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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]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* [[Ennealimmal]] (+4375/4374) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Subneutral]]'' (+274877906944/274658203125) → [[Luna family #Subneutral|Luna family]]
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec 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]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]
 
== Hemififths ==
{{Main| Hemififths }}
 
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
 
[[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|unchanged-interval (eigenmonzo) 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 ====
Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, maviloid, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  
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:
* 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 163: Line 41:
[[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}})
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
: [[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}})
: 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 178: Line 59:
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: {{mapping| 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 191: Line 74:
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: {{mapping| 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 204: Line 89:
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: {{mapping| 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 217: Line 104:
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: {{mapping| 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 230: Line 119:
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: {{mapping| 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 243: Line 134:
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: {{mapping| 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: {{mapping| 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 269: Line 166:
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: {{mapping| 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 282: Line 181:
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: {{mapping| 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: {{mapping| 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: {{mapping| 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 ====
{{Optimal ET sequence|legend=0| 31ei, 140, 311, 762g }}
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
Badness (Sintel): 0.858
 
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 }}
 
Badness: 0.009825


=== Hemitert ===
=== Hemitert ===
Line 373: Line 226:
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: {{mapping| 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 388: Line 242:
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: {{mapping| 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 401: Line 257:
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: {{mapping| 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 414: Line 272:
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: {{mapping| 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 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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
 
{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}


[[Badness]]: 0.035832
[[Badness]] (Sintel): 0.424


=== 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: {{mapping| 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: {{mapping| 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: {{mapping| 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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280
Badness (Sintel): 1.18


{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
== Hemififths ==
{{Main| Hemififths }}
 
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 ===
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338 }}
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 243/242, 364/363, 441/440
[[Badness]] (Sintel): 0.563


Mapping: {{mapping| 1 3 12 8 7 7 | 0 -6 -41 -22 -15 -14 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294
Comma list: 243/242, 441/440, 896/891


Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }}
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


Badness: 0.026942
Optimal tunings:  
* WE: ~2 = 1199.2845{{c}}, ~11/9 = 351.3110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4956{{c}}


== Emmthird ==
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 0.777


[[Comma list]]: 2401/2400, 14348907/14336000
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}
Comma list: 144/143, 196/195, 243/242, 364/363


: Mapping generators: ~2, ~2187/1372
Mapping: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}


{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
Badness (Sintel): 1.29


[[Badness]]: 0.037322
=== 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.  


== Unthirds ==
Subgroup: 2.3.5.7.11
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 in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.


The commas it tempers out include the [[breedsma]] (2401/2400), 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).
Comma list: 2401/2400, 4000/3993, 5120/5103


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
Comma list: 352/351, 847/845, 1575/1573, 2401/2400


[[Badness]]: 0.075253
Mapping: {{mapping| 1 -1 -30 -14 -28 -20 | 0 6 75 39 73 55 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1199.6427{{c}}, ~66/49 = 517.0035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1524{{c}}


Comma list: 2401/2400, 3025/3024, 4000/3993
{{Optimal ET sequence|legend=0| 58, 181, 239f }}


Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}
Badness (Sintel): 1.45


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718
== Osiris ==


Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.022926
[[Comma list]]: 2401/2400, 31381059609/31360000000


=== 13-limit ===
{{Mapping|legend=1| 1 13 33 21 | 0 32 86 51 }}
Subgroup: 2.3.5.7.11.13
: mapping generators: ~2, ~2187/1400


Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
[[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 }}


Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716
[[Badness]] (Sintel): 0.716


Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}
== 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.


Badness: 0.020888
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.
 
