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


{| class="wikitable center-all"
{{Flat}}{{natural}}{{sharp}}
! colspan="7" | Generator
 
! Cents
♭♮♯
! L
 
! s
<span class="bravura">&#xE260;</span>
! L/s
<span class="bravura">&#xE261;</span>
! Comments
<span class="bravura">&#xE262;</span>
 
<span class="heji">n</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
}}
 
== 5-limit rank-2 temperaments by TE simple badness ==
Breed's simple badness.
 
=== Junk temperaments ===
 
 
=== High-badness temperaments ===
{| class="wikitable sortable center-2 center-3 center-4"
|-
|-
| 4\7 || || || || || || || 685.714 || 1 || 1 || 1.000 ||
! Temperament !! Complexity !! Error (¢) !! Badness (moct) !! Mapping !! Comma list
|-
|-
| || || || || || || 27\47 || 689.362 || 7 || 6 || 1.167 ||
| Yo || .367 || 26.0 || 7.95 || {{mapping| 1 0 -1 | 0 1 -2 }} || 10/9
|-
|-
| || || || || || 23\40 || || 690.000 || 6 || 5 || 1.200 ||
| Antitonic || .395 || 27.0 || 8.89 || {{mapping| 2 3 0 | 0 0 1 }} || 9/8
|-
|-
| || || || || || || 42\73 || 690.411 || 11 || 9 || 1.222 ||
| Father || .443 || 13.2 || 4.87 || {{mapping| 1 0 4 | 0 1 -1 }} || 16/15
|-
|-
| || || || || 19\33 || || || 690.909 || 5 || 4 || 1.250 ||
| Bug || .602 || 11.6 || 5.81 || {{mapping| 1 0 0 | 0 2 3 }} || 27/25
|-
|-
| || || || || || || 53\92 || 691.304 || 14 || 11 || 1.273 ||
| Supersharp || 1.06 || 7.94 || 7.01 || {{mapping| 2 0 -5 | 0 1 3 }} || 800/729
|-
|-
| || || || || || 34\59 || || 691.525 || 9 || 7 || 1.286 ||
| Laconic || 1.24 || 6.52 || 6.74 || {{mapping| 1 1 1 | 0 3 7 }} || 2187/2000
|-
|-
| || || || || || || 49\85 || 691.765 || 13 || 10 || 1.300 ||
| Lafayette || 1.30 || 7.00 || 7.60 || {{mapping| 1 1 2 | 0 5 3 }} || 3456/3125
|-
|-
| || || || 15\26 || || || || 692.308 || 4 || 3 || 1.333 ||
| Symbolic || 1.31 || 6.09 || 6.66 || {{mapping| 1 3 4 | 0 -5 -6 }} || 2048/1875
|-
|-
| || || || || || || 56\97 || 692.784 || 15 || 11 || 1.364 ||
| Sixix || 1.37 || 4.57 || 5.22 || {{mapping| 1 3 2 | 0 -4 1 }} || 3125/2916
|-
|-
| || || || || || 41\71 || || 692.958 || 11 || 8 || 1.375 ||
| Uncle || 1.40 || 7.53 || 8.80 || {{mapping| 1 0 12 | 0 1 -6 }} || 4096/3645
|-
|-
| || || || || || || 67\116 || 693.103 || 18 || 13 || 1.385 ||
| Whitewood || 1.42 || 4.20 || 4.95 || {{mapping| 7 11 0 | 0 0 1 }} || 2187/2048
|-
|-
| || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 ||
| || 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"
|-
|-
| || || || || || || 63\109 || 693.578 || 17 || 12 || 1.417 ||
! Temperament !! Complexity !! Error (¢) !! Badness (moct) !! Mapping !! Comma list
|-
|-
| || || || || || 37\64 || || 693.750 || 10 || 7 || 1.429 ||
| Dicot || .521 || 7.09 || 3.08 || {{mapping| 1 1 2 | 0 2 1 }} || 25/24
|-
|-
| || || || || || || 48\83 || 693.976 || 13 || 9 || 1.444 ||
| Meantone || .711 || 1.58 || .937 || {{mapping| 1 0 -4 | 0 1 4 }} || 81/80
|-
|-
| || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
| Mavila || .795 || 6.06 || 4.02 || {{mapping| 1 0 7 | 0 1 -3 }} || 135/128
|-
|-
| || || || || || || 51\88 || 695.455 || 14 || 9 || 1.556 ||
| Augmented || .894 || 2.40 || 1.79 || {{mapping| 3 0 7 | 0 1 0 }} || 128/125
|-
|-
| || || || || || 40\69 || || 695.652 || 11 || 7 || 1.571 ||
| Porcupine || .960 || 2.68 || 2.14 || {{mapping| 1 2 3 | 0 -3 -5 }} || 250/243
|-
|-
| || || || || || || 69\119 || 695.798 || 19 || 12 || 1.583 ||
| Blackwood || 1.02 || 4.63 || 3.93 || {{mapping| 5 8 0 | 0 0 1 }} || 256/243
