || 5.500 ||
| Demibuzzard || 28.6 || .243 || 3.06 || {{mapping| 1 4 -1 | 0 8 -11 }} || {{monzo| -36 11 8 }}
|-
| || || || || || 19\32 || || 712.500 || 6 || 1 || 6.000 ||
|-
| || || || || || || 22\37 || 713.514 || 7 || 1 || 7.000 ||
|-
| 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
|}
|}


Version 2 (bounded by integer limit = 10)
30 < complexity < 60, badness < 3


{| class="wikitable center-all"
{| class="wikitable sortable center-2 center-3 center-4"
! colspan="10" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
|-
| 4\7 || || || || || || || || || || 685.714 || 1 || 1 || 1.000 ||
! Temperament !! Complexity !! Error (¢) !! Badness !! Mapping !! Comma list
|-
|-
| || || || || || || || || || 39\68 || 688.235 || 10 || 9 || 1.111 ||
| Luna || 33.5 || .0239 || .483 || {{mapping| 1 4 2 | 0 15 -2 }} || {{monzo| 38 -2 -15 }}
|-
|-
| || || || || || || || || 35\61 || || 688.525 || 9 || 8 || 1.125 ||
| Enneadecal || 35.8 || .0454 || 1.12 || {{mapping| 19 0 14 | 0 1 1 }} || {{monzo| -14 -19 19 }}
|-
|-
| || || || || || || || 31\54 || || || 688.889 || 8 || 7 || 1.143 ||
| Gammic || 37.2 || .0740 || 2.06 || {{mapping| 1 1 2 | 0 20 11 }} || {{monzo| -29 -11 20 }}
|-
|-
| || || || || || || 27\47 || || || || 689.362 || 7 || 6 || 1.167 ||
| Ennealimmal || 38.7 || .0128 || .403 || {{mapping| 9 1 1 | 0 2 3 }} || {{monzo| 1 -27 18 }}
|-
|-
| || || || || || 23\40 || || || || || 690.000 || 6 || 5 || 1.200 ||
| Alphatricot || 39.3 || .0330 || 1.08 || {{mapping| 1 0 -13 | 0 3 29 }} || {{monzo| 39 -29 3 }}
|-
|-
| || || || || 19\33 || || || || || || 690.909 || 5 || 4 || 1.250 ||
| Kwazy || 43.2 || .00762 || .331 || {{mapping| 2 1 6 | 0 8 -5 }} || {{monzo| -53 10 16 }}
|-
|-
| || || || || || 34\59 || || || || || 691.525 || 9 || 7 || 1.286 ||
| Minortone || 45.3 || .0139 || .698 || {{mapping| 1 16 32 | 0 17 35 }} || {{monzo| -16 35 -17 }}
|-
|-
| || || || 15\26 || || || || || || || 692.308 || 4 || 3 || 1.333 ||
| Monzismic || 51.7 || .00326 || .244 || {{mapping| 1 0 -27 | 0 2 37 }} || {{monzo| 54 -37 2 }}
|-
|-
| || || || || 26\45 || || || || || || 693.333 || 7 || 5 || 1.400 ||
| Vavoom || 54.2 || .0269 || 2.31 || {{mapping| 1 0 4 | 0 17 -18 }} || {{monzo| -68 18 17 }}
|-
|}
| || || || || || 37\64 || || || || || 693.750 || 10 || 7 || 1.429 ||
 
|-
== 5-limit rank-2 temperaments by TE simple badness ==
| || || 11\19 || || || || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
Breed's TE simple badness.  
 
