User:FloraC/Sandbox: Difference between revisions

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{{Flat}}{{natural}}{{sharp}}
♭♮♯
<span class="bravura">&#xE260;</span>
<span class="bravura">&#xE261;</span>
<span class="bravura">&#xE262;</span>
<span class="heji">n</span>
== Template: Temperament data ==
{{User:FloraC/Temperament data
{{User:FloraC/Temperament data
| Subgroup = 2.3.5.7
| Subgroup = 2.3.5.7
Line 140: Line 152:
|}
|}


== Tunings to be reviewed ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+style=white-space:nowrap | 5-limit Prime-Optimized Tunings
|+style=white-space:nowrap | 5-limit Prime-Optimized Tunings
|-
|-
! Weight-skew\Order !! Chebyshevian !! Euclidean !! Manhattan
! rowspan="2" |
|-
! rowspan="2" | Manhattan
| Tenney || CTC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CTE<br>~3/2 = 697.2143¢ || CTOP<br>~3/2 = 698.0197¢
! colspan="2" | Euclidean
! rowspan="2" | Chebyshevian
|-
|-
| Tenney-Weil || CTWC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CTWE<br>~3/2 = 696.6512¢ || CTWOP<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning)
! Unskewed
! Skewed
|-
|-
| Equilateral || CEC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CEE<br>~3/2 = 696.8947¢<br>Eigenmonzo basis: 2.1875 (4/17-comma tuning) || CEOP<br>~3/2 = 697.6537¢<br>Eigenmonzo basis: 2.15 (1/5-comma tuning)
! 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)
|-
|-
| Equilateral-Weil || CEWC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CEWE<br>~3/2 = 696.4534¢<br>Eigenmonzo basis: 2.48828125/3 (11/43-comma tuning) || CEWOP<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning)
! 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)
|-
|-
| Benedetti || CBC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CBE<br>~3/2 = 697.3738¢<br>Eigenmonzo basis: 2.{{monzo| 0 25 36 }} (36/169-comma tuning) || CBOP<br>~3/2 = 698.1598¢<br>Eigenmonzo basis: 2.30375 (3/17-comma tuning)
! Benedetti, <br>Wilson
|-
| CBOP: ~3/2 = 698.160¢<br>(3/17-comma)
| Benedetti-Weil || CBWC<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning) || CBWE<br>~3/2 = 696.7868¢<br>Eigenmonzo basis: 2.{{monzo| 0 5 31 }} (31/129-comma tuning) || CBWOP<br>~3/2 = 696.5784¢<br>Eigenmonzo basis: 2.5 (1/4-comma tuning)
| CBE: ~3/2 = 697.374¢<br>(36/169-comma)
| CSBE: ~3/2 = 696.787¢<br>(31/129-comma)
| CBC: ~3/2 = 696.578¢<br>(1/4-comma)
|}
|}