|-
|-
| || || || || 29\50 || || || 696.000 || 8 || 5 || 1.600 ||
| Diminished || 1.05 || 3.10 || 2.73 || {{mapping| 4 0 3 | 0 1 1 }} || 648/625
|-
|-
| || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || Golden meantone
| Srutal || 1.22 || .835 || .852 || {{mapping| 2 0 11 | 0 1 -2 }} || 2048/2025
|-
|-
| || || || || || 47\81 || || 696.296 || 13 || 8 || 1.625 ||
| Magic || 1.40 || 1.11 || 1.29 || {{mapping| 1 0 2 | 0 5 1 }} || 3125/3072
|-
|-
| || || || || || || 65\112 || 696.429 || 18 || 11 || 1.636 ||  
| Hanson || 1.55 || .274 || .353 || {{mapping| 1 0 1 | 0 6 5 }} || 15625/15552
|-
|-
| || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region
| Ripple || 1.56 || 2.82 || 3.66 || {{mapping| 1 2 3 | 0 -5 -8 }} || 6561/6250
|-
|-
| || || || || || || 61\105 || 697.143 || 17 || 10 || 1.700 ||
| Negri || 1.58 || 1.69 || 2.23 || {{mapping| 1 2 2 | 0 -4 3 }} || 16875/16384
|-
|-
| || || || || || 43\74 || || 697.297 || 12 || 7 || 1.714 ||
| Tetracot || 1.61 || .900 || 1.21 || {{mapping| 1 1 1 | 0 4 9 }} || 20000/19683
|-
|-
| || || || || || || 68\117 || 697.436 || 19 || 11 || 1.727 ||
| Superpyth || 1.70 || 2.11 || 2.99 || {{mapping| 1 0 -12 | 0 1 9 }} || 20480/19683
|-
|-
| || || || || 25\43 || || || 697.674 || 7 || 4 || 1.750 ||
| Helmholtz || 1.79 || .0570 || .0851 || {{mapping| 1 0 15 | 0 1 -8 }} || 32805/32768
|-
|-
| || || || || || || 57\98 || 697.959 || 16 || 9 || 1.778 ||
| Wesley || 1.91 || 2.75 || 4.37 || {{mapping| 1 4 3 | 0 -7 -2 }} || 78125/73728
|-
|-
| || || || || ||32\55 || || 698.182 || 9 || 5 || 1.800 ||
| Sensipent || 1.97 || .356 || .584 || {{mapping| 1 6 8 | 0 7 9 }} || 78732/78125
|-
|-
| || || || || || || 39\67 || 698.507 || 11 || 6 || 1.833 ||
| Stump || 2.02 || 1.88 || 3.16 || {{mapping| 1 0 6 | 0 3 -7 }} || 273375/262144
|-
|-
| || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)
| Passion || 2.02 || 1.57 || 2.64 || {{mapping| 1 2 2 | 0 -5 4 }} || 262144/253125
|-
|-
| || || || || || || 38\65 || 701.539 || 11 || 5 || 2.200 ||
| Doublewide || 2.06 || 2.00 || 3.43 || {{mapping| 2 1 3 | 0 4 3 }} || 390625/373248
|-
|-
| || || || || || 31\53 || || 701.887 || 9 || 4 || 2.250 ||
| Würschmidt || 2.29 || .262 || .499 || {{mapping| 1 7 3 | 0 8 1 }} || 393216/390625
|-
|-
| || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 ||
| Amity || 2.29 || .140 || .268 || {{mapping| 1 3 6 | 0 -5 -13 }} || 1600000/1594323
|-
|-
| || || || || 24\41 || || || 702.409 || 7 || 3 || 2.333 ||
| Valentine || 2.34 || .736 || 1.44 || {{mapping| 1 1 2 | 0 9 5 }} || 1990656/1953125
|-
|-
| || || || || || || 65\111 || 702.703 || 19 || 8 || 2.375 ||
| Immunity || 2.40 || 1.03 || 2.06 || {{mapping| 1 0 -8 | 0 2 13 }} || 1638400/1594323
|-
|-
| || || || || || 41\70 || || 702.857 || 12 || 5 || 2.400 ||
| Shibboleth || 2.42 || 1.24 || 2.50 || {{mapping| 1 4 5 | 0 -9 -10 }} || 1953125/1889568
|-
|-
| || || || || || || 58\99 || 703.030 || 17 || 7 || 2.428 ||
| Compton || 2.44 || .504 || 1.02 || {{mapping| 12 19 0 | 0 0 1 }} || 531441/524288
|-
|-
| || || || 17\29 || || || || 703.448 || 5 || 2 || 2.500 ||
| Orson || 2.44 || .215 || .438 || {{mapping| 1 0 3 | 0 7 -3 }} || 2109375/2097152
|-
|-
| || || || || || || 61\104 || 703.846 || 18 || 7 || 2.571 ||
| Unicorn || 2.52 || .725 || 1.52 || {{mapping| 1 2 3 | 0 -8 -13 }} || 1594323/1562500
|-
|-
| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
| Mynic || 2.60 || 1.10 || 2.37 || {{mapping| 1 9 9 | 0 -10 -9 }} || 10077696/9765625