=== Junk temperaments ===
=== High-badness temperaments ===
{| class="wikitable sortable center-2 center-3 center-4 mw-collapsible mw-collapsed"
|-
|-
| || || || || 29\50 || || || || || || 696.000 || 8 || 5 || 1.600 || Golden meantone
! Temperament !! Complexity !! Error (¢) !! Badness (moct) !! Mapping !! Comma list
|-
|-
| || || || 18\31 || || || || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region
| Yo || .367 || 26.0 || 7.95 || {{mapping| 1 0 -1 | 0 1 -2 }} || 10/9
|-
|-
| || || || || 25\43 || || || || || || 697.674 || 7 || 4 || 1.750 ||
| Antitonic || .395 || 27.0 || 8.89 || {{mapping| 2 3 0 | 0 0 1 }} || 9/8
|-
|-
| || || || || ||32\55 || || || || || 698.182 || 9 || 5 || 1.800 ||
| Father || .443 || 13.2 || 4.87 || {{mapping| 1 0 4 | 0 1 -1 }} || 16/15
|-
|-
| || 7\12 || || || || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)
| Bug || .602 || 11.6 || 5.81 || {{mapping| 1 0 0 | 0 2 3 }} || 27/25
|-
|-
| || || || || || 31\53 || || || || || 701.887 || 9 || 4 || 2.250 ||
| Supersharp || 1.06 || 7.94 || 7.01 || {{mapping| 2 0 -5 | 0 1 3 }} || 800/729
|-
|-
| || || || || 24\41 || || || || || || 702.409 || 7 || 3 || 2.333 ||
| Laconic || 1.24 || 6.52 || 6.74 || {{mapping| 1 1 1 | 0 3 7 }} || 2187/2000
|-
|-
| || || || 17\29 || || || || || || || 703.448 || 5 || 2 || 2.500 ||
| Lafayette || 1.30 || 7.00 || 7.60 || {{mapping| 1 1 2 | 0 5 3 }} || 3456/3125
|-
|-
| || || || || 27\46 || || || || || || 704.348 || 8 || 3 || 2.667 || Golden parapyth
| Symbolic || 1.31 || 6.09 || 6.66 || {{mapping| 1 3 4 | 0 5 6 }} || 2048/1875
|-
|-
| || || 10\17 || || || || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
| Sixix || 1.37 || 4.57 || 5.22 || {{mapping| 1 3 2 | 0 4 -1 }} || 3125/2916
|-
|-
| || || || || || 33\56 || || || || || 707.143 || 10 || 3 || 3.333 ||
| Uncle || 1.40 || 7.53 || 8.80 || {{mapping| 1 0 12 | 0 1 -6 }} || 4096/3645
|-
|-
| || || || || 23\39 || || || || || || 707.692 || 7 || 2 || 3.500 ||
| Whitewood || 1.42 || 4.20 || 4.95 || {{mapping| 7 11 0 | 0 0 1 }} || 2187/2048
|-
|-
| || || || 13\22 || || || || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region
| Progress || 1.66 || 4.13 || 5.72 || {{mapping| 1 0 5 | 0 3 -5 }} || 32768/30375
|-
| || || || || || 29\49 || || || || || 710.204 || 9 || 2 || 4.500 ||
|-
| || || || || 16\27 || || || || || || 711.111 || 5 || 1 || 5.000 ||
|-
| || || || || || 19\32 || || || || || 712.500 || 6 || 1 || 6.000 ||
|-
| || || || || || || 22\37 || || || || 713.514 || 7 || 1 || 7.000 ||
|-
| || || || || || || || 25\42 || || ||714.286 || 8 || 1 || 8.000 ||
|-
| || || || || || || || || 28\47 || ||714.894 || 9 || 1 || 9.000 ||
|-
| || || || || || || || || || 31\52 ||715.385 || 10 || 1 || 10.000 ||
|-
| 3\5 || || || || || || || || || || 720.000 || 1 || 0 || → inf ||
|}
|}


== Temperament pages ==
=== Main sequence ===
Databoxes has been canceled, but the cleanup will continue
{| class="wikitable sortable center-2 center-3 center-4 mw-collapsible mw-collapsed"
 
Note:
# Order: subgroup, [[comma list]], [[mapping]], mapping generators, gencom mapping, [[gencom]], map to lattice, lattice basis, [[wedgie]], [[minimax tuning]], [[tuning ranges]], [[algebraic generator]], vals, [[badness]], [[complexity spectrum]].
# Comma list shows the simplest commas sufficient to define the temperament, stated in [[Normal lists #Normal interval list]].
# Mapping generators should show all the ratios as used in the mapping, including the period.
# Minimax tuning are based on tonality diamond, so it should explicitly state the ''odd'' limit, or a diamond function of ratios.
# Use [[Template:Val list]].
# For subgroup temperaments, "mapping" becomes "sval mapping", add "gencom mapping" and "gencom". If TE is TE is TE (''sic''), simply show "POTE", otherwise show "POL2" or "POT2" instead of "POTE".
 