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|+style=white-space:nowrap | 7-limit Prime-Optimized Tunings
|+style=white-space:nowrap | 7-limit Prime-Optimized Tunings
|-
|-
! Weight-skew\Order !! Chebyshevian !! Euclidean !! Manhattan
! rowspan="2" |
! rowspan="2" | Manhattan
! colspan="2" | Euclidean
! rowspan="2" | Chebyshevian
|-
|-
| Tenney || CTC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CTE<br>~3/2 = 696.9521¢ || CTOP<br>~3/2 = 696.6458¢
! Unskewed
! Skewed
|-
|-
| Tenney-Weil || CTWC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CTWE<br>~3/2 = 696.6562¢ || CTWOP<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7
! Equilateral
| CEOP: ~3/2 = 697.344¢
| CEE: ~3/2 = 696.884¢
| CSEE: ~3/2 = 696.725¢
| CEC: ~3/2 = 696.883¢
|-
|-
| Equilateral || CEC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CEE<br>~3/2 = 696.8843¢<br>Eigenmonzo basis: 2.{{monzo| 0 1 4 10 }} || CEOP<br>~3/2 = 697.3437¢<br>Eigenmonzo basis: 2.21
! Tenney
| CTOP: ~3/2 = 696.646¢
| CTE: ~3/2 = 696.952¢
| CWE: ~3/2 = 696.656¢
| CTC: ~3/2 = 696.883¢
|-
|-
| Equilateral-Weil || CEWC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CEWE<br>~3/2 = 696.7248¢<br>Eigenmonzo basis: 2.4117715/9 || CEWOP<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7
! Benedetti, <br>Wilson
|-
| CBOP: ~3/2 = 697.842¢
| Benedetti || CBC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CBE<br>~3/2 = 697.0147¢<br>Eigenmonzo basis: 2.{{monzo| 0 1225 1764 2250 }} || CBOP<br>~3/2 = 697.8422¢<br>Eigenmonzo basis: 2.750141
| CBE: ~3/2 = 697.015¢
|-
| CSBE: ~3/2 = 696.631¢
| Benedetti-Weil || CBWC<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7 || CBWE<br>~3/2 = 696.6306¢<br>Eigenmonzo basis: 2.{{monzo| 0 -3290 3171 7215 }} || CBWOP<br>~3/2 = 696.8826¢<br>Eigenmonzo basis: 2.7
| CBC: ~3/2 = 696.883¢
|}
|}


== Temperament pages ==
== Temperament pages ==
Note:  
Note:  
# Order: [[subgroup]], [[comma list]], ([[sval]]) [[mapping]], (sval) mapping generators, gencom mapping, [[gencom]], lattice basis, [[wedgie]], [[optimal tuning]] ([[CTE]]), [[minimax tuning]], [[tuning ranges]], [[algebraic generator]], [[Optimal ET sequence]], [[badness]], [[complexity spectrum]], and others.  
# 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 should show 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.  
# Since minimax tunings are based on tonality diamond, 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.
# For subgroup temperaments, "mapping" becomes "sval mapping", add "gencom mapping" and "gencom". If TE is TE is TE (''sic''), simply show "CTE", otherwise show "subgroup CTE" or "inharmonic CTE" instead.  
# 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.  