|-
|-
| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden parapyth
| Ampersand || 2.78 || .594 || 1.38 || {{mapping| 1 1 3 | 0 6 -7 }} || 34171875/33554432
|-
|-
| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 ||
| Fifive || 2.91 || .643 || 1.56 || {{mapping| 2 2 3 | 0 5 7 }} || 9765625/9565938
|-
|-
| || || || || || || 64\109 || 704.587 || 19 || 7 || 2.714 ||
| Misty || 2.99 || .308 || .767 || {{mapping| 3 0 26 | 0 1 -4 }} || 67108864/66430125
|-
|-
| || || || || || 37\63 || || 704.762 || 11 || 4 || 2.750 ||
| Gravity || 2.99 || .269 || .669 || {{mapping| 1 5 12 | 0 -6 -17 }} || 129140163/128000000
|-
|-
| || || || || || || 47\80 || 705.000 || 14 || 5 || 2.800 ||
| Rodan || 3.10 || .433 || 1.12 || {{mapping| 1 1 -1 | 0 3 17 }} || {{monzo| 20 -17 3 }}
|-
|-
| || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
| || 3.11 || .757 || 1.96 || {{mapping| 1 0 -23 | 0 1 16 }} || {{monzo| -23 16 -1 }}
|-
|-
| || || || || || || 43\73 || 706.849 || 13 || 4 || 3.250 ||
| Mabila || 3.32 || .488 || 1.35 || {{mapping| 1 6 1 | 0 -10 3 }} || {{monzo| 28 -3 -10 }}
|-
|-
| || || || || || 33\56 || || 707.143 || 10 || 3 || 3.333 ||
| Parakleismic || 3.47 || .0798 || .231 || {{mapping| 1 5 6 | 0 -13 -14 }} || {{monzo| 8 14 -13 }}
|-
|-
| || || || || || || 56\95 || 707.368 || 17 || 5 || 3.400 ||
| Quartonic || 3.48 || .214 || .621 || {{mapping| 1 2 3 | 0 -11 -18 }} || {{monzo| 3 -18 11 }}
|-
|-
| || || || || 23\39 || || || 707.692 || 7 || 2 || 3.500 ||
| Escapade || 3.60 || .138 || .414 || {{mapping| 1 2 2 | 0 -9 7 }} || {{monzo| 32 -7 -9 }}
|-
|-
| || || || || || || 59\100 || 708.000 || 18 || 5 || 3.600 ||
| Ditonic || 3.68 || .258 || .792 || {{mapping| 1 6 3 | 0 -13 -2 }} || {{monzo| -27 -2 13 }}
|-
|-
| || || || || || 36\61 || || 708.197 || 11 || 3 || 3.667 ||
| Vishnuzmic || 3.71 || .0471 || .145 || {{mapping| 2 4 5 | 0 -7 -3 }} || {{monzo| 23 6 -14 }}
|-
|-
| || || || || || || 49\83 || 708.434 || 15 || 4 || 3.750 ||
| Vulture || 3.81 || .0576 || .183 || {{mapping| 1 0 -6 | 0 4 21 }} || {{monzo| 24 -21 4 }}
|-
|-
| || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region
| Trisedodge || 3.87 || .336 || 1.08 || {{mapping| 5 1 7 | 0 3 2 }} || {{monzo| 19 10 -15 }}
|-
|-
| || || || || || || 42\71 || 709.859 || 13 || 3 || 4.333 ||
| || 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 Prime-Optimized Tunings
|-
|-
| || || || || || 29\49 || || 710.204 || 9 || 2 || 4.500 ||
! rowspan="2" |  
! rowspan="2" | Manhattan
! colspan="2" | Euclidean
! rowspan="2" | Chebyshevian
|-
|-
| || || || || || || 45\76 || 710.526 || 14 || 3 || 4.667 ||
! Unskewed
! Skewed
|-
|-
| || || || || 16\27 || || || 711.111 || 5 || 1 || 5.000 ||
! Equilateral
| CEOP: ~3/2 = 697.654¢<br>(1/5-comma)
| CEE: ~3/2 = 696.895¢<br>(4/17-comma)
| CSEE: ~3/2 = 696.453¢<br>(11/43-comma)
| CEC: ~3/2 = 696.578¢<br>(1/4-comma)
|-
|-
| || || || || || || 35\59 || 711.864 || 11 || 2 || 5.500 ||
! Tenney
| CTOP: ~3/2 = 698.020¢
| CTE: ~3/2 = 697.214¢
| CWE: ~3/2 = 696.651¢
| CTC: ~3/2 = 696.578¢<br>(1/4-comma)
|-
|-
| || || || || || 19\32 || || 712.500 || 6 || 1 || 6.000 ||
! Benedetti, <br>Wilson
|-
| CBOP: ~3/2 = 698.160¢<br>(3/17-comma)
| || || || || || || 22\37 || 713.514 || 7 || 1 || 7.000 ||
| CBE: ~3/2 = 697.374¢<br>(36/169-comma)
|-
| CSBE: ~3/2 = 696.787¢<br>(31/129-comma)
| 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
| CBC: ~3/2 = 696.578¢<br>(1/4-comma)
|}
|}