Get a family for:
* <s>Ripple (3 different 7-limit extensions)</s>
* <s>Smate (2 different 7-limit extensions)</s>
* Maybe [[parakleismic]] (2 different 7-limit extensions)
* Maybe [[superpyth]] (2 different 7-limit extensions)
 
Who's next?
* <s>Meantone family</s>
* <s>Archytas clan</s>
* <s>Father family</s>
* <s>Trienstonic clan</s>
* <s>Septisemi temperaments</s>
* <s>Archytas family</s>
* <s>Slendro clan</s>
* <s>Semiphore family</s>
* <s>Marvel temperaments</s>
* <s>Marvel family</s>
* <s>Mint temperaments</s>
* <s>Mint family</s>
* [[Gamelismic clan]]
* <s>Gamelismic family</s>
* <s>Jubilismic clan</s>
* <s>Jubilismic family</s>
* <s>Didymus rank three family</s>
* <s>Kleismic family</s>
* [[Kleismic rank three family]]
* <s>Shibboleth family</s>
* <s>Keemic temperaments</s>
* <s>Keemic family</s>
* <s>Schismatic family</s>
* <s>Hemimean clan</s>
* <s>Hemimean family</s>
* <s>Luna family</s>
* <s>Canousmic temperaments</s>
* <s>Canou family</s>
* [[Starling temperaments]]
* [[Starling family]]
* <s>Sensipent family</s>
* [[Sensamagic clan]]
* [[Sensamagic family]]
* [[Magic family]]
 
== Septimal meantone ==
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
 
{{main| Meantone }}
{{see also| Wikipedia: Septimal meantone temperament }}
 
The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the tritone, and [[21/16]], C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 81/80, 126/125
 
[[Mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]
 
Mapping generators: ~2, ~3
 
{{Multival|legend=1| 1 4 10 4 13 12 }}
 
[[POTE generator]]: ~3/2 = 696.495
 
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo]]s: 2, 5
 
[[Tuning ranges]]:
* valid range: [694.737, 700.000] (11\19 to 7\12)
* nice range: [694.786, 701.955]
* strict range: [694.786, 700.000]
 
[[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.
 
{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
 
[[Badness]]: 0.0137
 
Scales: [[meantone5]], [[meantone7]], [[meantone12]]
 
== Archytas ==
{{main| Archytas }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 64/63
 
[[Mapping]]: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
 
Mapping generators: ~2, ~3, ~5
 
Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
 
Lattice basis:
: 3/2 length = 1.0508, 5/4 length = 2.3219
: Angle (3/2, 5/4) = 90 degrees
 
[[POTE generator]]s: ~3/2 = 709.3213, ~5/4 = 393.3747
 
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 1/3 0 -1/3 }}, {{monzo| 2 -2/3 1 -1/3 }}, {{monzo| 2 -2/3 0 2/3 }}]
: [[Eigenmonzo]]s: 2, 6/5, 7/5
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 }}, {{monzo| 3/2 -1/2 1 -1/4 }}, {{monzo| 3 -1 0 1/2 }}]
: [[Eigenmonzo]]s: 2, 6/5, 9/7
 
{{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 49 }}
 
Scales: [[archytas12]], [[archytas12synch]]
 
== Commas ==
 
41edo [[tempers out]] the following [[comma]]s using its patent [[val]], {{val| 41 65 95 115 142 152 168 174 185 199 203 }}.
 