Get a family for:  
Get a family for:  
Line 232: Line 268:
* <s>Kleismic family</s>
* <s>Kleismic family</s>
* <s>Kleismic rank three family</s>
* <s>Kleismic rank three family</s>
* [[Hemimean clan]]
* <s>Würschmidt family</s>
* [[Hemimean family]]
* <s>Unicorn family</s>
* <s>Unicorn family</s>
* <s>Shibboleth family</s>
* <s>Shibboleth family</s>
Line 243: Line 278:
* <s>Sycamore family</s>
* <s>Sycamore family</s>
* <s>Semicomma family</s>
* <s>Semicomma family</s>
* [[Orwellismic temperaments]]
* <s>Orwellismic temperaments</s>
* <s>Orwellismic family</s>
* <s>Orwellismic family</s>
* [[Hemimage temperaments]]
* <s>Hemimean clan</s>
* [[Hemimage family]]
* <s>Hemimean family</s>
* [[Porwell temperaments]]
* <s>Hemifamity temperaments</s>
* <s>Hemifamity family</s>
* <s>Porwell temperaments</s>
* <s>Porwell family</s>
* <s>Porwell family</s>
* [[Hemifamity temperaments]]
* <s>Hemimage temperaments</s>
* [[Hemifamity family]]
* <s>Hemimage family</s>
* [[Porcupine family]]
* <s>Porcupine family</s>
* [[Porcupine rank three family]]
* <s>Porcupine rank three family</s>
* [[Tetracot family]]
* <s>Tetracot family</s>
* [[Diaschismic family]]
* <s>Diaschismic family</s>
* [[Diaschismic rank three family]]
* <s>Diaschismic rank three family</s>
* [[Compton family]]
* <s>Compton family</s>
* [[Amity family]]
* <s>Amity family</s>
* [[Misty family]]
* <s>Misty family</s>
* [[Undim family]]
* <s>Undim family</s>
* [[Lehmerismic temperaments]]
* <s>Lehmerismic temperaments</s>
* [[Kalismic temperaments]]
* <s>Kalismic temperaments</s>
* [[Ragismic microtemperaments]]
* <s>Ragismic microtemperaments</s>
* <s>Ragismic family</s>
* <s>Ragismic family</s>
* [[Landscape microtemperaments]]
* <s>Landscape microtemperaments</s>
* [[Landscape family]]
* <s>Landscape family</s>
* [[Dimcomp family]]
* <s>Dimcomp family</s>
* [[Mirkwai clan]]
* <s>Mirkwai clan</s>
* [[Mirkwai family]]
* <s>Mirkwai family</s>
* [[Breedsmic temperaments]]
* <s>Quince clan</s>
* [[Breed family]]
* <s>Breedsmic temperaments</s>
* [[Escapade family]]
* <s>Breed family</s>
* [[Gravity family]]
* <s>Escapade family</s>
* [[Cataharry temperaments]]
* <s>Gravity family</s>
* [[Cataharry family]]
* <s>Cataharry temperaments</s>
* [[Varunismic temperaments]]
* <s>Cataharry family</s>
* <s>Varunismic temperaments</s>
* <s>Rastmic rank three clan</s>
* <s>Rastmic rank three clan</s>
* <s>Biyatismic clan</s>
* <s>Biyatismic clan</s>
* [[Ptolemismic clan]]
* <s>Ptolemismic clan</s>
* <s>Valinorsmic clan</s>
* <s>Valinorsmic clan</s>
* <s>Pentacircle clan</s>
* <s>Pentacircle clan</s>
* [[Keenanismic temperaments]]
* <s>Keenanismic temperaments</s>
* [[Werckismic temperaments]]
* <s>Werckismic temperaments</s>
* [[Swetismic temperaments]]
* <s>Swetismic temperaments</s>
* [[Wizmic microtemperaments]]
* <s>Wizmic microtemperaments</s>
* [[Metric microtemperaments]]
* <s>Metric microtemperaments</s>
* <s>Horwell temperaments</s>
* <s>Horwell family</s>
* <s>Luna family</s>
* <s>Luna family</s>
* [[Vulture family]]
* <s>Vulture family</s>
* [[Tricot family]]
* <s>Tricot family</s>
* [[Minortonic family]]
* <s>Minortonic family</s>
* [[Gammic family]]
* <s>Gammic family</s>
* [[Garischismic clan]]
* <s>Vishnuzmic family</s>
* [[Canousmic temperaments]]
* <s>Garischismic clan</s>
* [[Canou family]]
* <s>Canousmic temperaments</s>
* [[Semicanousmic clan]]
* <s>Canou family</s>
* [[Semiporwellismic clan]]
* <s>Semicanousmic clan</s>
* [[Olympic clan]]
* <s>Semiporwellismic clan</s>
* [[Alphaxenic rank three clan]]
* <s>Olympic clan</s>
* [[Mirwomo temperaments]]
* <s>Alphaxenic rank three clan</s>
* [[Horwell temperaments]]
* <s>Mirwomo temperaments</s>
* [[Horwell family]]
* <s>Gariboh clan</s>
* [[Gariboh clan]]
* <s>Gariboh family</s>
* [[Gariboh family]]
* <s>Octagar temperaments</s>
* [[Octagar temperaments]]
* <s>Octagar family</s>
* [[Octagar family]]
* <s>Wesley family</s>
* <s>Apotome family</s>
* [[Mabila family]]
* [[Maquila family]]
* [[Maja family]]
* [[Ditonmic family]]
* <s>Greenwoodmic temperaments</s>
* <s>Avicennmic temperaments</s>