Version 2 (bounded by integer limit = 10)
{| class="wikitable mw-collapsible mw-collapsed"
 
|+style=white-space:nowrap | 7-limit Prime-Optimized Tunings
{| class="wikitable center-all"
! colspan="10" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
| 4\7 || || || || || || || || || || 685.714 || 1 || 1 || 1.000 ||
|-
| || || || || || || || || || 39\68 || 688.235 || 10 || 9 || 1.111 ||
|-
| || || || || || || || || 35\61 || || 688.525 || 9 || 8 || 1.125 ||
|-
| || || || || || || || 31\54 || || || 688.889 || 8 || 7 || 1.143 ||
|-
| || || || || || || 27\47 || || || || 689.362 || 7 || 6 || 1.167 ||
|-
| || || || || || 23\40 || || || || || 690.000 || 6 || 5 || 1.200 ||
|-
| || || || || 19\33 || || || || || || 690.909 || 5 || 4 || 1.250 ||
|-
| || || || || || 34\59 || || || || || 691.525 || 9 || 7 || 1.286 ||
|-
| || || || 15\26 || || || || || || || 692.308 || 4 || 3 || 1.333 ||
|-
| || || || || 26\45 || || || || || || 693.333 || 7 || 5 || 1.400 ||
|-
| || || || || || 37\64 || || || || || 693.750 || 10 || 7 || 1.429 ||
|-
| || || 11\19 || || || || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
|-
| || || || || 29\50 || || || || || || 696.000 || 8 || 5 || 1.600 || Golden meantone
|-
| || || || 18\31 || || || || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region
|-
| || || || || 25\43 || || || || || || 697.674 || 7 || 4 || 1.750 ||
|-
| || || || || ||32\55 || || || || || 698.182 || 9 || 5 || 1.800 ||
|-
| || 7\12 || || || || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)
|-
| || || || || || 31\53 || || || || || 701.887 || 9 || 4 || 2.250 ||
|-
| || || || || 24\41 || || || || || || 702.409 || 7 || 3 || 2.333 ||
|-
| || || || 17\29 || || || || || || || 703.448 || 5 || 2 || 2.500 ||
|-
| || || || || 27\46 || || || || || || 704.348 || 8 || 3 || 2.667 || Golden parapyth
|-
| || || 10\17 || || || || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
|-
| || || || || || 33\56 || || || || || 707.143 || 10 || 3 || 3.333 ||
|-
| || || || || 23\39 || || || || || || 707.692 || 7 || 2 || 3.500 ||
|-
| || || || 13\22 || || || || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region
|-
| || || || || || 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 ||
! rowspan="2" |  
! rowspan="2" | Manhattan
! colspan="2" | Euclidean
! rowspan="2" | Chebyshevian
|-
|-
| || || || || || || || 25\42 || || ||714.286 || 8 || 1 || 8.000 ||
! Unskewed
! Skewed
|-
|-
| || || || || || || || || 28\47 || ||714.894 || 9 || 1 || 9.000 ||
! Equilateral
| CEOP: ~3/2 = 697.344¢
| CEE: ~3/2 = 696.884¢
| CSEE: ~3/2 = 696.725¢
| CEC: ~3/2 = 696.883¢
|-
|-
| || || || || || || || || || 31\52 ||715.385 || 10 || 1 || 10.000 ||
! Tenney
| CTOP: ~3/2 = 696.646¢
| CTE: ~3/2 = 696.952¢
| CWE: ~3/2 = 696.656¢
| CTC: ~3/2 = 696.883¢
|-
|-
| 3\5 || || || || || || || || || || 720.000 || 1 || 0 || → inf ||
! Benedetti, <br>Wilson
| CBOP: ~3/2 = 697.842¢
| CBE: ~3/2 = 697.015¢
| CSBE: ~3/2 = 696.631¢
| CBC: ~3/2 = 696.883¢
|}
|}


