Breedsmic temperaments: Difference between revisions

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This page discusses miscellaneous rank-2 temperaments 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]] [[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.
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:  
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{{Main| Hemififths }}
{{Main| Hemififths }}


Hemififths tempers out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator, with [[99edo|99EDO]] and [[140edo|140EDO]] providing good tunings, and [[239edo|239EDO]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5s, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7s. It may be called the 41&amp;58 temperament. 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}}.
Hemififths tempers out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator, with [[99edo]] and [[140edo]] providing 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 may be called the 41 &amp; 58 temperament. 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.
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 5120/5103
[[Comma list]]: 2401/2400, 5120/5103


[[Mapping]]: [{{val| 1 1 -5 -1 }}, {{val| 0 2 25 13 }}]
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}


{{Multival|legend=1| 2 25 13 35 15 -40 }}
{{Multival|legend=1| 2 25 13 35 15 -40 }}


[[POTE generator]]: ~49/40 = 351.477
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.477


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo|1/5 0 1/25}}
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
: [{{monzo|1 0 0 0}}, {{monzo|7/5 0 2/25 0}}, {{monzo|0 0 1 0}}, {{monzo|8/5 0 13/25 0}}]
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: Eigenmonzos: 2, 5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


[[Algebraic generator]]: (2 + sqrt(2))/2
[[Algebraic generator]]: (2 + sqrt(2))/2
Line 56: Line 56:
Comma list: 243/242, 441/440, 896/891
Comma list: 243/242, 441/440, 896/891


Mapping: [{{val| 1 1 -5 -1 2 }}, {{val| 0 2 25 13 5 }}]
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


POTE generator: ~11/9 = 351.521
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.521


{{Optimal ET sequence|legend=1| 17c, 41, 58, 99e }}
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99e }}
Line 69: Line 69:
Comma list: 144/143, 196/195, 243/242, 364/363
Comma list: 144/143, 196/195, 243/242, 364/363


Mapping: [{{val| 1 1 -5 -1 2 4 }}, {{val| 0 2 25 13 5 -1 }}]
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


POTE generator: ~11/9 = 351.573
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.573


{{Optimal ET sequence|legend=1| 17c, 41, 58, 99ef, 157eff }}
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99ef, 157eff }}
Line 82: Line 82:
Comma list: 2401/2400, 3388/3375, 5120/5103
Comma list: 2401/2400, 3388/3375, 5120/5103


Mapping: [{{val| 2 0 -35 -15 -47 }}, {{val| 0 2 25 13 34 }}]
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}


POTE generator: ~49/40 = 351.505
Optimal tuning (POTE): ~99/70 = 1\2, ~49/40 = 351.505


{{Optimal ET sequence|legend=1| 58, 140, 198 }}
{{Optimal ET sequence|legend=1| 58, 140, 198 }}
Line 95: Line 95:
Comma list: 352/351, 676/675, 847/845, 1716/1715
Comma list: 352/351, 676/675, 847/845, 1716/1715


Mapping: [{{val| 2 0 -35 -15 -47 -37 }}, {{val| 0 2 25 13 34 28 }}]
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


POTE generator: ~49/40 = 351.502
Optimal tuning (POTE): ~99/70 = 1\2, ~49/40 = 351.502


{{Optimal ET sequence|legend=1| 58, 140, 198, 536f }}
{{Optimal ET sequence|legend=1| 58, 140, 198, 536f }}
Line 110: Line 110:
Comma list: 2401/2400, 3025/3024, 5120/5103
Comma list: 2401/2400, 3025/3024, 5120/5103


Mapping: [{{val| 1 1 -5 -1 8 }}, {{val| 0 4 50 26 -31 }}]
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}


POTE generator: ~243/220 = 175.7378
Optimal tuning (POTE): ~2 = 1\1, ~243/220 = 175.7378


{{Optimal ET sequence|legend=1| 41, 157, 198, 239, 676b, 915be }}
{{Optimal ET sequence|legend=1| 41, 157, 198, 239, 676b, 915be }}
Line 123: Line 123:
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
Comma list: 352/351, 847/845, 2401/2400, 3025/3024


Mapping: [{{val| 1 1 -5 -1 8 10 }}, {{val| 0 4 50 26 -31 -43 }}]
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}


POTE generator: ~72/65 = 175.7470
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.7470