{| class="commatable wikitable center-1"
! [[Prime limit|Prime <br> limit]]
! [[Ratio]]<ref>Ratios with more than 9 digits are presented in monzos</ref>
! Name(s)
|-
|-
| 3
! Temperament !! Complexity !! Error (¢) !! Badness (moct) !! Mapping !! Comma list
| <abbr title="36893488147419103232/36472996377170786403">{{monzo| 65 -41 }}</abbr>
| [[41-comma]]
|-
|-
| 5
| Dicot || .521 || 7.09 || 3.08 || {{mapping| 1 1 2 | 0 2 1 }} || 25/24
| <abbr title="1953125/1889568">{{Monzo| -5 -10 9 }}</abbr>
| [[Shibboleth]]
|-
|-
| 5
| Meantone || .711 || 1.58 || .937 || {{mapping| 1 0 -4 | 0 1 4 }} || 81/80
| <abbr title="34171875/33554432">{{Monzo| -25 7 6 }}</abbr>
| [[Ampersand]]
|-
|-
| 5
| Mavila || .795 || 6.06 || 4.02 || {{mapping| 1 0 7 | 0 1 -3 }} || 135/128
| [[3125/3072]]
| Magic comma
|-
|-
| 5
| Augmented || .894 || 2.40 || 1.79 || {{mapping| 3 0 7 | 0 1 0 }} || 128/125
| <abbr title="20000/19683">{{Monzo| 5 -9 4 }}</abbr>
| [[Tetracot comma]]
|-
|-
| 5
| Porcupine || .960 || 2.68 || 2.14 || {{mapping| 1 2 3 | 0 3 5 }} || 250/243
| <abbr title="131072000/129140163">{{Monzo| 20 -17 3 }}</abbr>
| [[Roda]]
|-
|-
| 5
| Blackwood || 1.02 || 4.63 || 3.93 || {{mapping| 5 8 0 | 0 0 1 }} || 256/243
| <abbr title="32805/32768">{{Monzo| -15 8 1 }}</abbr>
| [[Schisma]]
|-
|-
| 7
| Diminished || 1.05 || 3.10 || 2.73 || {{mapping| 4 0 3 | 0 1 1 }} || 648/625
| <abbr title="15625/15309">{{Monzo| 0 -7 6 -1 }}</abbr>
| [[Great BP diesis]]
|-
|-
| 7
| Diaschismic || 1.22 || .835 || .852 || {{mapping| 2 0 11 | 0 1 -2 }} || 2048/2025
| <abbr title="854296875/843308032">{{Monzo| -10 7 8 -7 }}</abbr>
| [[Blackjackisma]]
|-
|-
| 7
| Magic || 1.40 || 1.11 || 1.29 || {{mapping| 1 0 2 | 0 5 1 }} || 3125/3072
| [[875/864]]
| Keema
|-
|-
| 7
| Hanson || 1.55 || .274 || .353 || {{mapping| 1 0 1 | 0 6 5 }} || 15625/15552
| [[3125/3087]]
| Gariboh
|-
|-
| 7
| Ripple || 1.56 || 2.82 || 3.66 || {{mapping| 1 2 3 | 0 5 8 }} || 6561/6250
| <abbr title="179200/177147">{{Monzo| 10 -11 2 1 }}</abbr>
| [[Tolerma]]
|-
|-
| 7
| Negri || 1.58 || 1.69 || 2.23 || {{mapping| 1 2 2 | 0 4 -3 }} || 16875/16384
| <abbr title="33075/32768">{{Monzo| -15 3 2 2 }}</abbr>
| [[Mirwomo comma]]
|-
|-
| 7
| Tetracot || 1.61 || .900 || 1.21 || {{mapping| 1 1 1 | 0 4 9 }} || 20000/19683
| [[245/243]]
| Sensamagic
|-
|-
| 7
| Superpyth || 1.70 || 2.11 || 2.99 || {{mapping| 1 0 -12 | 0 1 9 }} || 20480/19683
| [[4000/3969]]
| Octagar
|-
|-
| 7
| Helmholtz || 1.79 || .0570 || .0851 || {{mapping| 1 0 15 | 0 1 -8 }} || 32805/32768
| <abbr title="823543/819200">{{Monzo| -15 0 -2 7 }}</abbr>
| [[Quince]]
|-
|-
| 7
| Wesley || 1.91 || 2.75 || 4.37 || {{mapping| 1 4 3 | 0 7 2 }} || 78125/73728
| [[1029/1024]]
| Gamelisma
|-
|-
| 7
| Sensipent || 1.97 || .356 || .584 || {{mapping| 1 6 8 | 0 7 9 }} || 78732/78125
| [[225/224]]
| Marvel comma
|-
|-
| 7
| Stump || 2.02 || 1.88 || 3.16 || {{mapping| 1 0 6 | 0 3 -7 }} || 273375/262144
| <abbr title="16875/16807">{{Monzo| 0 3 4 -5 }}</abbr>
| [[Mirkwai]]
|-
|-
| 7
| Passion || 2.02 || 1.57 || 2.64 || {{mapping| 1 2 2 | 0 5 -4 }} || 262144/253125
| <abbr title="10976/10935">{{Monzo| 5 -7 -1 3 }}</abbr>
| [[Hemimage]]
|-
|-
| 7
| Doublewide || 2.06 || 2.00 || 3.43 || {{mapping| 2 1 3 | 0 4 3 }} || 390625/373248
| [[5120/5103]]
| Hemifamity
|-
|-
| 7
| Würschmidt || 2.29 || .262 || .499 || {{mapping| 1 7 3 | 0 8 1 }} || 393216/390625
| <abbr title="33554432/33480783">{{Monzo| 25 -14 0 -1 }}</abbr>
| [[Garischisma]]
|-
|-
| 7
| Amity || 2.29 || .140 || .268 || {{mapping| 1 3 6 | 0 5 13 }} || 1600000/1594323
| [[2401/2400]]
| Breedsma
|-
|-
| 11
| Valentine || 2.34 || .736 || 1.44 || {{mapping| 1 1 2 | 0 9 5 }} || 1990656/1953125
| <abbr title="163840/161051">{{Monzo| 15 0 1 0 -5 }}</abbr>
| [[Thuja comma]]
|-
|-