== Temperament pages ==
== Temperament pages ==
Databoxes has been canceled, but the cleanup will continue
Note:  
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]].  
# Order: [[subgroup]], [[comma list]], ([[sval]]) [[mapping]], (sval) mapping generators, gencom mapping, [[gencom]], lattice basis, [[wedgie]], [[optimal tuning]]s ([[CTE]], [[CWE]]/[[POTE]]), [[minimax tuning]], [[tuning ranges]], [[algebraic generator]], [[optimal ET sequence]], [[badness]], [[complexity spectrum]], and others.  
# Comma list shows the simplest commas sufficient to define the temperament, stated in [[Normal lists #Normal interval list]].  
# Comma list should show 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.  
# 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.
# Since minimax tunings are based on tonality diamond, 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 "CTE/CWE", otherwise show "subgroup/inharmonic CTE/CWE" instead.  
# 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:  
Get a family for:  
* <s>Ripple (3 different 7-limit extensions)</s>
* <s>Ripple (3 different 7-limit extensions)</s> done
* <s>Smate (2 different 7-limit extensions)</s>
* <s>Smate (2 different 7-limit extensions)</s> done
* Maybe [[parakleismic]] (2 different 7-limit extensions)
* <s>Passion (4 different 7-limit extensions, 3 strong and 1 weak)</s> done
* Maybe [[superpyth]] (2 different 7-limit extensions)
* <s>Undim</s> done
* <s>Quintaleap</s> done
* <s>Quindromeda</s> done
* Parakleismic (many reasonable but unnamed 7-limit extensions)
* Schismatic rank three family (perhaps)