{{Optimal ET sequence|legend=1| 41, 157, 198, 437f, 635bcff }}
{{Optimal ET sequence|legend=1| 41, 157, 198, 437f, 635bcff }}
Line 134: Line 134:
{{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|171EDO]] makes for an excellent tuning. 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 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. 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


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


[[Mapping]]: [{{val| 1 3 2 3 }}, {{val| 0 -22 5 -3 }}]
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}


{{Multival|legend=1| 22 -5 3 -59 -57 21 }}
{{Multival|legend=1| 22 -5 3 -59 -57 21 }}


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


{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
Line 155: Line 155:
Comma list: 243/242, 441/440, 65625/65536
Comma list: 243/242, 441/440, 65625/65536


Mapping: [{{val| 1 3 2 3 7 }}, {{val| 0 -22 5 -3 -55 }}]
Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}


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


{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171, 202 }}
{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171, 202 }}
Line 168: Line 168:
Comma list: 243/242, 441/440, 625/624, 3584/3575
Comma list: 243/242, 441/440, 625/624, 3584/3575


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


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


{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171 }}
{{Optimal ET sequence|legend=1| 31, 109e, 140e, 171 }}
Line 181: Line 181:
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: [{{val| 1 3 2 3 7 1 1 }}, {{val| 0 -22 5 -3 -55 42 48 }}]
Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}


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


{{Optimal ET sequence|legend=1| 31, 109eg, 140e, 171 }}
{{Optimal ET sequence|legend=1| 31, 109eg, 140e, 171 }}
Line 194: Line 194:
Comma list: 385/384, 1331/1323, 1375/1372
Comma list: 385/384, 1331/1323, 1375/1372


Mapping: [{{val| 1 3 2 3 5 }}, {{val| 0 -22 5 -3 -24 }}]
Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}


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


{{Optimal ET sequence|legend=1| 31, 109, 140, 171e, 311e }}
{{Optimal ET sequence|legend=1| 31, 109, 140, 171e, 311e }}
Line 207: Line 207:
Comma list: 352/351, 385/384, 625/624, 1331/1323
Comma list: 352/351, 385/384, 625/624, 1331/1323


Mapping: [{{val| 1 3 2 3 5 1 }}, {{val| 0 -22 5 -3 -24 42 }}]
Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}


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


{{Optimal ET sequence|legend=1| 31, 109, 140, 311e, 451ee }}
{{Optimal ET sequence|legend=1| 31, 109, 140, 311e, 451ee }}
Line 220: Line 220:
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: [{{val| 1 3 2 3 5 1 1 }}, {{val| 0 -22 5 -3 -24 42 48 }}]
Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}


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


{{Optimal ET sequence|legend=1| 31, 109g, 140, 311e, 451ee }}
{{Optimal ET sequence|legend=1| 31, 109g, 140, 311e, 451ee }}
Line 233: Line 233:
Comma list: 2401/2400, 6250/6237, 65625/65536
Comma list: 2401/2400, 6250/6237, 65625/65536


Mapping: [{{val|1 3 2 3 -4}}, {{val|0 -22 5 -3 116}}]
Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}


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


{{Optimal ET sequence|legend=1| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
{{Optimal ET sequence|legend=1| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
Line 246: Line 246:
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400


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


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


{{Optimal ET sequence|legend=1| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
{{Optimal ET sequence|legend=1| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
Line 259: Line 259:
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: [{{val|1 3 2 3 -4 1 1}}, {{val|0 -22 5 -3 116 42 48}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}


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


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Line 272: Line 272:
Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197
Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197


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


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


{{Optimal ET sequence|legend=1| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
{{Optimal ET sequence|legend=1| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
Line 285: Line 285:
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215


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


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


{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfgg }}
{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfgg }}
Line 298: Line 298:
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3 1}}, {{val|0 -22 5 -3 116 42 48 -105 117 60}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}


POTE generator: ~23/22 = 77.167
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167


{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfggj }}
{{Optimal ET sequence|legend=1| 140, 311, 762g, 1073g, 1384cfggj }}
Line 311: Line 311:
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3 1 11}}, {{val|0 -22 5 -3 116 42 48 -105 117 60 -94}}]
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}


POTE generator: ~23/22 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Line 324: Line 324:
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014