| 11
| Immunity || 2.40 || 1.03 || 2.06 || {{mapping| 1 0 -8 | 0 2 13 }} || 1638400/1594323
| [[245/242]]
| Cassacot
|-
|-
| 11
| Shibboleth || 2.42 || 1.24 || 2.50 || {{mapping| 1 4 5 | 0 9 10 }} || 1953125/1889568
| [[100/99]]
| Ptolemisma
|-
|-
| 11
| Compton || 2.44 || .504 || 1.02 || {{mapping| 12 19 0 | 0 0 1 }} || 531441/524288
| [[1344/1331]]
| Hemimin
|-
|-
| 11
| Orson || 2.44 || .215 || .438 || {{mapping| 1 0 3 | 0 7 -3 }} || 2109375/2097152
| [[896/891]]
| Pentacircle
|-
|-
| 11
| Unicorn || 2.52 || .725 || 1.52 || {{mapping| 1 2 3 | 0 8 13 }} || 1594323/1562500
| <abbr title="65536/65219">{{Monzo| 16 0 0 -2 -3 }}</abbr>
| [[Orgonisma]]
|-
|-
| 11
| Mynic || 2.60 || 1.10 || 2.37 || {{mapping| 1 9 9 | 0 10 9 }} || 10077696/9765625
| [[243/242]]
| Rastma
|-
|-
| 11
| Ampersand || 2.78 || .594 || 1.38 || {{mapping| 1 1 3 | 0 6 -7 }} || 34171875/33554432
| [[385/384]]
| Keenanisma
|-
|-
| 11
| Fifive || 2.91 || .643 || 1.56 || {{mapping| 2 2 3 | 0 5 7 }} || 9765625/9565938
| [[441/440]]
| Werckisma
|-
|-
| 11
| Misty || 2.99 || .308 || .767 || {{mapping| 3 0 26 | 0 1 -4 }} || 67108864/66430125
| [[1375/1372]]
| Moctdel
|-
|-
| 11
| Gravity || 2.99 || .269 || .669 || {{mapping| 1 5 12 | 0 6 17 }} || 129140163/128000000
| [[540/539]]
| Swetisma
|-
|-
| 11
| Rodan || 3.10 || .433 || 1.12 || {{mapping| 1 1 -1 | 0 3 17 }} || {{monzo| 20 -17 3 }}
| [[3025/3024]]
| Lehmerisma
|-
|-
| 11
| || 3.11 || .757 || 1.96 || {{mapping| 1 0 -23 | 0 1 16 }} || {{monzo| -23 16 -1 }}
| <abbr title="151263/151250">{{Monzo| -1 2 -4 5 -2 }}</abbr>
| [[Odiheim]]
|-
|-
| 13
| Mabila || 3.32 || .488 || 1.35 || {{mapping| 1 6 1 | 0 10 -3 }} || {{monzo| 28 -3 -10 }}
| [[343/338]]
|  
|-
|-
| 13
| Parakleismic || 3.47 || .0798 || .231 || {{mapping| 1 5 6 | 0 13 14 }} || {{monzo| 8 14 -13 }}
| [[105/104]]
| Animist comma
|-
|-
| 13
| Quartonic || 3.48 || .214 || .621 || {{mapping| 1 2 3 | 0 11 18 }} || {{monzo| 3 -18 11 }}
| <abbr title="28672/28431">{{Monzo| 12 -7 0 1 0 -1 }}</abbr>
| [[Secorian]]
|-
|-
| 13
| Escapade || 3.60 || .138 || .414 || {{mapping| 1 2 2 | 0 9 -7 }} || {{monzo| 32 -7 -9 }}
| [[275/273]]
| Gassorma
|-
|-
| 13
| Ditonic || 3.68 || .258 || .792 || {{mapping| 1 6 3 | 0 13 2 }} || {{monzo| -27 -2 13 }}
| [[144/143]]
| Grossma
|-
|-
| 13
| Vishnuzmic || 3.71 || .0471 || .145 || {{mapping| 2 4 5 | 0 7 3 }} || {{monzo| 23 6 -14 }}
| [[196/195]]
| Mynucuma
|-
|-
| 13
| Vulture || 3.81 || .0576 || .183 || {{mapping| 1 0 -6 | 0 4 21 }} || {{monzo| 24 -21 4 }}
| [[640/637]]
| Huntma
|-
|-
| 13
| Trisedodge || 3.87 || .336 || 1.08 || {{mapping| 5 1 7 | 0 3 2 }} || {{monzo| 19 10 -15 }}
| [[1188/1183]]
| Kestrel comma
|-
|-
| 13
| || 4.00 || .157 || .524 || {{mapping| 1 4 -1 | 0 8 -11 }} || {{monzo| -36 11 8 }}
| [[325/324]]
|}
| Marveltwin
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+style=white-space:nowrap | 5-limit norm-based tunings
|-
|-
| 13
! rowspan="2" |  
| [[352/351]]
! rowspan="2" | Manhattan
| Minthma
! colspan="2" | Euclidean
! rowspan="2" | Chebyshevian
|-
|-
| 13
! Constrained
| [[364/363]]
! Constrained & skewed
| Gentle comma
|-
|-
| 13
! Equilateral
| [[847/845]]
| CEOP: ~3/2 = 697.654{{c}}<br>(1/5-comma)
| Cuthbert
| 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)
|-
|-
| 13
! Tenney
| [[729/728]]
| CTOP: ~3/2 = 698.020{{c}}
| Squbema
| CTE: ~3/2 = 697.214{{c}}
| CWE: ~3/2 = 696.651{{c}}
| CTC: ~3/2 = 696.578{{c}}<br>(1/4-comma)
|-
|-
| 13
! Benedetti, <br>Wilson
| [[4096/4095]]
| CBOP: ~3/2 = 698.160{{c}}<br>(3/17-comma)
| Schismina
| CBE: ~3/2 = 697.374{{c}}<br>(36/169-comma)
|-
| CSBE: ~3/2 = 696.787{{c}}<br>(31/129-comma)
| 13
| CBC: ~3/2 = 696.578{{c}}<br>(1/4-comma)
| <abbr title="10648/10647">{{Monzo| 3 -2 0 -1 3 -2 }}</abbr>
|}
| [[Harmonisma]]
 
