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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/>
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