== Regular temperament properties ==
== Regular temperament properties ==
Line 367: Line 414:
; 12-tone
; 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]]
: [[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]]
(Bounded by branch depth = 7)
{| class="wikitable center-all"
! colspan="7" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
| 4\7 || || || || || || || 685.714 || 1 || 1 || 1.000 ||
|-
| || || || || || || 27\47 || 689.362 || 7 || 6 || 1.167 ||
|-
| || || || || || 23\40 || || 690.000 || 6 || 5 || 1.200 ||
|-
| || || || || || || 42\73 || 690.411 || 11 || 9 || 1.222 ||
|-
| || || || || 19\33 || || || 690.909 || 5 || 4 || 1.250 ||
|-
| || || || || || || 53\92 || 691.304 || 14 || 11 || 1.273 ||
|-
| || || || || || 34\59 || || 691.525 || 9 || 7 || 1.286 ||
|-
| || || || || || || 49\85 || 691.765 || 13 || 10 || 1.300 ||
|-
| || || || 15\26 || || || || 692.308 || 4 || 3 || 1.333 ||
|-
| || || || || || || 56\97 || 692.784 || 15 || 11 || 1.364 ||
|-
| || || || || || 41\71 || || 692.958 || 11 || 8 || 1.375 ||
|-
| || || || || || || 67\116 || 693.103 || 18 || 13 || 1.385 ||
|-
| || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 ||
|-
| || || || || || || 63\109 || 693.578 || 17 || 12 || 1.417 ||
|-
| || || || || || 37\64 || || 693.750 || 10 || 7 || 1.429 ||
|-
| || || || || || || 48\83 || 693.976 || 13 || 9 || 1.444 ||
|-
| || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
|-
| || || || || || || 51\88 || 695.455 || 14 || 9 || 1.556 ||
|-
| || || || || || 40\69 || || 695.652 || 11 || 7 || 1.571 ||
|-
| || || || || || || 69\119 || 695.798 || 19 || 12 || 1.583 ||
|-
| || || || || 29\50 || || || 696.000 || 8 || 5 || 1.600 ||
|-
| || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || Golden meantone
|-
| || || || || || 47\81 || || 696.296 || 13 || 8 || 1.625 ||
|-
| || || || || || || 65\112 || 696.429 || 18 || 11 || 1.636 ||
|-
| || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region
|-
| || || || || || || 61\105 || 697.143 || 17 || 10 || 1.700 ||
|-
| || || || || || 43\74 || || 697.297 || 12 || 7 || 1.714 ||
|-
| || || || || || || 68\117 || 697.436 || 19 || 11 || 1.727 ||
|-
| || || || || 25\43 || || || 697.674 || 7 || 4 || 1.750 ||
|-
| || || || || || || 57\98 || 697.959 || 16 || 9 || 1.778 ||
|-
| || || || || ||32\55 || || 698.182 || 9 || 5 || 1.800 ||
|-
| || || || || || || 39\67 || 698.507 || 11 || 6 || 1.833 ||
|-
| || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)
|-
| || || || || || || 38\65 || 701.539 || 11 || 5 || 2.200 ||
|-
| || || || || || 31\53 || || 701.887 || 9 || 4 || 2.250 ||
|-
| || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 ||
|-
| || || || || 24\41 || || || 702.409 || 7 || 3 || 2.333 ||
|-
| || || || || || || 65\111 || 702.703 || 19 || 8 || 2.375 ||
|-
| || || || || || 41\70 || || 702.857 || 12 || 5 || 2.400 ||
|-
| || || || || || || 58\99 || 703.030 || 17 || 7 || 2.428 ||
|-
| || || || 17\29 || || || || 703.448 || 5 || 2 || 2.500 ||
|-
| || || || || || || 61\104 || 703.846 || 18 || 7 || 2.571 ||
|-
| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
|-
| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden neogothic
|-
| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 ||
|-
| || || || || || || 64\109 || 704.587 || 19 || 7 || 2.714 ||
|-
| || || || || || 37\63 || || 704.762 || 11 || 4 || 2.750 ||
|-
| || || || || || || 47\80 || 705.000 || 14 || 5 || 2.800 ||
|-
| || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
|-
| || || || || || || 43\73 || 706.849 || 13 || 4 || 3.250 ||
|-
| || || || || || 33\56 || || 707.143 || 10 || 3 || 3.333 ||
|-
| || || || || || || 56\95 || 707.368 || 17 || 5 || 3.400 ||
|-
| || || || || 23\39 || || || 707.692 || 7 || 2 || 3.500 ||
|-
| || || || || || || 59\100 || 708.000 || 18 || 5 || 3.600 ||
|-
| || || || || || 36\61 || || 708.197 || 11 || 3 || 3.667 ||
|-
| || || || || || || 49\83 || 708.434 || 15 || 4 || 3.750 ||
|-
| || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region
|-
| || || || || || || 42\71 || 709.859 || 13 || 3 || 4.333 ||
|-
| || || || || || 29\49 || || 710.204 || 9 || 2 || 4.500 ||
|-
| || || || || || || 45\76 || 710.526 || 14 || 3 || 4.667 ||
|-
| || || || || 16\27 || || || 711.111 || 5 || 1 || 5.000 ||
|-
| || || || || || || 35\59 || 711.864 || 11 || 2 || 5.500 ||
|-
| || || || || || 19\32 || || 712.500 || 6 || 1 || 6.000 ||
|-
| || || || || || || 22\37 || 713.514 || 7 || 1 || 7.000 ||
|-
| 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
|}