Who's next?  
Who's next?  
* <s>Meantone family</s>
* <s>Meantone family</s>
* <s>Didymus rank three family</s>
* <s>Archytas clan</s>
* <s>Archytas clan</s>
* <s>Archytas family</s>
* <s>Father family</s>
* <s>Father family</s>
* <s>Trienstonic clan</s>
* <s>Trienstonic clan</s>
* <s>Septisemi temperaments</s>
* <s>Septisemi temperaments</s>
* <s>Archytas family</s>
* <s>Slendro clan</s>
* <s>Slendro clan</s>
* <s>Semiphore family</s>
* <s>Semiphore family</s>
* <s>Jubilismic clan</s>
* <s>Jubilismic family</s>
* <s>Mint temperaments</s>
* <s>Mint family</s>
* <s>Ripple family</s>
* <s>Smate family</s>
* <s>Augmented family</s>
* <s>Dimipent family</s>
* <s>Dicot family</s>
* <s>Bug family</s>
* <s>Pelogic family</s>
* <s>Marvel temperaments</s>
* <s>Marvel temperaments</s>
* <s>Marvel family</s>
* <s>Marvel family</s>
* <s>Mint temperaments</s>
* <s>Gamelismic clan</s>
* <s>Mint family</s>
* [[Gamelismic clan]]
* <s>Gamelismic family</s>
* <s>Gamelismic family</s>
* <s>Jubilismic clan</s>
* <s>Starling temperaments</s>
* <s>Jubilismic family</s>
* <s>Starling family</s>
* <s>Didymus rank three family</s>
* <s>Sensamagic clan</s>
* <s>Kleismic family</s>
* <s>Sensamagic family</s>
* [[Kleismic rank three family]]
* <s>Magic family</s>
* <s>Shibboleth family</s>
* <s>Sensipent family</s>
* <s>Keemic temperaments</s>
* <s>Keemic temperaments</s>
* <s>Keemic family</s>
* <s>Keemic family</s>
* <s>Sengic family</s>
* <s>Schismatic family</s>
* <s>Schismatic family</s>
* <s>Kleismic family</s>
* <s>Kleismic rank three family</s>
* <s>Würschmidt family</s>
* <s>Unicorn family</s>
* <s>Shibboleth family</s>
* <s>Immunity family</s>
* <s>Fifive family</s>
* <s>Trisedodge family</s>
* <s>Quintosec family</s>
* <s>Pental family</s>
* <s>Sycamore family</s>
* <s>Semicomma family</s>
* <s>Orwellismic temperaments</s>
* <s>Orwellismic family</s>
* <s>Hemimean clan</s>
* <s>Hemimean clan</s>
* <s>Hemimean family</s>
* <s>Hemimean family</s>
* <s>Hemifamity temperaments</s>
* <s>Hemifamity family</s>
* <s>Porwell temperaments</s>
* <s>Porwell family</s>
* <s>Hemimage temperaments</s>
* <s>Hemimage family</s>
* <s>Porcupine family</s>
* <s>Porcupine rank three family</s>
* <s>Tetracot family</s>
* <s>Diaschismic family</s>
* <s>Diaschismic rank three family</s>
* <s>Compton family</s>
* <s>Amity family</s>
* <s>Misty family</s>
* <s>Undim family</s>
* <s>Lehmerismic temperaments</s>
* <s>Kalismic temperaments</s>
* <s>Ragismic microtemperaments</s>
* <s>Ragismic family</s>
* <s>Landscape microtemperaments</s>
* <s>Landscape family</s>
* <s>Dimcomp family</s>
* <s>Mirkwai clan</s>
* <s>Mirkwai family</s>
* <s>Quince clan</s>
* <s>Breedsmic temperaments</s>
* <s>Breed family</s>
* <s>Escapade family</s>
* <s>Gravity family</s>
* <s>Cataharry temperaments</s>
* <s>Cataharry family</s>
* <s>Varunismic temperaments</s>
* <s>Rastmic rank three clan</s>
* <s>Biyatismic clan</s>
* <s>Ptolemismic clan</s>
* <s>Valinorsmic clan</s>
* <s>Pentacircle clan</s>
* <s>Keenanismic temperaments</s>
* <s>Werckismic temperaments</s>
* <s>Swetismic temperaments</s>
* <s>Wizmic microtemperaments</s>
* <s>Metric microtemperaments</s>
* <s>Horwell temperaments</s>
* <s>Horwell family</s>
* <s>Luna family</s>
* <s>Luna family</s>
* <s>Vulture family</s>
* <s>Tricot family</s>
* <s>Minortonic family</s>
* <s>Gammic family</s>
* <s>Vishnuzmic family</s>
* <s>Garischismic clan</s>
* <s>Canousmic temperaments</s>
* <s>Canousmic temperaments</s>
* <s>Canou family</s>
* <s>Canou family</s>
* [[Starling temperaments]]
* <s>Semicanousmic clan</s>
* [[Starling family]]
* <s>Semiporwellismic clan</s>
* <s>Sensipent family</s>
* <s>Olympic clan</s>
* [[Sensamagic clan]]
* <s>Alphaxenic rank three clan</s>
* [[Sensamagic family]]
* <s>Mirwomo temperaments</s>
* [[Magic family]]
* <s>Gariboh clan</s>
* <s>Gariboh family</s>
* <s>Octagar temperaments</s>
* <s>Octagar family</s>
* <s>Wesley family</s>
* <s>Apotome family</s>
* [[Mabila family]]
* [[Maquila family]]
* [[Maja family]]
* [[Ditonmic family]]
* <s>Greenwoodmic temperaments</s>
* <s>Avicennmic temperaments</s>


