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{{Technical data page}}
{{Technical data page}}
This page discusses miscellaneous [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[breedsma]], {{monzo| -5 -1 -2 4 }} = 2401/2400. This is the amount by which two [[49/40]] intervals exceed [[3/2]], and by which two [[60/49]] intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system ([[12edo]], for example) which does not possess a neutral third cannot be tempering out the breedsma.
This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[breedsma]] ({{monzo|legend=1| -5 -1 -2 4 }}, [[ratio]]: 2401/2400). This is the amount by which two [[49/40]] intervals exceed [[3/2]], and by which two [[60/49]] intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system ([[12edo]], for example) which does not possess a neutral third cannot be tempering out the breedsma.


The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: (10000/2401)⋅(2401/2400) = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)⋅(10/7) = 7/4 and (49/40)⋅(10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Decimal]]'' (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* ''[[Beatles]]'' (+64/63) → [[Archytas clan #Beatles|Archytas clan]]
* ''[[Beatles]]'' (+64/63 or 686/675) → [[Archytas clan #Beatles|Archytas clan]]
* ''[[Newt]]'' (+33554432/33480783) → [[Garischismic clan #Beatles|Garischismic clan]]
* [[Decimal]] (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* [[Ennealimmal]] (+4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* [[Ennealimmal]] (+4375/4374) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* ''[[Catafourth]]'' (+78732/78125) → [[Sensipent family #Catafourth|Sensipent family]]
* ''[[Neptune (temperament)|Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* ''[[Maviloid]]'' (+1224440064/1220703125) → [[Parakleismic family #Maviloid|Parakleismic family]]
 
* ''[[Subneutral]]'' (+274877906944/274658203125) → [[Luna family #Subneutral|Luna family]]
== Hemififths ==
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]
{{Main| Hemififths }}
* ''[[Hemicountercomp]]'' (+52613349376/52301766015) → [[Countercomp family #Hemicountercomp|Countercomp family]]
 
Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.
 
By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 5120/5103
 
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
 
: mapping generators: ~2, ~49/40
 
{{Multival|legend=1| 2 25 13 35 15 -40 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
: [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }}
* [[POTE]]: ~2 = 1200.0000, ~49/40 = 351.4774
: error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
 
[[Algebraic generator]]: (2 + sqrt(2))/2
 
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }}
 
[[Badness]] (Smith): 0.022243
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 896/891
 
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
 
Badness (Smith): 0.023498
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 243/242, 364/363
 
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
 
Badness (Smith): 0.019090
 
=== Semihemi ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3388/3375, 5120/5103
 
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
 
: mapping generators: ~99/70, ~400/231
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047
 
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
 
Badness (Smith): 0.042487
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 676/675, 847/845, 1716/1715
 
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019
 
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
 
Badness (Smith): 0.021188
 
=== Quadrafifths ===
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5120/5103
 
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
 
: Mapping generators: ~2, ~243/220


Optimal tunings:
Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, lockerbie, unthirds, neominor, cotritone, fibo, quasimoha, mintone, gorgik, retrotonic, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
 
Badness (Smith): 0.040170
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
 
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}
 
Badness (Smith): 0.031144


== Tertiaseptal ==
== Tertiaseptal ==
{{Main| Tertiaseptal }}
{{Main| Tertiaseptal }}


Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31 &amp; 171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] is especially notable. The 15- or 16-note mos can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the {{nowrap| 31 & 171 }} temperament, and [[256/245]], [[1029/1024]] less than [[21/20]], serves as its generator. Three of these fall short of [[8/7]] by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure).
 
[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and in very high limits [[311edo]] ({{nowrap| {{=}} 140 + 171 }}) is especially notable. The 15- or 16-note [[mos]] can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 165: Line 44:
[[Comma list]]: 2401/2400, 65625/65536
[[Comma list]]: 2401/2400, 65625/65536


{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
{{Mapping|legend=1| 1 -19 7 0 | 0 22 -5 3 }}
: mapping generators: ~2, ~245/128


: Mapping generators: ~2, ~256/245
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.1004{{c}}, ~245/128 = 1122.9024{{c}} (~256/245 = 77.1979{{c}})
{{Multival|legend=1| 22 -5 3 -59 -57 21 }}
: [[error map]]: {{val| +0.100 -0.008 -0.123 -0.119 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8101{{c}} (~256/245 = 77.1899{{c}})
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
: error map: {{val| 0.000 -0.133 -0.364 -0.396 }}


{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}


[[Badness]]: 0.012995
[[Badness]] (Sintel): 0.329


=== 11-limit ===
=== 11-limit ===
Line 182: Line 62:
Comma list: 243/242, 441/440, 65625/65536
Comma list: 243/242, 441/440, 65625/65536


Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}
{{Mapping|legend=0| 1 -19 7 0 -48 | 0 22 -5 3 55 }}


Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
Optimal tunings:
* WE: ~2 = 1200.1034{{c}}, ~245/128 = 1122.8694{{c}} (~256/245 = 77.2340{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.7743{{c}} (~256/245 = 77.2257{{c}})


Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}


Badness: 0.035576
Badness (Sintel): 1.18


==== 13-limit ====
==== 13-limit ====
Line 195: Line 77:
Comma list: 243/242, 441/440, 625/624, 3584/3575
Comma list: 243/242, 441/440, 625/624, 3584/3575


Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}
{{Mapping|legend=0| 1 -19 7 0 -48 43 | 0 22 -5 3 55 -42 }}


Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
Optimal tunings:
* WE: ~2 = 1199.8783{{c}}, ~224/117 = 1122.6835{{c}} (~117/112 = 77.1948{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.7968{{c}} (~117/112 = 77.2032{{c}})


Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}
{{Optimal ET sequence|legend=0| 31, 140e, 171, 373ef }}


Badness: 0.036876
Badness (Sintel): 1.52


==== 17-limit ====
==== 17-limit ====
Line 208: Line 92:
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575


Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}
{{Mapping|legend=0| 1 -19 7 0 -48 43 49 | 0 22 -5 3 55 -42 -48 }}


Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201
Optimal tunings:
* WE: ~2 = 1199.8677{{c}}, ~65/34 = 1122.6748{{c}} (~68/65 = 77.1929{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.7985{{c}} (~68/65 = 77.2015{{c}})


Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}
{{Optimal ET sequence|legend=0| 31, 140e, 171 }}


Badness: 0.027398
Badness (Sintel): 1.40


=== Tertia ===
=== Tertia ===
Line 221: Line 107:
Comma list: 385/384, 1331/1323, 1375/1372
Comma list: 385/384, 1331/1323, 1375/1372


Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}
{{Mapping|legend=0| 1 -19 7 0 -19 | 0 22 -5 3 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173
Optimal tunings:
* WE: ~2 = 1200.2336{{c}}, ~21/11 = 1123.0454{{c}} (~22/21 = 77.1882{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8311{{c}} (~22/21 = 77.1689{{c}})


Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}
{{Optimal ET sequence|legend=0| 31, 109, 140, 171e, 311e }}


Badness: 0.030171
Badness (Sintel): 0.997


==== 13-limit ====
==== 13-limit ====
Line 234: Line 122:
Comma list: 352/351, 385/384, 625/624, 1331/1323
Comma list: 352/351, 385/384, 625/624, 1331/1323


Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}
{{Mapping|legend=0| 1 -19 7 0 -19 43 | 0 22 -5 3 24 -42 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158
Optimal tunings:
* WE: ~2 = 1200.1395{{c}}, ~21/11 = 1122.9727{{c}} (~22/21 = 77.1669{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8426{{c}} (~22/21 = 77.1574{{c}})


Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}
{{Optimal ET sequence|legend=0| 31, 78f, 109, 140 }}


Badness: 0.028384
Badness (Sintel): 1.17


==== 17-limit ====
==== 17-limit ====
Line 247: Line 137:
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714


Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}
{{Mapping|legend=0| 1 -19 7 0 -19 43 49 | 0 22 -5 3 24 -42 -48 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162
Optimal tunings:
* WE: ~2 = 1200.1655{{c}}, ~21/11 = 1122.9926{{c}} (~22/21 = 77.1729{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8376{{c}} (~22/21 = 77.1624{{c}})


Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}
{{Optimal ET sequence|legend=0| 31, 78fg, 109g, 140 }}


Badness: 0.022416
Badness (Sintel): 1.14


=== Tertiaseptia ===
=== Tertiaseptia ===
This extension was considered by [[Gene Ward Smith]] as a 41-limit temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9274.html Yahoo! Tuning Group | ''A 41-limit temperament'']</ref>. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 6250/6237, 65625/65536
Comma list: 2401/2400, 6250/6237, 65625/65536


Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}
{{Mapping|legend=0| 1 -19 7 0 112 | 0 22 -5 3 -116 }}


Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169
Optimal tunings:
* WE: ~2 = 1200.0053{{c}}, ~245/128 = 1122.8357{{c}} (~256/245 = 77.1696{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8308{{c}} (~256/245 = 77.1692{{c}})


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}


Badness: 0.056926
Badness (Sintel): 1.88


==== 13-limit ====
==== 13-limit ====
Line 273: Line 169:
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400


Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}
{{Mapping|legend=0| 1 -19 7 0 112 43 | 0 22 -5 3 -116 -42 }}


Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168
Optimal tunings:
* WE: ~2 = 1199.9823{{c}}, ~224/117 = 1122.8150{{c}} (~117/112 = 77.1673{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8316{{c}} (~117/112 = 77.1684{{c}})


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311, 1073 }}


Badness: 0.027474
Badness (Sintel): 1.14


==== 17-limit ====
==== 17-limit ====
Line 286: Line 184:
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197


Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}
{{Mapping|legend=0| 1 -19 7 0 112 43 49 | 0 22 -5 3 -116 -42 -48 }}


Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
Optimal tunings:
* WE: ~2 = 1200.0092{{c}}, ~65/34 = 1122.8392{{c}} (~68/65 = 77.1700{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8305{{c}} (~68/65 = 77.1695{{c}})


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}


Badness: 0.018773
Badness (Sintel): 0.956


==== 19-limit ====
==== 2.3.5.7.11.13.17.23 subgroup ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.23


Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197
Comma list: 595/594, 625/624, 833/832, 1105/1104, 1156/1155, 2200/2197


Mapping: {{mapping| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}
{{Mapping|legend=0| 1 -19 7 0 112 43 49 114 | 0 22 -5 3 -116 -42 -48 -117 }}


Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
Optimal tunings:
* WE: ~2 = 1200.0047{{c}}, ~44/23 = 1122.8363{{c}} (~23/22 = 77.1684{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8319{{c}} (~23/22 = 77.1681{{c}})


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
{{Optimal ET sequence|legend=0| 31ei, 140, 171, 311 }}


Badness: 0.017653
Badness (Sintel): 0.944


==== 23-limit ====
==== 2.3.5.7.11.13.17.23.29 subgroup ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.23.29


Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215
Comma list: 595/594, 625/624, 784/783, 833/832, 1015/1014, 1105/1104, 1156/1155


Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}
{{Mapping|legend=0| 1 -19 7 0 112 43 49 114 61 | 0 22 -5 3 -116 -42 -48 -117 -60 }}


Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168
Optimal tunings:
 
* WE: ~2 = 1199.9945{{c}}, ~44/23 = 1122.8270{{c}} (~23/22 = 77.1675{{c}})
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8322{{c}} (~23/22 = 77.1678{{c}})
 
Badness: 0.015123
 
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167
 
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}
 
Badness: 0.012181
 
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
 
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.012311
 
==== 37-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
 
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.010949
 
==== 41-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
 
Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 6 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169


Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 31ei, 140, 311, 762g }}


Badness: 0.009825
Badness (Sintel): 0.858


=== Hemitert ===
=== Hemitert ===
Line 377: Line 229:
Comma list: 2401/2400, 3025/3024, 65625/65536
Comma list: 2401/2400, 3025/3024, 65625/65536


Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}
{{Mapping|legend=0| 1 -41 12 -3 -73 | 0 44 -10 6 79 }}
: mapping generators: ~2, ~88/45


: Mapping generators: ~2, ~45/44
Optimal tunings:  
 
* WE: ~2 = 1200.1008{{c}}, ~88/45 = 1161.5020{{c}} (~45/44 = 38.5988{{c}})
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4053{{c}} (~45/44 = 38.5947{{c}})


Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}
{{Optimal ET sequence|legend=0| 31, …, 280, 311, 342, 2021cde, 2363cde, …, 3389ccddee, 3731ccddee }}


Badness: 0.015633
Badness (Sintel): 0.517


==== 13-limit ====
==== 13-limit ====
Line 392: Line 245:
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095


Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}
{{Mapping|legend=0| 1 -41 12 -3 -73 85 | 0 44 -10 6 79 -84 }}


Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588
Optimal tunings:
* WE: ~2 = 1199.9822{{c}}, ~88/45 = 1161.3952{{c}} (~45/44 = 38.5871{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4123{{c}} (~45/44 = 38.5877{{c}})


Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }}
{{Optimal ET sequence|legend=0| 31, 280, 311 }}


Badness: 0.033573
Badness (Sintel): 1.39


==== 17-limit ====
==== 17-limit ====
Line 405: Line 260:
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095


Mapping: {{mapping| 1 3 2 3 6 1 1 | 0 -44 10 -6 -79 84 96 }}
{{Mapping|legend=0| 1 -41 12 -3 -73 85 97| 0 44 -10 6 79 -84 -96 }}


Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589
Optimal tunings:
* WE: ~2 = 1200.0042{{c}}, ~88/45 = 1161.4149{{c}} (~45/44 = 38.5893{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4109{{c}} (~45/44 = 38.5891{{c}})


Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }}
{{Optimal ET sequence|legend=0| 31, 280, 311, 653f }}


Badness: 0.025298
Badness (Sintel): 1.29


=== Semitert ===
=== Semitert ===
Line 418: Line 275:
Comma list: 2401/2400, 9801/9800, 65625/65536
Comma list: 2401/2400, 9801/9800, 65625/65536


Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}
{{Mapping|legend=0| 2 -16 9 3 47 | 0 22 -5 3 -46 }}
: mapping generators: ~99/70, ~693/512


: Mapping generators: ~99/70, ~256/245
Optimal tunings:  
* WE: ~99/70 = 600.0548{{c}}, ~693/512 = 522.8547{{c}} (~256/245 = 77.2002{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~693/512 = 522.8069{{c}} (~256/245 = 77.1931{{c}})


Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
{{Optimal ET sequence|legend=0| 62e, 140, 202, 342 }}


Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}
Badness (Sintel): 0.853


Badness: 0.025790
== Emmthird ==
Emmthird tempers out the [[scheme comma]] and may be described as the {{nowrap| 58 & 171 }} temperament. The generator for emmthird is flatter than [[81/64]] by a lee comma, [[177147/175616]], and sharper than [[5/4]] by the hemimage comma, [[10976/10935]]. The [[ploidacot]] for this temperament is delta-14-cot.  


== Quasiorwell ==
The [[11-limit]] version, which tempers out [[243/242]] and [[441/440]], has much lower accuracy and is [[support]]ed by much fewer equal temperaments.  
In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 22 -1 -10 1 }}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 &amp; 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.
 
Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 29360128/29296875
[[Comma list]]: 2401/2400, 14348907/14336000


{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}
{{Mapping|legend=1| 1 -3 -17 -8 | 0 14 59 33 }}
: mapping generators: ~2, ~2744/2187


: Mapping generators: ~2, ~875/512
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0435{{c}}, ~2744/2187 = 393.0021{{c}}
: [[error map]]: {{val| +0.043 -0.057 +0.069 -0.106 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2744/2187 = 392.9887{{c}}
: error map: {{val| 0.000 -0.113 +0.022 -0.197 }}


{{Multival|legend=1| 38 -3 8 -93 -94 27 }}
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107


{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
[[Badness]] (Sintel): 0.424
 
[[Badness]]: 0.035832


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 3025/3024, 5632/5625
Comma list: 243/242, 441/440, 1792000/1771561


Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}
{{Mapping|legend=0| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}


Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111
Optimal tunings:
* WE: ~2 = 1199.8090{{c}}, ~1372/1089 = 392.9286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1372/1089 = 392.9870{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


Badness: 0.017540
Badness (Sintel): 1.73


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
Comma list: 243/242, 364/363, 441/440, 2200/2197


Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}
{{Mapping|legend=0| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}


Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107
Optimal tunings:
* WE: ~2 = 1199.7756{{c}}, ~180/143 = 392.9154{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~180/143 = 392.9840{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


Badness: 0.017921
Badness (Sintel): 1.11


== Neominor ==
=== 17-limit ===
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.
Subgroup: 2.3.5.7.11.13.17


[[Subgroup]]: 2.3.5.7
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197


[[Comma list]]: 2401/2400, 177147/175616
{{Mapping|legend=0| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}


{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}
Optimal tunings:
* WE: ~2 = 1199.8396{{c}}, ~64/51 = 392.9322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/51 = 392.9826{{c}}


: Mapping generators: ~2, ~189/160
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


{{Multival|legend=1| 6 41 22 51 18 -64 }}
Badness (Sintel): 1.18


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280
== Hemififths ==
{{Main| Hemififths }}


{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.


[[Badness]]: 0.088221
By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.


=== 11-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 35937/35840
[[Comma list]]: 2401/2400, 5120/5103


Mapping: {{mapping| 1 3 12 8 7 | 0 -6 -41 -22 -15 }}
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
: mapping generators: ~2, ~49/40


Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7412{{c}}, ~49/40 = 351.4016{{c}}
: [[error map]]: {{val| -0.259 +0.590 +0.021 -0.346 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4671{{c}}
: error map: {{val| 0.000 +0.979 +0.364 +0.246 }}


Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


Badness: 0.027959
[[Algebraic generator]]: (2 + sqrt(2))/2
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 243/242, 364/363, 441/440
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338 }}


Mapping: {{mapping| 1 3 12 8 7 7 | 0 -6 -41 -22 -15 -14 }}
[[Badness]] (Sintel): 0.563


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }}
Comma list: 243/242, 441/440, 896/891


Badness: 0.026942
{{Mapping|legend=0| 1 1 -5 -1 2 | 0 2 25 13 5 }}


== Emmthird ==
Optimal tunings:
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
* WE: ~2 = 1199.2845{{c}}, ~11/9 = 351.3110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4956{{c}}


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}


[[Comma list]]: 2401/2400, 14348907/14336000
Badness (Sintel): 0.777


{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


: Mapping generators: ~2, ~2187/1372
Comma list: 144/143, 196/195, 243/242, 364/363


{{Multival|legend=1| 14 59 33 61 13 -8 9 }}
{{Mapping|legend=0| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
Optimal tunings:
* WE: ~2 = 1198.8875{{c}}, ~11/9 = 351.2475{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.5438{{c}}


{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}


[[Badness]]: 0.016736
Badness (Sintel): 0.789


=== 11-limit ===
=== Semihemi ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 1792000/1771561
Comma list: 2401/2400, 3388/3375, 5120/5103


Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}
{{Mapping|legend=0| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
: mapping generators: ~99/70, ~400/231


Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991
Optimal tunings:
* WE: ~99/70 = 599.8556{{c}}, ~400/231 = 951.2757{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~400/231 = 951.4939{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
{{Optimal ET sequence|legend=0| 58, 140, 198 }}


Badness: 0.052358
Badness (Sintel): 1.40


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 364/363, 441/440, 2200/2197
Comma list: 352/351, 676/675, 847/845, 1716/1715


Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }}
{{Mapping|legend=0| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989
Optimal tunings:
* WE: ~99/70 = 599.8513{{c}}, ~26/15 = 951.2662{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/15 = 951.4905{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}


Badness: 0.026974
Badness (Sintel): 0.876


=== 17-limit ===
=== Quadrafifths ===
Subgroup: 2.3.5.7.11.13.17
This has been catalogued as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.  


Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}
Comma list: 2401/2400, 3025/3024, 5120/5103


Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985
{{Mapping|legend=0| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
: mapping generators: ~2, ~243/220


Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
Optimal tunings:  
* WE: ~2 = 1199.7520{{c}}, ~243/220 = 175.7015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7360{{c}}


Badness: 0.023205
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}


== Quinmite ==
Badness (Sintel): 1.33
The generator for quinmite is quasi-tempered minor third [[25/21]], flatter than 6/5 by the starling comma, [[126/125]]. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.