Mapping: [{{val|1 3 2 3 -4 1 1 11 -3 1 11 0}}, {{val|0 -22 5 -3 116 42 48 -105 117 60 -94 81}}]
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 }}


POTE generator: ~23/22 = 77.170
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Line 337: Line 337:
Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930
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: [{{val|1 3 2 3 -4 1 1 11 -3 1 11 0 6}}, {{val|0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10}}]
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 }}


POTE generator: ~23/22 = 77.169
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169


{{Optimal ET sequence|legend=1| 140, 171, 311 }}
{{Optimal ET sequence|legend=1| 140, 171, 311 }}
Line 350: Line 350:
Comma list: 2401/2400, 3025/3024, 65625/65536
Comma list: 2401/2400, 3025/3024, 65625/65536


Mapping: [{{val| 1 3 2 3 6 }}, {{val| 0 -44 10 -6 -79 }}]
Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}


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


{{Optimal ET sequence|legend=1| 31, 280, 311, 342 }}
{{Optimal ET sequence|legend=1| 31, 280, 311, 342 }}
Line 363: Line 363:
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095


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


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


{{Optimal ET sequence|legend=1| 31, 280, 311, 964f, 1275f, 1586cff }}
{{Optimal ET sequence|legend=1| 31, 280, 311, 964f, 1275f, 1586cff }}
Line 376: Line 376:
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: [{{val| 1 3 2 3 6 1 1 }}, {{val| 0 -44 10 -6 -79 84 96 }}]
Mapping: {{mapping| 1 3 2 3 6 1 1 | 0 -44 10 -6 -79 84 96 }}


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


{{Optimal ET sequence|legend=1| 31, 280, 311, 653f, 964f }}
{{Optimal ET sequence|legend=1| 31, 280, 311, 653f, 964f }}
Line 389: Line 389:
Comma list: 2401/2400, 9801/9800, 65625/65536
Comma list: 2401/2400, 9801/9800, 65625/65536


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


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


{{Optimal ET sequence|legend=1| 62e, 140, 202, 342 }}
{{Optimal ET sequence|legend=1| 62e, 140, 202, 342 }}
Line 398: Line 398:


== Quasiorwell ==
== Quasiorwell ==
In addition to 2401/2400, quasiorwell tempers out 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 7s, or 384<sup>1/38</sup>, giving pure fifths.
In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 22 -1 -10 1 }}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 &amp; 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.


Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning.
Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 31 0 9}}, {{val|0 -38 3 -8}}]
{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}


{{Multival|legend=1| 38 -3 8 -93 -94 27 }}
{{Multival|legend=1| 38 -3 8 -93 -94 27 }}


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


{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
Line 421: Line 421:
Comma list: 2401/2400, 3025/3024, 5632/5625
Comma list: 2401/2400, 3025/3024, 5632/5625


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


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


{{Optimal ET sequence|legend=1| 31, 208, 239, 270 }}
{{Optimal ET sequence|legend=1| 31, 208, 239, 270 }}
Line 434: Line 434:
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095


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


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


{{Optimal ET sequence|legend=1| 31, 239, 270, 571, 841, 1111 }}
{{Optimal ET sequence|legend=1| 31, 239, 270, 571, 841, 1111 }}
Line 492: Line 492:
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 3 12 8}}, {{val|0 -6 -41 -22}}]
{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}


{{Multival|legend=1|6 41 22 51 18 -64}}
{{Multival|legend=1| 6 41 22 51 18 -64 }}


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


{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
Line 511: Line 511:
Comma list: 243/242, 441/440, 35937/35840
Comma list: 243/242, 441/440, 35937/35840


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


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


{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
Line 524: Line 524:
Comma list: 169/168, 243/242, 364/363, 441/440
Comma list: 169/168, 243/242, 364/363, 441/440


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


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


{{Optimal ET sequence|legend=1| 72, 161f, 233f }}
{{Optimal ET sequence|legend=1| 72, 161f, 233f }}
Line 535: Line 535:
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 -3 -17 -8}}, {{val|0 14 59 33}}]
{{Mapping|legend=1| 1 -3 -17 -8 | 0 14 59 33 }}


{{Multival|legend=1|14 59 33 61 13 -89}}
{{Multival|legend=1|14 59 33 61 13 -89}}


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


{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
Line 554: Line 554:
Comma list: 243/242, 441/440, 1792000/1771561
Comma list: 243/242, 441/440, 1792000/1771561


