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== Septimal meantone ==
Period: 1\1


Optimal ([[POTE]]) generator: ~3/2 = 696.495
{{Flat}}{{natural}}{{sharp}}


EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
♭♮♯


Scales: [[Meantone5]], [[Meantone7]], [[Meantone12]]
<span class="bravura">&#xE260;</span>
<span class="bravura">&#xE261;</span>
<span class="bravura">&#xE262;</span>


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<span class="heji">n</span>
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
 
<div class="mw-collapsible-content">
== Template: Temperament data ==
{| class="wikitable right-1 right-2 sortable"
{{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"
|-
|-
! #Gens up
! Temperament !! Complexity !! Error (¢) !! Badness (moct) !! Mapping !! Comma list
! Cents <ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>
|-
|-
| 0
| Yo || .367 || 26.0 || 7.95 || {{mapping| 1 0 -1 | 0 1 -2 }} || 10/9
| 0.00
| 1/1
|-
|-
| 1
| Antitonic || .395 || 27.0 || 8.89 || {{mapping| 2 3 0 | 0 0 1 }} || 9/8
| 696.5
| 3/2
|-
|-
| 2
| Father || .443 || 13.2 || 4.87 || {{mapping| 1 0 4 | 0 1 -1 }} || 16/15
| 193.0
| 9/8, 10/9
|-
|-
| 3
| Bug || .602 || 11.6 || 5.81 || {{mapping| 1 0 0 | 0 2 3 }} || 27/25
| 889.5
| 5/3
|-
|-
| 4
| Supersharp || 1.06 || 7.94 || 7.01 || {{mapping| 2 0 -5 | 0 1 3 }} || 800/729
| 386.0
| 5/4
|-
|-
| 5
| Laconic || 1.24 || 6.52 || 6.74 || {{mapping| 1 1 1 | 0 3 7 }} || 2187/2000
| 1082.5
| 15/8, 28/15
|-
|-
| 6
| Lafayette || 1.30 || 7.00 || 7.60 || {{mapping| 1 1 2 | 0 5 3 }} || 3456/3125
| 579.0
| 7/5
|-
|-
| 7
| Symbolic || 1.31 || 6.09 || 6.66 || {{mapping| 1 3 4 | 0 -5 -6 }} || 2048/1875
| 75.5
| 21/20, 25/24, 28/27
|-
|-
| 8
| Sixix || 1.37 || 4.57 || 5.22 || {{mapping| 1 3 2 | 0 -4 1 }} || 3125/2916
| 772.0
| 14/9, 25/16
|-
|-
| 9
| Uncle || 1.40 || 7.53 || 8.80 || {{mapping| 1 0 12 | 0 1 -6 }} || 4096/3645
| 268.5
| 7/6
|-
|-
| 10
| Whitewood || 1.42 || 4.20 || 4.95 || {{mapping| 7 11 0 | 0 0 1 }} || 2187/2048
| 965.0
| 7/4
|-
|-
| 11
| || 1.66 || 4.13 || 5.72 || {{mapping| 1 0 5 | 0 3 -5 }} || 32768/30375
| 461.4
| 21/16
|}
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
[[Comma]]s: 81/80, 126/125
7 and [[9-odd-limit|9-limit]] minimax
[{{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 of Regular Temperaments|Valid range]]: [694.737, 700.000] (19 to 12)


Nice range: [694.786, 701.955]
=== Main sequence ===
 
{| class="wikitable sortable center-2 center-3 center-4"
Strict range: [694.786, 700.000]
 