[[Subgroup]]: 2.3.5.7
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 2401/2400, 1959552/1953125
Comma list: 352/351, 847/845, 2401/2400, 3025/3024


{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}
{{Mapping|legend=0| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}


: Mapping generators: ~2, ~42/25
Optimal tunings:  
* WE: ~2 = 1199.6502{{c}}, ~72/65 = 175.6957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7461{{c}}


{{Multival|legend=1| 34 29 23 -33 -59 -28 }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
Badness (Sintel): 1.29


{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
=== Cutefourths ===
This extension splits the neutral third plus an octave in three, with a ploidacot signature of beta-hexacot. The generator is an acute fourth in size (but not representing [[27/20]]), hence the name.


[[Badness]]: 0.037322
Subgroup: 2.3.5.7.11


== Unthirds ==
Comma list: 2401/2400, 4000/3993, 5120/5103
The generator for unthirds temperament is undecimal major third, 14/11.


[[Subgroup]]: 2.3.5.7
{{Mapping|legend=0| 1 -1 -30 -14 -28 | 0 6 75 39 73 }}
: mapping generators: ~2, ~66/49


[[Comma list]]: 2401/2400, 68359375/68024448
Optimal tunings:  
* WE: ~2 = 1199.7345{{c}}, ~66/49 = 517.0436{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1543{{c}}


{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}
{{Optimal ET sequence|legend=0| 58, 181, 239, 1014bcee }}


: Mapping generators: ~2, ~6125/3888
Badness (Sintel): 1.71


{{Multival|legend=1| 42 47 34 -23 -64 -53 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
Comma list: 352/351, 847/845, 1575/1573, 2401/2400


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
{{Mapping|legend=0| 1 -1 -30 -14 -28 -20 | 0 6 75 39 73 55 }}
 
Optimal tunings:
* WE: ~2 = 1199.6427{{c}}, ~66/49 = 517.0035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1524{{c}}


[[Badness]]: 0.075253
{{Optimal ET sequence|legend=0| 58, 181, 239f }}


=== 11-limit ===
Badness (Sintel): 1.45
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 3025/3024, 4000/3993
== Osiris ==
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}
[[Comma list]]: 2401/2400, 31381059609/31360000000


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718
{{Mapping|legend=1| 1 13 33 21 | 0 32 86 51 }}
: mapping generators: ~2, ~2187/1400


Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0285{{c}}, ~2187/1400 = 771.9522{{c}}
: [[error map]]: {{val| +0.028 -0.025 +0.068 -0.117 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2187/1400 = 771.9343{{c}}
: error map: {{val| 0.000 -0.056 +0.039 -0.175 }}


Badness: 0.022926
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}


=== 13-limit ===
[[Badness]] (Sintel): 0.716
Subgroup: 2.3.5.7.11.13


Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
=== Semiosiris ===
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}
Comma list: 2401/2400, 9801/9800, 1771561/1771470


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716
{{Mapping|legend=0| 2 -6 -20 -9 -14 | 0 32 86 51 73 }}
: mapping generators: ~99/70, ~243/220


Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}
Optimal tunings:  
* WE: ~99/70 = 600.0152{{c}}, ~243/220 = 171.9383{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~243/220 = 171.9344{{c}}


Badness: 0.020888
{{Optimal ET sequence|legend=0| 328, 342, 1012, 1354, 1696 }}


== Newt ==
Badness (Sintel): 0.703
Newt has a generator of a neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]]. It can be described as the 41 & 270 temperament, and extends naturally to the no-17 19-limit, a.k.a. '''neonewt'''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] work much better.


[[Subgroup]]: 2.3.5.7
== Quasiorwell ==
In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 ({{monzo| 22 -1 -10 1 }}). It has a generator 1024/875, which is [[6144/6125]] more than [[7/6]]. It may be described as the {{nowrap| 31 & 270 }} temperament, and its [[ploidacot]] is eta-38-cot (or omega-triseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure). As one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.


[[Comma list]]: 2401/2400, 33554432/33480783
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.


{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
[[Subgroup]]: 2.3.5.7


: mapping generators: ~2, ~49/40
[[Comma list]]: 2401/2400, 29360128/29296875


{{Multival|legend=1| 2 -57 -28 -95 -50 95 }}
{{Mapping|legend=1| 1 -7 3 1 | 0 38 -3 8 }}
: mapping generators: ~2, ~1024/875


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9403{{c}}, ~1024/875 = 271.0935{{c}}
: [[error map]]: {{val| -0.060 +0.018 +0.226 -0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1024/875 = 271.1064{{c}}
: error map: {{val| 0.000 +0.087 +0.367 +0.025 }}


{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
{{Optimal ET sequence|legend=1| 31, …, 177, 208, 239, 270, 571, 841, 1111 }}


[[Badness]]: 0.041878
[[Badness]] (Sintel): 0.907


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 3025/3024, 19712/19683
Comma list: 2401/2400, 3025/3024, 5632/5625


Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
{{Mapping|legend=0| 1 -7 3 1 -11 | 0 38 -3 8 64 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115
Optimal tunings:
* WE: ~2 = 1199.9484{{c}}, ~90/77 = 271.0989{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1099{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}
{{Optimal ET sequence|legend=0| 31, …, 177e, 208, 239, 270 }}


Badness: 0.019461
Badness (Sintel): 0.580


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095


Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
{{Mapping|legend=0| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
Optimal tunings:
* WE: ~2 = 1199.9916{{c}}, ~90/77 = 271.1051{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1070{{c}}


Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}
{{Optimal ET sequence|legend=0| 31, 239, 270, 571, 841, 1111 }}


Badness: 0.013830
Badness (Sintel): 0.741


=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
== Quinmite ==
Subgroup: 2.3.5.7.11.13.19
Quinmite may be described as the {{nowrap| 99 & 103 }} temperament. The generator for quinmite is the quasi-tempered minor third [[25/21]], sharper than [[32/27]] by the marvel comma, [[225/224]]. It is also generated by 1/5 of the minor tenth [[12/5]], and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>. Its [[ploidacot]] is eta-34-cot.  


Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
[[Comma list]]: 2401/2400, 1959552/1953125


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
{{Mapping|legend=1| 1 -7 -5 -3 | 0 34 29 23 }}
: mapping generators: ~2, ~25/21


Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9361{{c}}, ~25/21 = 302.9808{{c}}
: [[error map]]: {{val| -0.064 -0.162 +0.448 -0.077 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 302.9953{{c}}
: error map: {{val| 0.000 -0.116 +0.549 +0.065 }}
 
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c }}
 
[[Badness]] (Sintel): 0.945


== Septidiasemi ==
== Septidiasemi ==
{{Main| Septidiasemi }}
{{Main| Septidiasemi }}


Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit, and may be described as the {{nowrap| 10 & 171 }} temperament. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]), with a [[ploidacot]] of beta-26-cot. It is an excellent temperament for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 707: Line 626:
[[Comma list]]: 2401/2400, 2152828125/2147483648
[[Comma list]]: 2401/2400, 2152828125/2147483648


{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}
{{Mapping|legend=1| 1 -1 6 4 | 0 26 -37 -12 }}
: mapping generators: ~2, ~15/14


: Mapping generators: ~2, ~28/15
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1043{{c}}, ~15/14 = 119.3076{{c}}
: [[error map]]: {{val| +0.104 -0.061 -0.070 -0.100 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 119.2971{{c}}
: error map: {{val| 0.000 -0.230 -0.307 -0.391 }}


{{Multival|legend=1| 26 -37 -12 -119 -92 76 }}
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, …, 5633bbccddd, 5804bbccddd }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
[[Badness]] (Sintel): 1.12
 
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
 
[[Badness]]: 0.044115


=== Sedia ===
=== Sedia ===
The ''sedia'' temperament (10&amp;161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.
The sedia temperament (10 & 161) is an 11-limit extension of the septidiasemi, which tempers out [[243/242]] and [[441/440]].


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 726: Line 646:
Comma list: 243/242, 441/440, 939524096/935859375
Comma list: 243/242, 441/440, 939524096/935859375


Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}
{{Mapping|legend=0| 1 -1 6 4 -3 | 0 26 -37 -12 65 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279
Optimal tunings:
* WE: ~2 = 1199.9635{{c}}, ~15/14 = 119.2755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2791{{c}}


Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }}
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332 }}


Badness: 0.090687
Badness (Sintel): 3.00


==== 13-limit ====
==== 13-limit ====
Line 739: Line 661:
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
Comma list: 243/242, 441/440, 2200/2197, 3584/3575


Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}
{{Mapping|legend=0| 1 -1 6 4 -3 4 | 0 26 -37 -12 65 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
Optimal tunings:
* WE: ~2 = 1199.8922{{c}}, ~15/14 = 119.2700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2804{{c}}


Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }}
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332 }}


Badness: 0.045773
Badness (Sintel): 1.89


==== 17-limit ====
==== 17-limit ====
Line 752: Line 676:
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575


Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 0 -26 37 12 -65 3 -21 }}
{{Mapping|legend=0| 1 -1 6 4 -3 4 2 | 0 26 -37 -12 65 -3 21 }}
 
Optimal tunings:
* WE: ~2 = 1199.9088{{c}}, ~15/14 = 119.2719{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2808{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332, 503ef }}


Optimal ET sequence: {{Optimal ET sequence|l 10, 151, 161, 171, 332, 503ef, 835eeff }}
Badness (Sintel): 1.39


Badness: 0.027322
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''


== Maviloid ==
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[~]][[77/60]] from the 11-limit onwards, and 74 generator steps give the interval class of [[3/1|3]]; its [[ploidacot]] is 26-sheared 74-cot. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.
{{See also| Ragismic microtemperaments #Parakleismic }}
 
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament has a [[temperament merging|join]] 103 & 270, hence the name. The name was proposed in 2022 by [[Eliora]], who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.
 
Lockerbie also has a unique extension that adds the [[41/1|41st]] [[harmonic]] such that the generator is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 1224440064/1220703125
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}


{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
: mapping generators: ~2, ~3828125/2985984


: Mapping generators: ~2, ~1296/875
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9950{{c}}, ~3828125/2985984 = 431.1055{{c}}
: [[error map]]: {{val| -0.005 -0.024 +0.146 -0.120 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3828125/2985984 = 431.1072{{c}}
: error map: {{val| 0.0000 -0.020 +0.155 -0.108 }}


{{Multival|legend=1| 52 56 41 -32 -81 -62 }}
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913, 1183 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
[[Badness]] (Sintel): 1.51


{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.057632
Comma list: 2401/2400, 3025/3024, 766656/765625


== Subneutral ==
{{Mapping|legend=0| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}
{{See also| Luna family }}


[[Subgroup]]: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.0199{{c}}, ~77/60 = 431.1147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1078{{c}}


[[Comma list]]: 2401/2400, 274877906944/274658203125
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}


{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}
Badness (Sintel): 0.865


: Mapping generators: ~2, ~57344/46875
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Multival|legend=1| 60 -8 11 -152 -151 48 }}
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
{{Mapping|legend=0| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}


{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
Optimal tunings:
* WE: ~2 = 1200.0707{{c}}, ~77/60 = 431.1316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1069{{c}}


[[Badness]]: 0.045792
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}


== Osiris ==
Badness (Sintel): 0.662
{{See also| Metric microtemperaments #Geb }}


[[Subgroup]]: 2.3.5.7
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 2401/2400, 31381059609/31360000000
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224


{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}
{{Mapping|legend=0| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}


: Mapping generators: ~2, ~2800/2187
Optimal tunings:  
* WE: ~2 = 1199.9639{{c}}, ~77/60 = 431.0957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1083{{c}}


{{Multival|legend=1| 32 86 51 62 -9 -123 }}
{{Optimal ET sequence|legend=0| 103, 167, 270 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
Badness (Sintel): 1.07


{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
== Unthirds ==
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[temperament merging|join]] of two tuning systems well-known for their high accuracy. It is generated by the interval of [[14/11]] (<u>un</u>decimal major <u>third</u>, hence the name) tuned less than a cent flat, 42 of which [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]. Its [[ploidacot]] is 14-sheared 42-cot. The 23-note [[mos]] from the generator serves as a well temperament of, of all things, [[23edo]]. The 49-note mos is needed to access the [[3/1|3rd]], [[5/1|5th]], [[7/1|7th]], and [[11/1|11th]] [[harmonic]]s.


[[Badness]]: 0.028307
The commas it tempers out in the 11-limit include the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).


