View source for Breedsmic temperaments

{{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.

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: 
* ''[[Decimal]]'' (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* ''[[Beatles]]'' (+64/63 or 686/675) → [[Archytas clan #Beatles|Archytas clan]]
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic 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]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture 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 ====
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 ==
{{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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 65625/65536

{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}

: Mapping generators: ~2, ~256/245

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191

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

[[Badness]]: 0.012995

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

Comma list: 243/242, 441/440, 65625/65536

Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227

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

Badness: 0.035576

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

Comma list: 243/242, 441/440, 625/624, 3584/3575

Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}

Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203

Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}

Badness: 0.036876

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

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201

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

Badness: 0.027398

=== Tertia ===
Subgroup:2.3.5.7.11

Comma list: 385/384, 1331/1323, 1375/1372

Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}

Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173

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

Badness: 0.030171

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

Comma list: 352/351, 385/384, 625/624, 1331/1323

Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}

Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158

Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}

Badness: 0.028384

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

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162

Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}

Badness: 0.022416

=== Tertiaseptia ===
Subgroup: 2.3.5.7.11

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

Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169

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

Badness: 0.056926

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

Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400

Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}

Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168

Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}

Badness: 0.027474

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

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169

Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}

Badness: 0.018773

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: {{mapping| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}

Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169

Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}

Badness: 0.017653

==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168

Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}

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 }}

Badness: 0.009825

=== Hemitert ===
Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 65625/65536

Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}

: Mapping generators: ~2, ~45/44

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596

Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}

Badness: 0.015633

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

Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095

Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588

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

Badness: 0.033573

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

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589

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

Badness: 0.025298

=== Semitert ===
Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 65625/65536

Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}

: Mapping generators: ~99/70, ~256/245

Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193

Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}

Badness: 0.025790

== 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 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

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

{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}

: Mapping generators: ~2, ~875/512

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107

{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}

[[Badness]]: 0.035832

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

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

Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}

Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111

Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}

Badness: 0.017540

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

Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095

Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}

Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107

Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}

Badness: 0.017921

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

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}

: Mapping generators: ~2, ~189/160

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

{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}

[[Badness]]: 0.088221

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

Comma list: 243/242, 441/440, 35937/35840

Mapping: {{mapping| 1 3 12 8 7 | 0 -6 -41 -22 -15 }}

Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276

Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}

Badness: 0.027959

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

Comma list: 169/168, 243/242, 364/363, 441/440

Mapping: {{mapping| 1 3 12 8 7 7 | 0 -6 -41 -22 -15 -14 }}

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

Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }}

Badness: 0.026942

== Emmthird ==
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}

: Mapping generators: ~2, ~2187/1372

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988

{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}

[[Badness]]: 0.016736

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

Comma list: 243/242, 441/440, 1792000/1771561

Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}

Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991

Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}

Badness: 0.052358

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

Comma list: 243/242, 364/363, 441/440, 2200/2197

Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }}

Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989

Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}

Badness: 0.026974

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

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

Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}

Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985

Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}

Badness: 0.023205

== Quinmite ==
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

[[Comma list]]: 2401/2400, 1959552/1953125

{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}

: Mapping generators: ~2, ~42/25

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

{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}

[[Badness]]: 0.037322

== 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 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).

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}

: Mapping generators: ~2, ~6125/3888

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717

{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}

[[Badness]]: 0.075253

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

Comma list: 2401/2400, 3025/3024, 4000/3993

Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}

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

Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}

Badness: 0.022926

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

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

Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716

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

Badness: 0.020888

== 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

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

{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}

: mapping generators: ~2, ~49/40

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113

{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}

[[Badness]]: 0.041878

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

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

Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115

Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}

Badness: 0.019461

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

Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095

Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}

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

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

Badness: 0.013830

=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400

Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}

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

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

== 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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 2152828125/2147483648