Mapping: [{{val|1 -3 -17 -8 -8}}, {{val|0 14 59 33 35}}]
Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}


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


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
{{Optimal ET sequence|legend=1| 58, 113, 171 }}
Line 567: Line 567:
Comma list: 243/242, 364/363, 441/440, 2200/2197
Comma list: 243/242, 364/363, 441/440, 2200/2197


Mapping: [{{val|1 -3 -17 -8 -8 -13}}, {{val|0 14 59 33 35 51}}]
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}


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


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
{{Optimal ET sequence|legend=1| 58, 113, 171 }}
Line 580: Line 580:
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197


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


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


{{Optimal ET sequence|legend=1| 58, 113, 171 }}
{{Optimal ET sequence|legend=1| 58, 113, 171 }}
Line 589: Line 589:


== Quinmite ==
== 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>.
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
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 -7 -5 -3}}, {{val|0 34 29 23}}]
{{Mapping|legend=1| 1 -7 -5 -3 | 0 34 29 23 }}


{{Multival|legend=1|34 29 23 -33 -59 -28}}
{{Multival|legend=1| 34 29 23 -33 -59 -28 }}


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


{{Optimal ET sequence|legend=1| 95, 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
{{Optimal ET sequence|legend=1| 95, 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
Line 608: Line 608:
The generator for unthirds temperament is undecimal major third, 14/11.
The generator for unthirds temperament is undecimal major third, 14/11.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 -13 -14 -9}}, {{val|0 42 47 34}}]
{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}


{{Multival|legend=1|42 47 34 -23 -64 -53}}
{{Multival|legend=1| 42 47 34 -23 -64 -53 }}


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


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
Line 627: Line 627:
Comma list: 2401/2400, 3025/3024, 4000/3993
Comma list: 2401/2400, 3025/3024, 4000/3993


Mapping: [{{val|1 -13 -14 -9 -8}}, {{val|0 42 47 34 33}}]
Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}


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


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 1316c }}
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 1316c }}
Line 640: Line 640:
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400


Mapping: [{{val|1 -13 -14 -9 -9 -47}}, {{val|0 42 47 34 33 146}}]
Mapping: {{mapping| 1 -13 -14 -9 -9 -47 | 0 42 47 34 33 146 }}


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


{{Optimal ET sequence|legend=1| 72, 311, 694, 1005c, 1699cd }}
{{Optimal ET sequence|legend=1| 72, 311, 694, 1005c, 1699cd }}
Line 651: Line 651:
This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]].
This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


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


{{Multival|legend=1|2 -57 -28 -95 -50 95}}
{{Multival|legend=1| 2 -57 -28 -95 -50 95 }}


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


{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bbcc }}
{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bbcc }}
Line 670: Line 670:
Comma list: 2401/2400, 3025/3024, 19712/19683
Comma list: 2401/2400, 3025/3024, 19712/19683


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


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


{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}
{{Optimal ET sequence|legend=1| 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b }}
Line 683: Line 683:
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095


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


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


{{Optimal ET sequence|legend=1| 41, 229, 270, 581, 851, 2283b, 3134b }}
{{Optimal ET sequence|legend=1| 41, 229, 270, 581, 851, 2283b, 3134b }}
Line 696: Line 696:
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. 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
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 -1 6 4 }}, {{val| 0 26 -37 -12 }}]
{{Mapping|legend=1| 1 -1 6 4 | 0 26 -37 -12 }}


{{Multival|legend=1|26 -37 -12 -119 -92 76}}
{{Multival|legend=1| 26 -37 -12 -119 -92 76 }}


[[POTE generator]]: ~15/14 = 119.297
[[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 }}
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
Line 717: Line 717:
Comma list: 243/242, 441/440, 939524096/935859375
Comma list: 243/242, 441/440, 939524096/935859375


Mapping: [{{val| 1 -1 6 4 -3 }}, {{val| 0 26 -37 -12 65 }}]
Mapping: {{mapping| 1 -1 6 4 -3 | 0 26 -37 -12 65 }}


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


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332 }}
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332 }}
Line 730: Line 730:
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
Comma list: 243/242, 441/440, 2200/2197, 3584/3575


Mapping: [{{val| 1 -1 6 4 -3 4 }}, {{val| 0 26 -37 -12 65 -3 }}]
Mapping: {{mapping| 1 -1 6 4 -3 4 | 0 26 -37 -12 65 -3 }}