Mapping generator: ~3
 
Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
 
[[Wedgie]]: &lt;&lt;1 4 10 4 13 12||
 
[[Val]]s: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb
 
[[Badness]]: 0.0137
</div></div>
 
== 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)
! Comment(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
| Srutal || 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 Prime-Optimized Tunings
|-
|-
| 13
! rowspan="2" |  
| [[352/351]]
! rowspan="2" | Manhattan
| Minthma
! colspan="2" | Euclidean
|  
! rowspan="2" | Chebyshevian
|-
|-
| 13
! Unskewed
| [[364/363]]
! Skewed
| Gentle comma
|
|-
|-
| 13
! Equilateral
| [[847/845]]
| CEOP: ~3/2 = 697.654¢<br>(1/5-comma)
| Cuthbert
| 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)
|-
|-
| 13
! Tenney
| [[729/728]]
| CTOP: ~3/2 = 698.020¢
| Squbema
| CTE: ~3/2 = 697.214¢
|  
| CWE: ~3/2 = 696.651¢
| CTC: ~3/2 = 696.578¢<br>(1/4-comma)
|-
|-
| 13
! Benedetti, <br>Wilson
| [[4096/4095]]
| CBOP: ~3/2 = 698.160¢<br>(3/17-comma)
| Schismina
| 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)
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+style=white-space:nowrap | 7-limit Prime-Optimized Tunings
|-
|-
| 13
! rowspan="2" |  
| <abbr title="10648/10647">{{Monzo| 3 -2 0 -1 3 -2 }}</abbr>
! rowspan="2" | Manhattan
| [[Harmonisma]]
! colspan="2" | Euclidean
|  
! rowspan="2" | Chebyshevian
|-
|-
| 17
! Unskewed
| [[2187/2176]]
! Skewed
| Septendecimal schisma
|
|-
|-
| 17
! Equilateral
| [[256/255]]
| CEOP: ~3/2 = 697.344¢
| Septendecimal kleisma
| CEE: ~3/2 = 696.884¢
|  
| CSEE: ~3/2 = 696.725¢
| CEC: ~3/2 = 696.883¢
|-
|-
| 17
! Tenney
| [[715/714]]
| CTOP: ~3/2 = 696.646¢
| Septendecimal bridge comma
| CTE: ~3/2 = 696.952¢
|  
| CWE: ~3/2 = 696.656¢
| CTC: ~3/2 = 696.883¢
|-
|-
| 19
! Benedetti, <br>Wilson
| [[210/209]]
| CBOP: ~3/2 = 697.842¢
| Spleen comma
| CBE: ~3/2 = 697.015¢
|  
| CSBE: ~3/2 = 696.631¢
| CBC: ~3/2 = 696.883¢
|}
 
== Temperament pages ==
Note:
# 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]].
# 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.
# 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:
* <s>Ripple (3 different 7-limit extensions)</s> done
* <s>Smate (2 different 7-limit extensions)</s> done
* <s>Passion (4 different 7-limit extensions, 3 strong and 1 weak)</s> done
* <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?
* <s>Meantone family</s>
* <s>Didymus rank three family</s>
* <s>Archytas clan</s>
* <s>Archytas family</s>
* <s>Father family</s>
* <s>Trienstonic clan</s>
* <s>Septisemi temperaments</s>
* <s>Slendro clan</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 family</s>
* <s>Gamelismic clan</s>
* <s>Gamelismic family</s>
* <s>Starling temperaments</s>
* <s>Starling family</s>
* <s>Sensamagic clan</s>
* <s>Sensamagic family</s>
* <s>Magic family</s>
* <s>Sensipent family</s>
* <s>Keemic temperaments</s>
* <s>Keemic family</s>
* <s>Sengic 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 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>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>Canou family</s>
* <s>Semicanousmic clan</s>
* <s>Semiporwellismic clan</s>
* <s>Olympic clan</s>
* <s>Alphaxenic rank three clan</s>
* <s>Mirwomo temperaments</s>
* <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>
 
== Regular temperament properties ==
{| 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
|-
|-
| 19
! [[TE error|Absolute]] (¢)
| [[361/360]]
! [[TE simple badness|Relative]] (%)
| Go comma
|
|-
|-
| 19
| 2.3
| [[513/512]]
| {{monzo| -49 31 }}
| Undevicesimal comma
| {{mapping| 31 49 }}
|  
| +1.63
| 1.64
| 4.22
|-
|-
| 19
| 2.3.5
| [[1216/1215]]
| 81/80, 393216/390625
| Eratosthenes' comma
| {{mapping| 31 49 72 }}
|  
| +0.98
| 1.63
| 4.20
|-
|-
| 23
| 2.3.5.7
| [[736/729]]
| 81/80, 126/125, 1029/1024
| Vicesimotertial comma
| {{mapping| 31 49 72 87 }}
|  
| +0.83
| 1.43
| 3.70
|-
|-
| 29
| 2.3.5.7.11
| [[145/144]]
| 81/80, 99/98, 121/120, 126/125
| 29th-partial chroma
| {{mapping| 31 49 72 87 107 }}
|  
| +1.21
| 1.49
| 3.84
|}
|}
<references/>
 
== Scale tree ==
; 6-tone
: <s>1L 5s</s>, [[2L 4s]], <s>3L 3s</s>, [[4L 2s]], <s>5L 1s</s>
 
; 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>
 
; 8-tone
: [[1L 7s]], <s>2L 6s</s>, [[3L 5s]], <s>4L 4s</s>, <s>5L 3s</s>, <s>6L 2s</s>, [[7L 1s]]
 
; 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]]
 
; 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]]
 
; 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]]
 
; 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]]

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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