|-
{| class="wikitable mw-collapsible mw-collapsed"
| 17
|+style=white-space:nowrap | 7-limit norm-based tunings
| [[2187/2176]]
| Septendecimal schisma
|-
| 17
| [[256/255]]
| Septendecimal kleisma
|-
| 17
| [[715/714]]
| Septendecimal bridge comma
|-
| 19
| [[210/209]]
| Spleen comma
|-
|-
| 19
! rowspan="2" |  
| [[361/360]]
! rowspan="2" | Manhattan
| Go comma
! colspan="2" | Euclidean
! rowspan="2" | Chebyshevian
|-
|-
| 19
! Constrained
| [[513/512]]
! Constrained & skewed
| Undevicesimal comma
|-
|-
| 19
! Equilateral
| [[1216/1215]]
| CEOP: ~3/2 = 697.344{{c}}
| Eratosthenes' comma
| CEE: ~3/2 = 696.884{{c}}
| CSEE: ~3/2 = 696.725{{c}}
| CEC: ~3/2 = 696.883{{c}}
|-
|-
| 23
! Tenney
| [[736/729]]
| CTOP: ~3/2 = 696.646{{c}}
| Vicesimotertial comma
| CTE: ~3/2 = 696.952{{c}}
| CWE: ~3/2 = 696.656{{c}}
| CTC: ~3/2 = 696.883{{c}}
|-
|-
| 29
! Benedetti, <br>Wilson
| [[145/144]]
| CBOP: ~3/2 = 697.842{{c}}
| 29th-partial chroma
| CBE: ~3/2 = 697.015{{c}}
| CSBE: ~3/2 = 696.631{{c}}
| CBC: ~3/2 = 696.883{{c}}
|}
|}
<references/>

Latest revision as of 12:07, 19 May 2026

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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

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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