== Gorgik ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 28672/28125
[[Comma list]]: 2401/2400, 68359375/68024448


{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}
{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
: mapping generators: ~2, ~3969/3125


: Mapping generators: ~2, ~8/7
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0859{{c}}, ~3969/3125 = 416.7465{{c}}
: [[error map]]: {{val| +0.086 +0.281 -0.431 -0.218 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3969/3125 = 416.7184{{c}}
: error map: {{val| 0.000 +0.220 -0.547 -0.399 }}


{{Multival|legend=1| 18 -7 1 -53 -49 22 }}
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
 
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}


[[Badness]]: 0.158384
[[Badness]] (Sintel): 1.90


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 2401/2400, 2560/2541
Comma list: 2401/2400, 3025/3024, 4000/3993


Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}
{{Mapping|legend=0| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500
Optimal tunings:
* WE: ~2 = 1200.0246{{c}}, ~14/11 = 416.7270{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7190{{c}}


Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}


Badness: 0.059260
Badness (Sintel): 0.758


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 196/195, 364/363, 512/507
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400


Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}
{{Mapping|legend=0| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493
Optimal tunings:
* WE: ~2 = 1200.0536{{c}}, ~14/11 = 416.7343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7164{{c}}


Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}
{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}


Badness: 0.032205
Badness (Sintel): 0.863
 
== Neominor ==
Neominor tempers out [[177147/175616]] and may be described as the {{nowrap| 72 & 89 }} temperament. The generator is a neogothic minor third, which represents [[13/11]][[~]][[20/17]], or its [[octave complement]], which represents [[17/10]]~[[22/13]]. The latter stacked six times [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]], and the temperament has a [[ploidacot]] of delta-hexacot. [[72edo]] and [[89edo]] can be used as tunings.


== Fibo ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 341796875/339738624
[[Comma list]]: 2401/2400, 177147/175616


{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}
{{Mapping|legend=1| 1 -3 -29 -14 | 0 6 41 22 }}
: mapping generators: ~2, ~320/189


: Mapping generators: ~2, ~125/96
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4276{{c}}, ~320/189 = 917.0471{{c}}
: [[error map]]: {{val| +0.428 -0.955 +0.216 +0.224 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/189 = 916.7320{{c}}
: error map: {{val| 0.000 -1.563 -0.301 -0.722 }}


{{Multival|legend=1| 46 15 19 -83 -99 2 }}
{{Optimal ET sequence|legend=1| 17c, 55c, 72, 161, 233, 305 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
[[Badness]] (Sintel): 2.23
 
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
 
Badness: 0.100511


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1375/1372, 43923/43750
Comma list: 243/242, 441/440, 35937/35840


Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}
{{Mapping|legend=0| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
Optimal tunings:
* WE: ~2 = 1200.3466{{c}}, ~56/33 = 916.9889{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/33 = 916.7330{{c}}


Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
{{Optimal ET sequence|legend=0| 17c, 55c, 72, 161, 233, 305 }}


Badness: 0.056514
Badness (Sintel): 0.924


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 385/384, 625/624, 847/845, 1375/1372
Comma list: 169/168, 243/242, 364/363, 441/440
 
{{Mapping|legend=0| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}


Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}
Optimal tunings:  
* WE: ~2 = 1200.6874{{c}}, ~22/13 = 917.2313{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 916.7228{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316
{{Optimal ET sequence|legend=0| 17c, 55cf, 72 }}


Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
Badness (Sintel): 1.11


Badness: 0.027429
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== Mintone ==
Comma list: 169/168, 221/220, 243/242, 273/272, 364/363
In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo| -3 11 -5 -1 }} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 &amp; 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.
 
{{Mapping|legend=0| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}
 
Optimal tunings:
* WE: ~2 = 1200.6905{{c}}, ~17/10 = 917.2356{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 916.7252{{c}}


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=0| 17cg, 55cfg, 72 }}


[[Comma list]]: 2401/2400, 177147/175000
Badness (Sintel): 0.918


{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}
== Cotritone ==
[[Subgroup]]: 2.3.5.7


: Mapping generators: ~2, ~10/9
[[Comma list]]: 2401/2400, 390625/387072


{{Multival|legend=1| 22 43 27 17 -19 -58 }}
{{Mapping|legend=1| 1 -13 -4 -4 | 0 30 13 14 }}
: mappping generators: ~2, ~7/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9278{{c}}, ~7/5 = 583.5994{{c}}
: [[error map]]: {{val| +0.441 +0.289 -1.287 -0.200 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/5 = 583.3956{{c}}
: error map: {{val| 0.000 -0.086 -2.170 -1.287 }}


{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
{{Optimal ET sequence|legend=1| 35, 37, 72, 181, 253, 325c }}


[[Badness]]: 0.125672
[[Badness]] (Sintel): 2.49


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 43923/43750
Comma list: 385/384, 1375/1372, 4000/3993


Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}
{{Mapping|legend=0| 1 -13 -4 -4 2 | 0 30 13 14 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345
Optimal tunings:
* WE: ~2 = 1200.4058{{c}}, ~7/5 = 583.5845{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 583.3950{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}
{{Optimal ET sequence|legend=0| 35, 37, 72, 181, 253, 325c }}


Badness: 0.039962
Badness (Sintel): 1.07


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 351/350, 441/440, 847/845
Comma list: 169/168, 364/363, 385/384, 625/624
 
{{Mapping|legend=0| 1 -13 -4 -4 2 -7 | 0 30 13 14 3 22 }}
 
Optimal tunings:
* WE: ~2 = 1200.6111{{c}}, ~7/5 = 583.6837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 583.3987{{c}}
 
{{Optimal ET sequence|legend=0| 35f, 37, 72, 181f, 253ff }}


Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}
Badness (Sintel): 1.19


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347
== Fibo ==
[[Subgroup]]: 2.3.5.7


Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
[[Comma list]]: 2401/2400, 341796875/339738624


Badness: 0.021849
{{Mapping|legend=1| 1 -27 -7 -9 | 0 46 15 19 }}
: mapping generators: ~2, ~192/125


=== 17-limit ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17
* [[WE]]: ~2 = 1200.2050{{c}}, ~192/125 = 745.8170{{c}}
: [[error map]]: {{val| +0.205 +0.094 -0.493 -0.147 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/125 = 745.6927{{c}}
: error map: {{val| 0.000 -0.092 -0.924 -0.665 }}


Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}


Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}
Badness (Sintel): 2.54


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
Comma list: 385/384, 1375/1372, 43923/43750


Badness: 0.020295
{{Mapping|legend=0| 1 -27 -7 -9 -4 | 0 46 15 19 12 }}


== Catafourth ==
Optimal tunings:
{{See also| Sensipent family }}
* WE: ~2 = 1200.4064{{c}}, ~77/50 = 745.9349{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/50 = 745.6876{{c}}


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}


[[Comma list]]: 2401/2400, 78732/78125
Badness (Sintel): 1.87


{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


: Mapping generators: ~2, ~250/189
Comma list: 385/384, 625/624, 847/845, 1375/1372


{{Multival|legend=1| 28 36 25 -8 -39 -43 }}
{{Mapping|legend=0| 1 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
Optimal tunings:
* WE: ~2 = 1200.3728{{c}}, ~20/13 = 745.9152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 745.6879{{c}}


{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}


Badness: 0.079579
Badness (Sintel): 1.13


=== 11-limit ===
== Quasimoha ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasimoha]].''