== Newt ==
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 33554432/33480783
[[Comma list]]: 2401/2400, 29360128/29296875


{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
{{Mapping|legend=1| 1 -7 3 1 | 0 38 -3 8 }}
: mapping generators: ~2, ~1024/875


: mapping generators: ~2, ~49/40
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
{{Optimal ET sequence|legend=1| 31, …, 177, 208, 239, 270, 571, 841, 1111 }}


{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
[[Badness]] (Sintel): 0.907
 
[[Badness]]: 0.041878


=== 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: {{mapping| 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 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}
 
Optimal tunings:
* WE: ~2 = 1199.9916{{c}}, ~90/77 = 271.1051{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1070{{c}}


Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
{{Optimal ET sequence|legend=0| 31, 239, 270, 571, 841, 1111 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
Badness (Sintel): 0.741


Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}
== Quinmite ==
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.


Badness: 0.013830
[[Subgroup]]: 2.3.5.7


=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
[[Comma list]]: 2401/2400, 1959552/1953125
Subgroup: 2.3.5.7.11.13.19


Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
{{Mapping|legend=1| 1 -7 -5 -3 | 0 34 29 23 }}
: mapping generators: ~2, ~25/21


Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
[[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 tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c }}


Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}
[[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 693: Line 608:
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, …, 5633bbccddd, 5804bbccddd }}


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
[[Badness]] (Sintel): 1.12
 
[[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 710: Line 628:
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: {{mapping| 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 723: Line 643:
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: {{mapping| 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 736: Line 658:
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: {{mapping| 1 -1 6 4 -3 4 2 | 0 26 -37 -12 65 -3 21 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
Optimal tunings:
* WE: ~2 = 1199.9088{{c}}, ~15/14 = 119.2719{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2808{{c}}


Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332, 503ef }}


Badness: 0.027322
Badness (Sintel): 1.39


== Maviloid ==
== Maviloid ==
Line 751: Line 675:
[[Comma list]]: 2401/2400, 1224440064/1220703125
[[Comma list]]: 2401/2400, 1224440064/1220703125


{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}
{{Mapping|legend=1| 1 -21 -22 -15 | 0 52 56 41 }}
: mapping generators: ~2, ~875/648


: Mapping generators: ~2, ~1296/875
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.9863{{c}}, ~875/648 = 521.1837{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
: [[error map]]: {{val| -0.014 -0.115 +0.274 -0.089 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~875/648 = 521.1894{{c}}
: error map: {{val| 0.000 -0.106 +0.293 -0.060 }}


{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}


[[Badness]]: 0.057632
[[Badness]] (Sintel): 1.46


== Subneutral ==
== Lockerbie ==
{{See also| Luna family }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 274877906944/274658203125
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.


{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}
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.


: Mapping generators: ~2, ~57344/46875
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.
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
 
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
 
[[Badness]]: 0.045792
 
== Osiris ==
{{See also| Metric microtemperaments #Geb }}


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


[[Comma list]]: 2401/2400, 31381059609/31360000000
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}


{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
: mapping generators: ~2, ~3828125/2985984


: Mapping generators: ~2, ~2800/2187
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913, 1183 }}


{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
[[Badness]] (Sintel): 1.51


[[Badness]]: 0.028307
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Gorgik ==
Comma list: 2401/2400, 3025/3024, 766656/765625
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 28672/28125
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}


{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}
Optimal tunings:
* WE: ~2 = 1200.0199{{c}}, ~77/60 = 431.1147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1078{{c}}


: Mapping generators: ~2, ~8/7
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
Badness (Sintel): 0.865


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.158384
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224


=== 11-limit ===
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}
Subgroup: 2.3.5.7.11


Comma list: 176/175, 2401/2400, 2560/2541
Optimal tunings:  
* WE: ~2 = 1200.0707{{c}}, ~77/60 = 431.1316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1069{{c}}


Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500
Badness (Sintel): 0.662


Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.059260
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224


=== 13-limit ===
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 196/195, 364/363, 512/507
Optimal tunings:  
* WE: ~2 = 1199.9639{{c}}, ~77/60 = 431.0957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1083{{c}}


Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}
{{Optimal ET sequence|legend=0| 103, 167, 270 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493
Badness (Sintel): 1.07


Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}
== 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: 0.032205
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).