Latest revision as of 19:05, 15 January 2025

♭♮♯

n

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

5-limit rank-2 temperaments by TE simple badness

Breed's 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
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
Srutal 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 Prime-Optimized Tunings
Manhattan Euclidean Chebyshevian
Unskewed 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 Prime-Optimized Tunings
Manhattan Euclidean Chebyshevian
Unskewed 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¢

Temperament pages

Note:

  1. Order: subgroup, comma list, (sval) mapping, (sval) mapping generators, gencom mapping, gencom, lattice basis, wedgie, optimal tunings (CTE, CWE/POTE), minimax tuning, tuning ranges, algebraic generator, optimal ET sequence, badness, complexity spectrum, and others.
  2. Comma list should show the simplest commas sufficient to define the temperament, stated in Normal lists #Normal interval list.
  3. Mapping generators should show all the ratios as used in the mapping, including the period.
  4. Since minimax tunings are based on tonality diamond, it should explicitly state the odd limit, or a diamond function of ratios.
  5. 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.

Get a family for:

  • Ripple (3 different 7-limit extensions) done
  • Smate (2 different 7-limit extensions) done
  • Passion (4 different 7-limit extensions, 3 strong and 1 weak) done
  • Undim done
  • Quintaleap done
  • Quindromeda done
  • Parakleismic (many reasonable but unnamed 7-limit extensions)
  • Schismatic rank three family (perhaps)

Who's next?

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

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-49 31⟩ [⟨31 49]] +1.63 1.64 4.22
2.3.5 81/80, 393216/390625 [⟨31 49 72]] +0.98 1.63 4.20
2.3.5.7 81/80, 126/125, 1029/1024 [⟨31 49 72 87]] +0.83 1.43 3.70
2.3.5.7.11 81/80, 99/98, 121/120, 126/125 [⟨31 49 72 87 107]] +1.21 1.49 3.84

Scale tree

6-tone
1L 5s, 2L 4s, 3L 3s, 4L 2s, 5L 1s
7-tone
1L 6s, 2L 5s, 3L 4s, 4L 3s, 5L 2s, 6L 1s
8-tone
1L 7s, 2L 6s, 3L 5s, 4L 4s, 5L 3s, 6L 2s, 7L 1s
9-tone
1L 8s, 2L 7s, 3L 6s, 4L 5s, 5L 4s, 6L 3s, 7L 2s, 8L 1s
10-tone
1L 9s, 2L 8s, 3L 7s, 4L 6s, 5L 5s, 6L 4s, 7L 3s, 8L 2s, 9L 1s
11-tone
1L 10s, 2L 9s, 3L 8s, 4L 7s, 5L 6s, 6L 5s, 7L 4s, 8L 3s, 9L 2s, 10L 1s
12-tone
1L 11s, 2L 10s, 3L 9s, 4L 8s, 5L 7s, 6L 6s, 7L 5s, 8L 4s, 9L 3s, 10L 2s, 11L 1s
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