== Septimal meantone ==
== Regular temperament properties ==
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -49 31 }}
| {{mapping| 31 49 }}
| +1.63
| 1.64
| 4.22
|-
| 2.3.5
| 81/80, 393216/390625
| {{mapping| 31 49 72 }}
| +0.98
| 1.63
| 4.20
|-
| 2.3.5.7
| 81/80, 126/125, 1029/1024
| {{mapping| 31 49 72 87 }}
| +0.83
| 1.43
| 3.70
|-
| 2.3.5.7.11
| 81/80, 99/98, 121/120, 126/125
| {{mapping| 31 49 72 87 107 }}
| +1.21
| 1.49
| 3.84
|}


{{main| Meantone }}
== Scale tree ==
{{see also| Wikipedia: Septimal meantone temperament }}
; 6-tone
 
: <s>1L 5s</s>, [[2L 4s]], <s>3L 3s</s>, [[4L 2s]], <s>5L 1s</s>
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]]
; 7-tone
: <s>1L 6s</s>, <s>2L 5s</s>, <s>3L 4s</s>, <s>4L 3s</s>, <s>5L 2s</s>, <s>6L 1s</s>


== Archytas ==
; 8-tone
{{main| Archytas }}
: [[1L 7s]], <s>2L 6s</s>, [[3L 5s]], <s>4L 4s</s>, <s>5L 3s</s>, <s>6L 2s</s>, [[7L 1s]]


Subgroup: 2.3.5.7
; 9-tone
: [[1L 8s]], [[2L 7s]], <s>3L 6s</s>, <s>4L 5s</s>, <s>5L 4s</s>, <s>6L 3s</s>, [[7L 2s]], [[8L 1s]]


[[Comma list]]: 64/63
; 10-tone
: [[1L 9s]], <s>2L 8s</s>, <s>3L 7s</s>, <s>4L 6s</s>, <s>5L 5s</s>, <s>6L 4s</s>, <s>7L 3s</s>, [[8L 2s]], [[9L 1s]]


[[Mapping]]: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
; 11-tone
: [[1L 10s]], [[2L 9s]], [[3L 8s]], <s>4L 7s</s>, <s>5L 6s</s>, <s>6L 5s</s>, <s>7L 4s</s>, [[8L 3s]], [[9L 2s]], [[10L 1s]]