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


[[Comma list]]: 2401/2400, 341796875/339738624
[[Comma list]]: 2401/2400, 68359375/68024448


{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}
{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
: mapping generators: ~2, ~3969/3125


: Mapping generators: ~2, ~125/96
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
 
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}


Badness: 0.100511
[[Badness]] (Sintel): 1.90


=== 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: 2401/2400, 3025/3024, 4000/3993


Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}
Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}


Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
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| 37, 66b, 103, 140, 243e }}
{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}


Badness: 0.056514
Badness (Sintel): 0.758


=== 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: 625/624, 1575/1573, 2080/2079, 2401/2400


Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}
Mapping: {{mapping| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316
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| 37, 66b, 103, 140, 243e }}
{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}


Badness: 0.027429
Badness (Sintel): 0.863


== Mintone ==
== Neominor ==
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.
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.  


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


[[Comma list]]: 2401/2400, 177147/175000
[[Comma list]]: 2401/2400, 177147/175616


{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}
{{Mapping|legend=1| 1 -3 -29 -14 | 0 6 41 22 }}
: mapping generators: ~2, ~320/189


: Mapping generators: ~2, ~10/9
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.4276{{c}}, ~320/189 = 917.0471{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
: [[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 }}


{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
{{Optimal ET sequence|legend=1| 17c, 55c, 72, 161, 233, 305 }}


[[Badness]]: 0.125672
[[Badness]] (Sintel): 2.23


=== 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: 243/242, 441/440, 35937/35840


Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}
Mapping: {{mapping| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345
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| 58, 103, 161, 425b, 586b, 747bc }}
{{Optimal ET sequence|legend=0| 17c, 55c, 72, 161, 233, 305 }}


Badness: 0.039962
Badness (Sintel): 0.924


=== 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, 243/242, 364/363, 441/440


Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347
Optimal tunings:
* WE: ~2 = 1200.6874{{c}}, ~22/13 = 917.2313{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 916.7228{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
{{Optimal ET sequence|legend=0| 17c, 55cf, 72 }}


Badness: 0.021849
Badness (Sintel): 1.11


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
Comma list: 169/168, 221/220, 243/242, 273/272, 364/363


Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348
Optimal tunings:
* WE: ~2 = 1200.6905{{c}}, ~17/10 = 917.2356{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 916.7252{{c}}


Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
{{Optimal ET sequence|legend=0| 17cg, 55cfg, 72 }}


Badness: 0.020295
Badness (Sintel): 0.918


== Catafourth ==
== Catafourth ==
Line 940: Line 883:
[[Comma list]]: 2401/2400, 78732/78125
[[Comma list]]: 2401/2400, 78732/78125


{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}
{{Mapping|legend=1| 1 -15 -19 -12 | 0 28 36 25 }}
: mapping generators: ~2, ~189/125


: Mapping generators: ~2, ~250/189
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.9278{{c}}, ~189/125 = 710.7220{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
: [[error map]]: {{val| -0.072 -0.656 +1.050 +0.091 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~189/125 = 710.7626{{c}}
: error map: {{val| 0.000 -0.603 +1.139 +0.238 }}


{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}


Badness: 0.079579
[[Badness]] (Sintel): 2.01


=== 11-limit ===
=== 11-limit ===
Line 955: Line 901:
Comma list: 243/242, 441/440, 78408/78125
Comma list: 243/242, 441/440, 78408/78125


Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}
Mapping: {{mapping| 1 -15 -19 -12 -38 | 0 28 36 25 70 }}


Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252
Optimal tunings:
* WE: ~2 = 1200.0219{{c}}, ~189/125 = 710.7610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~189/125 = 710.7487{{c}}


Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363, 493e }}


Badness: 0.036785
Badness (Sintel): 1.22


=== 13-limit ===
=== 13-limit ===
Line 968: Line 916:
Comma list: 243/242, 351/350, 441/440, 10985/10976
Comma list: 243/242, 351/350, 441/440, 10985/10976


Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}
Mapping: {{mapping| 1 -15 -19 -12 -38 -4 | 0 28 36 25 70 13 }}


Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256
Optimal tunings:
* WE: ~2 = 1200.1023{{c}}, ~98/65 = 710.8043{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~98/65 = 710.7459{{c}}


Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363 }}


Badness: 0.021694
Badness (Sintel): 0.896


== Cotritone ==
== Cotritone ==
Line 981: Line 931:
[[Comma list]]: 2401/2400, 390625/387072
[[Comma list]]: 2401/2400, 390625/387072


{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
{{Mapping|legend=1| 1 -13 -4 -4 | 0 30 13 14 }}
: mappping generators: ~2, ~7/5


: Mappping generators: ~2, ~10/7
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.9278{{c}}, ~7/5 = 583.5994{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
: [[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| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=1| 35, 37, 72, 181, 253, 325c }}


[[Badness]]: 0.098322
[[Badness]] (Sintel): 2.49


=== 11-limit ===
=== 11-limit ===
Line 996: Line 949:
Comma list: 385/384, 1375/1372, 4000/3993
Comma list: 385/384, 1375/1372, 4000/3993


Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}
Mapping: {{mapping| 1 -13 -4 -4 2 | 0 30 13 14 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
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| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=0| 35, 37, 72, 181, 253, 325c }}


Badness: 0.032225
Badness (Sintel): 1.07


=== 13-limit ===
=== 13-limit ===
Line 1,009: Line 964:
Comma list: 169/168, 364/363, 385/384, 625/624
Comma list: 169/168, 364/363, 385/384, 625/624


Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}
Mapping: {{mapping| 1 -13 -4 -4 2 -7 | 0 30 13 14 3 22 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
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: {{Optimal ET sequence| 37, 72, 109, 181f }}
{{Optimal ET sequence|legend=0| 35f, 37, 72, 181f, 253ff }}


Badness: 0.028683
Badness (Sintel): 1.19
 
== Fibo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 341796875/339738624
 
{{Mapping|legend=1| 1 -27 -7 -9 | 0 46 15 19 }}
: mapping generators: ~2, ~192/125
 
[[Optimal tuning]]s:
* [[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 }}
 
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
 
Badness (Sintel): 2.54
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 43923/43750
 
Mapping: {{mapping| 1 -27 -7 -9 -4 | 0 46 15 19 12 }}
 
Optimal tunings:
* WE: ~2 = 1200.4064{{c}}, ~77/50 = 745.9349{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/50 = 745.6876{{c}}
 
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}
 
Badness (Sintel): 1.87
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 625/624, 847/845, 1375/1372
 
Mapping: {{mapping| 1 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}
 
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=0| 37, 66b, 103, 140, 243e }}
 
Badness (Sintel): 1.13


== Quasimoha ==
== Quasimoha ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasimoha]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,025: Line 1,030:


{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
: mapping generators: ~2, ~49/40


: Mapping generators: ~2, ~49/40
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603
{{Optimal ET sequence|legend=1| 24c, 31, 117c, 148bc, 179bcd }}


{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
[[Badness]] (Sintel): 2.80
 
[[Badness]]: 0.110820


=== 11-limit ===
=== 11-limit ===
Line 1,041: Line 1,049:
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639
Optimal tunings:
* WE: ~2 = 1201.7630{{c}}, ~11/9 = 349.1510{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.6050{{c}}


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


Badness: 0.046181
Badness (Sintel): 1.53


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


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


[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}
[[Comma list]]: 2401/2400, 177147/175000
 