Comma list: 243/242, 441/440, 78408/78125
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}
[[Comma list]]: 2401/2400, 3645/3584


Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
: mapping generators: ~2, ~49/40


Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5059{{c}}, ~49/40 = 348.0409{{c}}
: [[error map]]: {{val| +1.506 -2.367 -0.702 +0.759 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 348.5582{{c}}
: error map: {{val| 0.000 -4.839 -3.152 -2.966 }}


Badness: 0.036785
{{Optimal ET sequence|legend=1| 24c, 31, 117c, 148bc, 179bcd }}


=== 13-limit ===
[[Badness]] (Sintel): 2.80
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 351/350, 441/440, 10985/10976
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}
Comma list: 243/242, 441/440, 1815/1792


Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256
{{Mapping|legend=0| 1 1 9 6 2 | 0 2 -23 -11 5 }}


Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}
Optimal tunings:  
* WE: ~2 = 1201.7630{{c}}, ~11/9 = 349.1510{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.6050{{c}}


Badness: 0.021694
{{Optimal ET sequence|legend=0| 24c, 31, 86ce, 117ce, 148bce }}


== Cotritone ==
Badness (Sintel): 1.53
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 390625/387072
== Mintone ==
In addition to 2401/2400, mintone tempers out 177147/175000 ({{monzo| -3 11 -5 -1 }}) in the 7-limit; [[243/242]], [[441/440]], and [[43923/43750]] in the 11-limit. It may be described as the {{nowrap| 58 & 103 }} temperament. It has a generator of [[~]][[10/9]], tuned to around [[49/44]]. Note that in the data below, the generator is its [[octave complement]], ~[[9/5]], so that 22 of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]. Its [[ploidacot]] is 18-sheared 22-cot. As one might expect, 25\161 makes for an excellent tuning choice.


{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
[[Subgroup]]: 2.3.5.7


: Mappping generators: ~2, ~10/7
[[Comma list]]: 2401/2400, 177147/175000


{{Multival|legend=1| 30 13 14 -49 -62 -4 }}
{{Mapping|legend=1| 1 -17 -34 -20 | 0 22 43 27 }}
: mapping generators: ~2, ~9/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1458{{c}}, ~9/5 = 1013.7798{{c}}
: [[error map]]: {{val| +0.146 -1.277 +1.263 +0.314 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6611{{c}}
: error map: {{val| 0.000 -1.410 +1.116 +0.025 }}


{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=1| 45, 58, 103, 161 }}


[[Badness]]: 0.098322
[[Badness]] (Sintel): 3.18


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1375/1372, 4000/3993
Comma list: 243/242, 441/440, 43923/43750


Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}
{{Mapping|legend=0| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
Optimal tunings:
* WE: ~2 = 1200.1491{{c}}, ~9/5 = 1013.7809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6593{{c}}


Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=0| 45e, 58, 103, 161, 425b }}


Badness: 0.032225
Badness (Sintel): 1.32


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 364/363, 385/384, 625/624
Comma list: 243/242, 351/350, 441/440, 847/845
 
{{Mapping|legend=0| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}
 
Optimal tunings:
* WE: ~2 = 1200.0928{{c}}, ~9/5 = 1013.7311{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6556{{c}}
 
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}
 
Badness (Sintel): 0.903
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845


Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}
{{Mapping|legend=0| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
Optimal tunings:
* WE: ~2 = 1200.1085{{c}}, ~9/5 = 1013.7433{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6537{{c}}


Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }}
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}


Badness: 0.028683
Badness (Sintel): 1.03


== Quasimoha ==
== Gorgik ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''
{{See also| Llywelynsmic clan }}
 
Gorgik may be described as the {{nowrap| 21 & 37 }} temperament, with a [[ploidacot]] of 14-sheared 18-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). [[58edo]] makes for a strong tuning for this temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 3645/3584
[[Comma list]]: 2401/2400, 28672/28125


{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
{{Mapping|legend=1| 1 -13 8 2 | 0 18 -7 1 }}
: mapping generators: ~2, ~7/4


: Mapping generators: ~2, ~49/40
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.5503{{c}}, ~7/4 = 971.3132{{c}} (~8/7 = 227.2371{{c}})
: [[error map]]: {{val| -1.450 +0.528 +2.896 -0.412 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 972.4675{{c}} (~8/7 = 227.5325{{c}})
: error map: {{val| 0.000 +2.460 +6.414 +3.642 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}


{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
[[Badness]] (Sintel): 4.01
 
[[Badness]]: 0.110820


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 1815/1792
Comma list: 176/175, 2401/2400, 2560/2541
 
{{Mapping|legend=0| 1 -13 8 2 14 | 0 18 -7 1 -13 }}
 
Optimal tunings:
* WE: ~2 = 1198.4615{{c}}, ~7/4 = 971.2535{{c}} (~8/7 = 227.2079{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.4918{{c}} (~8/7 = 227.5082{{c}})
 
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
 
Badness (Sintel): 1.96
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}
Comma list: 176/175, 196/195, 364/363, 512/507


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639
{{Mapping|legend=0| 1 -13 8 2 14 11 | 0 18 -7 1 -13 -9 }}


Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }}
Optimal tunings:  
* WE: ~2 = 1198.4012{{c}}, ~7/4 = 971.2110{{c}} (~8/7 = 227.1903{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.5030{{c}} (~8/7 = 227.4970{{c}})


Badness: 0.046181
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bcef, 211bccdeeff }}


== Lockerbie ==
Badness (Sintel): 1.33
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''
 
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.  
== Retrotonic ==
Retrotonic may be described as the {{nowrap| 58 & 311 }} temperament, with a [[ploidacot]] of 14-sheared 18-cot, the same as [[#Gorgik|gorgik]], but is strongly a 58-form. It is the generated by the retroptolemaic whole tone, [[~]][[729/640]], whose [[octave complement]] is the retroptolemaic minor seventh, ~[[1280/729]], given in the data below.  


The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.  
Retrotonic becomes much more interesting in the [[11-limit]], where it tempers out important commas such as the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), and the [[symbiotic comma]] (19712/19683). This leads to a 13-limit extension where [[1575/1573]], [[2080/2079]], and [[2200/2197]] are tempered out, enabling their corresponding [[essentially tempered chord]]s. Recommended retrotonic tunings include [[311edo]], [[369edo]] (in the 369f val) and [[680edo]] (in the 680ef val).  


Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.
Retrotonic was named by [[Flora Canou]] in 2026 after one of its possible generators, the retroptolemaic whole tone.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}
[[Comma list]]: 2401/2400, {{monzo| 29 -28 3 3 }}
 
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}


: mapping generators: ~2, ~3828125/2985984
{{Mapping|legend=1| 1 -13 -86 -45 | 0 18 109 59 }}
: mapping generators: ~2, ~1280/729


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
* [[WE]]: ~2 = 1199.9156{{c}}, ~1280/729 = 972.2848{{c}}
: [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }}
: [[error map]]: {{val| -0.084 +0.269 -0.012 -0.225 }}
* [[CWE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1280/729 = 972.3523{{c}}
: error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }}
: error map: {{val| 0.000 +0.386 +0.088 -0.040 }}


{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913 }}
{{Optimal ET sequence|legend=1| 58, 195c, 253, 311, 369, 680 }}


[[Badness]] (Smith): 0.0597
[[Badness]] (Sintel): 4.25


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 3025/3024, 766656/765625
Comma list: 2401/2400, 4000/3993, 19712/19683


Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}
{{Mapping|legend=0| 1 -13 -86 -45 -80 | 0 18 109 59 103 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* WE: ~2 = 1199.9156{{c}}, ~135/77 = 972.1718{{c}}
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078
: error map: {{val| -0.099 +0.221 -0.196 -0.346 +0.577 }}
* CWE: ~2 = 1200.0000{{c}}, ~135/77 = 972.3507{{c}}
: error map: {{val| 0.000 +0.358 -0.083 -0.132 +0.808 }}


{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
{{Optimal ET sequence|legend=0| 58, 195ce, 253, 311, 680e, 991e }}


Badness (Smith): 0.0262
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224
Comma list: 1575/1573, 2080/2079, 2200/2197, 2401/2400


Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}
{{Mapping|legend=0| 1 -13 -86 -45 -80 -83 | 0 18 109 59 103 107 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
* WE: ~2 = 1199.8883{{c}}, ~135/77 = 972.2604{{c}}
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069
: error map: {{val| -0.112 +0.184 -0.325 -0.436 +0.438 +0.605 }}
 
* CWE: ~2 = 1200.0000{{c}}, ~135/77 = 972.3497{{c}}
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
: error map: {{val| 0.000 +0.339 -0.201 -0.196 +0.697 +0.886 }}


Badness (Smith): 0.0160
{{Optimal ET sequence|legend=0| 58, 195cef, 253, 311, 680ef, 991ef }}


=== 17-limit ===
Badness (Sintel): 1.10
Subgroup: 2.3.5.7.11.13.17


Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
== Hemigoldis ==
Hemigoldis may be described as the {{nowrap| 68 & 89 }} temperament. Though fairly complex in the [[7-limit]], it does a lot better in badness metrics than pure [[5-limit]] [[goldis]], and yet again has many possible extensions to higher primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.


Mapping: {{Val|1 49 35 31 56 21 31}}, {{Val|0 -74 -51 -44 -82 -27 -42}}
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~120/77 = 768.893
[[Comma list]]: 2401/2400, 549755813888/533935546875


=== 2.3.5.7.11.13.17.41 subgroup ===
{{Mapping|legend=1| 1 21 -9 2 | 0 24 -14 -1 }}
Subgroup: 2.3.5.7.11.13.17.41
: mapping generators: ~2, ~8/7


Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.2264{{c}}, ~8/7 = 229.1679{{c}}
: [[error map]]: {{val| -0.774 +0.394 +1.468 -0.314 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.3103{{c}}
: error map: {{val| 0.000 +1.491 +3.343 +1.864 }}


Mapping: {{Val|1 49 35 31 56 21 31 6}}, {{Val|0 -74 -51 -44 -82 -27 -42 -1}}
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}


Optimal tuning (POTE): ~64/41 = 768.893
[[Badness]] (Sintel): 4.40


== Surmarvelpyth ==
== Surmarvelpyth ==
''Surmarvelpyth'' is named for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2. It can be described as the 311 & 431 temperament, starting with the 7-limit to the 19-limit.
Surmarvelpyth can be described as the {{nowrap| 311 & 431 }} temperament, starting with the 7-limit to the 19-limit. Its [[ploidacot]] is 28-sheared 70-cot. It was named by [[Eliora]] in 2022 for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,160: Line 1,207:
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}


{{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }}
{{Mapping|legend=1| 1 -27 55 22 | 0 70 -129 -47 }}
: mapping generators: ~2, ~896/675


: Mapping generators: ~2, ~675/448
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.0051{{c}}, ~896/675 = 490.0303{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719
: [[error map]]: {{val| +0.005 +0.025 +0.063 -0.136 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~896/675 = 490.0282{{c}}
: error map: {{val| 0.000 +0.017 +0.052 -0.150 }}


{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}


[[Badness]]: 0.202249
[[Badness]] (Sintel): 5.12


=== 11-limit ===
=== 11-limit ===
Line 1,175: Line 1,225:
Comma list: 2401/2400, 820125/819896, 2097152/2096325
Comma list: 2401/2400, 820125/819896, 2097152/2096325


Mapping: {{mapping| 1 43 -74 -25 36 | 0 -70 129 47 -55 }}
{{Mapping|legend=0| 1 -27 55 22 -19 | 0 70 -129 -47 55 }}


Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720
Optimal tunings:
* WE: ~2 = 1199.9901{{c}}, ~896/675 = 490.0239{{c}}
* CWE: ~2 = 1200.000{{c}}, ~896/675 = 490.0279{{c}}


Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795 }}
{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795 }}


Badness: 0.052308
Badness (Sintel): 1.73


=== 13-limit ===
=== 13-limit ===
Line 1,188: Line 1,240:
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167


Mapping: {{mapping| 1 43 -74 -25 36 25 | 0 -70 129 47 -55 -36 }}
{{Mapping|legend=0| 1 -27 55 22 -19 -11 | 0 70 -129 -47 55 36 }}


Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723
Optimal tunings:
* WE: ~2 = 1199.9701{{c}}, ~65/49 = 490.0155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0277{{c}}


Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795f }}
{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795f }}


Badness: 0.032503
Badness (Sintel): 1.34


=== 17-limit ===
=== 17-limit ===
Line 1,201: Line 1,255:
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619


Mapping: {{mapping| 1 43 -74 -25 36 25 -103 | 0 -70 129 47 -55 -36 181 }}
{{Mapping|legend=0| 1 -27 55 22 -19 -11 78 | 0 70 -129 -47 55 36 -181 }}


Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
Optimal tunings:
* WE: ~2 = 1199.9726{{c}}, ~65/49 = 490.0164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0276{{c}}


Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}


Badness: 0.020995
Badness (Sintel): 1.07


=== 19-limit ===
=== 19-limit ===
Line 1,214: Line 1,270:
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984


Mapping: {{mapping| 1 43 -74 -25 36 25 -103 -49 | 0 -70 129 47 -55 -36 181 90 }}
{{Mapping|legend=0| 1 -27 55 22 -19 -11 78 41 | 0 70 -129 -47 55 36 -181 -90 }}


Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
Optimal tunings:
* WE: ~2 = 1199.9756{{c}}, ~65/49 = 490.0176{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0276{{c}}


Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}


Badness: 0.013771
Badness (Sintel): 0.838


== Notes ==
== References ==


[[Category:Temperament collections]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament collections]]
[[Category:Catalogs of rank-2 temperaments]]
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