Mapping generators: ~2, ~3, ~5
; 12-tone
 
: [[1L 11s]], [[2L 10s]], [[3L 9s]], [[4L 8s]], <s>5L 7s</s>, <s>6L 6s</s>, <s>7L 5s</s>, [[8L 4s]], [[9L 3s]], [[10L 2s]], [[11L 1s]]
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
| <abbr title="36893488147419103232/36472996377170786403">{{monzo| 65 -41 }}</abbr>
| [[41-comma]]
|-
| 5
| <abbr title="1953125/1889568">{{Monzo| -5 -10 9 }}</abbr>
| [[Shibboleth]]
|-
| 5
| <abbr title="34171875/33554432">{{Monzo| -25 7 6 }}</abbr>
| [[Ampersand]]
|-
| 5
| [[3125/3072]]
| Magic comma
|-
| 5
| <abbr title="20000/19683">{{Monzo| 5 -9 4 }}</abbr>
| [[Tetracot comma]]
|-
| 5
| <abbr title="131072000/129140163">{{Monzo| 20 -17 3 }}</abbr>
| [[Roda]]
|-
| 5
| <abbr title="32805/32768">{{Monzo| -15 8 1 }}</abbr>
| [[Schisma]]
|-
| 7
| <abbr title="15625/15309">{{Monzo| 0 -7 6 -1 }}</abbr>
| [[Great BP diesis]]
|-
| 7
| <abbr title="854296875/843308032">{{Monzo| -10 7 8 -7 }}</abbr>
| [[Blackjackisma]]
|-
| 7
| [[875/864]]
| Keema
|-
| 7
| [[3125/3087]]
| Gariboh
|-
| 7
| <abbr title="179200/177147">{{Monzo| 10 -11 2 1 }}</abbr>
| [[Tolerma]]
|-
| 7
| <abbr title="33075/32768">{{Monzo| -15 3 2 2 }}</abbr>
| [[Mirwomo comma]]
|-
| 7
| [[245/243]]
| Sensamagic
|-
| 7
| [[4000/3969]]
| Octagar
|-
| 7
| <abbr title="823543/819200">{{Monzo| -15 0 -2 7 }}</abbr>
| [[Quince]]
|-
| 7
| [[1029/1024]]
| Gamelisma
|-
| 7
| [[225/224]]
| Marvel comma
|-
| 7
| <abbr title="16875/16807">{{Monzo| 0 3 4 -5 }}</abbr>
| [[Mirkwai]]
|-
| 7
| <abbr title="10976/10935">{{Monzo| 5 -7 -1 3 }}</abbr>
| [[Hemimage]]
|-
| 7
| [[5120/5103]]
| Hemifamity
|-
| 7
| <abbr title="33554432/33480783">{{Monzo| 25 -14 0 -1 }}</abbr>
| [[Garischisma]]
|-
| 7
| [[2401/2400]]
| Breedsma
|-
| 11
| <abbr title="163840/161051">{{Monzo| 15 0 1 0 -5 }}</abbr>
| [[Thuja comma]]
|-
| 11
| [[245/242]]
| Cassacot
|-
| 11
| [[100/99]]
| Ptolemisma
|-
| 11
| [[1344/1331]]
| Hemimin
|-
| 11
| [[896/891]]
| Pentacircle
|-
| 11
| <abbr title="65536/65219">{{Monzo| 16 0 0 -2 -3 }}</abbr>
| [[Orgonisma]]
|-
| 11
| [[243/242]]
| Rastma
|-
| 11
| [[385/384]]
| Keenanisma
|-
| 11
| [[441/440]]
| Werckisma
|-
| 11
| [[1375/1372]]
| Moctdel
|-
| 11
| [[540/539]]
| Swetisma
|-
| 11
| [[3025/3024]]
| Lehmerisma
|-
| 11
| <abbr title="151263/151250">{{Monzo| -1 2 -4 5 -2 }}</abbr>
| [[Odiheim]]
|-
| 13
| [[343/338]]
|
|-
| 13
| [[105/104]]
| Animist comma
|-
| 13
| <abbr title="28672/28431">{{Monzo| 12 -7 0 1 0 -1 }}</abbr>
| [[Secorian]]
|-
| 13
| [[275/273]]
| Gassorma
|-
| 13
| [[144/143]]
| Grossma
|-
| 13
| [[196/195]]
| Mynucuma
|-
| 13
| [[640/637]]
| Huntma
|-
| 13
| [[1188/1183]]
| Kestrel comma
|-
| 13
| [[325/324]]
| Marveltwin
|-
| 13
| [[352/351]]
| Minthma
|-
| 13
| [[364/363]]
| Gentle comma
|-
| 13
| [[847/845]]
| Cuthbert
|-
| 13
| [[729/728]]
| Squbema
|-
| 13
| [[4096/4095]]
| Schismina
|-
| 13
| <abbr title="10648/10647">{{Monzo| 3 -2 0 -1 3 -2 }}</abbr>
| [[Harmonisma]]
|-
| 17
| [[2187/2176]]
| Septendecimal schisma
|-
| 17
| [[256/255]]
| Septendecimal kleisma
|-
| 17
| [[715/714]]
| Septendecimal bridge comma
|-
| 19
| [[210/209]]
| Spleen comma
|-
| 19
| [[361/360]]
| Go comma
|-
| 19
| [[513/512]]
| Undevicesimal comma
|-
| 19
| [[1216/1215]]
| Eratosthenes' comma
|-
| 23
| [[736/729]]
| Vicesimotertial comma
|-
| 29
| [[145/144]]
| 29th-partial chroma
|}
<references/>
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