{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}

: Mapping generators: ~2, ~28/15

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297

{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}

[[Badness]]: 0.044115

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

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 939524096/935859375

Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279

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

Badness: 0.090687

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

Comma list: 243/242, 441/440, 2200/2197, 3584/3575

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

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281

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

Badness: 0.045773

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

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281

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

Badness: 0.027322

== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 1224440064/1220703125

{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}

: Mapping generators: ~2, ~1296/875

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

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

[[Badness]]: 0.057632

== Subneutral ==
{{See also| Luna family }}

[[Subgroup]]: 2.3.5.7

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

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

: Mapping generators: ~2, ~57344/46875

[[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

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

{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}

: Mapping generators: ~2, ~2800/2187

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

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

[[Badness]]: 0.028307

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

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

{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}

: Mapping generators: ~2, ~8/7

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

{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}

[[Badness]]: 0.158384

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

Comma list: 176/175, 2401/2400, 2560/2541

Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}

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

Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}

Badness: 0.059260

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

Comma list: 176/175, 196/195, 364/363, 512/507

Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}

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

Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}

Badness: 0.032205

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

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

{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}

: Mapping generators: ~2, ~125/96

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310

{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}

Badness: 0.100511

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

Comma list: 385/384, 1375/1372, 43923/43750

Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}

Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318

Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}

Badness: 0.056514

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

Comma list: 385/384, 625/624, 847/845, 1375/1372

Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316

Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}

Badness: 0.027429

== 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 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.

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}

: Mapping generators: ~2, ~10/9

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343

{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}

[[Badness]]: 0.125672

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

Comma list: 243/242, 441/440, 43923/43750

Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345

Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}

Badness: 0.039962

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

Comma list: 243/242, 351/350, 441/440, 847/845

Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347

Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}

Badness: 0.021849

=== 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 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348

Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}

Badness: 0.020295

== Catafourth ==
{{See also| Sensipent family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 78732/78125

{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}

: Mapping generators: ~2, ~250/189

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235

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

Badness: 0.079579

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

Comma list: 243/242, 441/440, 78408/78125

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

Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252

Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}

Badness: 0.036785

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

Comma list: 243/242, 351/350, 441/440, 10985/10976

Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}

Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256

Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}

Badness: 0.021694

== Cotritone ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 390625/387072

{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}

: Mappping generators: ~2, ~10/7

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385

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

[[Badness]]: 0.098322

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

Comma list: 385/384, 1375/1372, 4000/3993

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

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387

Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}

Badness: 0.032225

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

Comma list: 169/168, 364/363, 385/384, 625/624

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

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387

Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }}

Badness: 0.028683

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

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}

: Mapping generators: ~2, ~49/40

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603

{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}

[[Badness]]: 0.110820

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

Comma list: 243/242, 441/440, 1815/1792

Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639

Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }}

Badness: 0.046181

== Lockerbie ==
: ''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. 

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

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

{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}

: Mapping generators: ~2, ~3828125/2985984

[[Optimal tuning]]s: 
* [[CTE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
: [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }}
* [[CWE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
: error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }}

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

[[Badness]] (Smith): 0.0597

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

Comma list: 2401/2400, 3025/3024, 766656/765625

Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}

Optimal tunings: 
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078

{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}

Badness (Smith): 0.0262

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

Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224

Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}

Optimal tunings: 
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069

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

Badness (Smith): 0.0160

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

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

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

Optimal tunings: 
* CTE: ~2 = 1200.000, ~77/60 = 431.107
* CWE: ~2 = 1200.000, ~77/60 = 431.108

{{Optimal ET sequence|legend=0| 103, 167, 270 }}

Badness (Smith): 0.0210

=== 2.3.5.7.11.13.17.41 subgroup ===
Subgroup: 2.3.5.7.11.13.17.41

Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224

Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}

Optimal tunings: 
* CTE: ~2 = 1200.000, ~41/32 = 431.107
* CWE: ~2 = 1200.000, ~41/32 = 431.111

{{Optimal ET sequence|legend=0| 103, 167, 270 }}

== Hemigoldis ==
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''

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

[[Comma list]]: 2401/2400, 549755813888/533935546875

{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}

: mapping generators: ~2, ~7/4

[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690

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

[[Badness]] (Sintel): 4.40

== 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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}

{{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }}

: Mapping generators: ~2, ~675/448

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719

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

[[Badness]]: 0.202249

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

Comma list: 2401/2400, 820125/819896, 2097152/2096325

Mapping: {{mapping| 1 43 -74 -25 36 | 0 -70 129 47 -55 }}

Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720

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

Badness: 0.052308

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

Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167

Mapping: {{mapping| 1 43 -74 -25 36 25 | 0 -70 129 47 -55 -36 }}

Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723

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

Badness: 0.032503

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

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 }}

Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722

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

Badness: 0.020995

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

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 }}

Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722

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

Badness: 0.013771

== Notes ==

[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 2]]