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}


: Mapping generators: ~2, ~3828125/2985984
{{Mapping|legend=1| 1 -17 -34 -20 | 0 22 43 27 }}
: mapping generators: ~2, ~9/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
* [[WE]]: ~2 = 1200.1458{{c}}, ~9/5 = 1013.7798{{c}}
: [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }}
: [[error map]]: {{val| +0.146 -1.277 +1.263 +0.314 }}
* [[CWE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6611{{c}}
: error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }}
: error map: {{val| 0.000 -1.410 +1.116 +0.025 }}


{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913 }}
{{Optimal ET sequence|legend=1| 45, 58, 103, 161 }}


[[Badness]] (Smith): 0.0597
[[Badness]] (Sintel): 3.18


=== 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: 243/242, 441/440, 43923/43750


Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}
Mapping: {{mapping| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* WE: ~2 = 1200.1491{{c}}, ~9/5 = 1013.7809{{c}}
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6593{{c}}


{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
{{Optimal ET sequence|legend=0| 45e, 58, 103, 161, 425b }}


Badness (Smith): 0.0262
Badness (Sintel): 1.32


=== 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: 243/242, 351/350, 441/440, 847/845


Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}
Mapping: {{mapping| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
* WE: ~2 = 1200.0928{{c}}, ~9/5 = 1013.7311{{c}}
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6556{{c}}


{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}


Badness (Smith): 0.0160
Badness (Sintel): 0.903


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
 
Mapping: {{mapping| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}
 
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|legend=0| 45ef, 58, 103, 161 }}
 
Badness (Sintel): 1.03
 
== Gorgik ==
{{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
 
[[Comma list]]: 2401/2400, 28672/28125
 
{{Mapping|legend=1| 1 -13 8 2 | 0 18 -7 1 }}
: mapping generators: ~2, ~7/4
 
[[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 ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
 
[[Badness]] (Sintel): 4.01
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 2401/2400, 2560/2541


Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
Mapping: {{mapping| 1 -13 8 2 14 | 0 18 -7 1 -13 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~77/60 = 431.107
* WE: ~2 = 1198.4615{{c}}, ~7/4 = 971.2535{{c}} (~8/7 = 227.2079{{c}})
* CWE: ~2 = 1200.000, ~77/60 = 431.108
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.4918{{c}} (~8/7 = 227.5082{{c}})


{{Optimal ET sequence|legend=0| 103, 167, 270 }}
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}


Badness (Smith): 0.0210
Badness (Sintel): 1.96


=== 2.3.5.7.11.13.17.41 subgroup ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13.17.41
Subgroup: 2.3.5.7.11.13


Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Comma list: 176/175, 196/195, 364/363, 512/507


Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}
Mapping: {{mapping| 1 -13 8 2 14 11 | 0 18 -7 1 -13 -9 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~41/32 = 431.107
* WE: ~2 = 1198.4012{{c}}, ~7/4 = 971.2110{{c}} (~8/7 = 227.1903{{c}})
* CWE: ~2 = 1200.000, ~41/32 = 431.111
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.5030{{c}} (~8/7 = 227.4970{{c}})
 
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bcef, 211bccdeeff }}


{{Optimal ET sequence|legend=0| 103, 167, 270 }}
Badness (Sintel): 1.33


== Hemigoldis ==
== Hemigoldis ==
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''
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.
 
Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other 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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,140: Line 1,181:
[[Comma list]]: 2401/2400, 549755813888/533935546875
[[Comma list]]: 2401/2400, 549755813888/533935546875


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


: mapping generators: ~2, ~7/4
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.2264{{c}}, ~8/7 = 229.1679{{c}}
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690
: [[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 }}


{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
Line 1,151: Line 1,195:


== 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 1,157: Line 1,201:
[[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 1,172: Line 1,219:
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: {{mapping| 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 1,185: Line 1,234:
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: {{mapping| 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 1,198: Line 1,249:
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: {{mapping| 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 1,211: Line 1,264:
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: {{mapping| 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:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"