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


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 835eeff }}
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 835eeff }}
Line 743: Line 743:
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: [{{val| 1 -1 6 4 -3 4 2 }}, {{val| 0 26 -37 -12 65 -3 21 }}]
Mapping: {{mapping| 1 -1 6 4 -3 4 2 | 0 26 -37 -12 65 -3 21 }}


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


{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 503ef, 835eeff }}
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 332, 503ef, 835eeff }}
Line 752: Line 752:


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 31 34 26 }}, {{val| 0 -52 -56 -41 }}]
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}


{{Multival|legend=1|52 56 41 -32 -81 -62}}
{{Multival|legend=1| 52 56 41 -32 -81 -62 }}


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


{{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 }}
Line 771: Line 771:
{{See also| Luna family }}
{{See also| Luna family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 19 0 6 }}, {{val| 0 -60 8 -11 }}]
{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}


{{Multival|legend=1|60 -8 11 -152 -151 48}}
{{Multival|legend=1| 60 -8 11 -152 -151 48 }}


[[POTE generator]]: ~57344/46875 = 348.301
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301


{{Optimal ET sequence|legend=1| 31, 348, 379, 410, 441, 1354, 1795, 2236 }}
{{Optimal ET sequence|legend=1| 31, 348, 379, 410, 441, 1354, 1795, 2236 }}
Line 788: Line 788:
{{See also| Metric microtemperaments #Geb }}
{{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, 31381059609/31360000000


[[Mapping]]: [{{val| 1 13 33 21 }}, {{val| 0 -32 -86 -51 }}]
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}


{{Multival|legend=1|32 86 51 62 -9 -123}}
{{Multival|legend=1| 32 86 51 62 -9 -123 }}


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


{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696, 6955dd }}
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696, 6955dd }}
Line 803: Line 803:


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


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


[[Mapping]]: [{{val| 1 5 1 3 }}, {{val| 0 -18 7 -1 }}]
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}


{{Multival|legend=1|18 -7 1 -53 -49 22}}
{{Multival|legend=1| 18 -7 1 -53 -49 22 }}


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


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
Line 822: Line 822:
Comma list: 176/175, 2401/2400, 2560/2541
Comma list: 176/175, 2401/2400, 2560/2541


Mapping: [{{val| 1 5 1 3 1 }}, {{val| 0 -18 7 -1 13 }}]
Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}


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


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
Line 835: Line 835:
Comma list: 176/175, 196/195, 364/363, 512/507
Comma list: 176/175, 196/195, 364/363, 512/507


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


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


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bcef, 211bccdeeff }}
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bcef, 211bccdeeff }}
Line 844: Line 844:


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


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


[[Mapping]]: [{{val| 1 19 8 10 }}, {{val| 0 -46 -15 -19 }}]
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}


{{Multival|legend=1|46 15 19 -83 -99 2}}
{{Multival|legend=1| 46 15 19 -83 -99 2 }}


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


{{Optimal ET sequence|legend=1| 37, 103, 140, 243, 383, 1009cd, 1392ccd }}
{{Optimal ET sequence|legend=1| 37, 103, 140, 243, 383, 1009cd, 1392ccd }}
Line 863: Line 863:
Comma list: 385/384, 1375/1372, 43923/43750
Comma list: 385/384, 1375/1372, 43923/43750


Mapping: [{{val| 1 19 8 10 8 }}, {{val| 0 -46 -15 -19 -12 }}]
Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}


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


{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
Line 876: Line 876:
Comma list: 385/384, 625/624, 847/845, 1375/1372
Comma list: 385/384, 625/624, 847/845, 1375/1372


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


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


{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
{{Optimal ET sequence|legend=1| 37, 103, 140, 243e }}
Line 885: Line 885:


== Mintone ==
== 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.
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
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 5 9 7 }}, {{val| 0 -22 -43 -27 }}]
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}


{{Multival|legend=1|22 43 27 17 -19 -58}}
{{Multival|legend=1| 22 43 27 17 -19 -58 }}


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


{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
Line 906: Line 906:
Comma list: 243/242, 441/440, 43923/43750
Comma list: 243/242, 441/440, 43923/43750


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


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


{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586b, 747bc }}
{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586b, 747bc }}
Line 919: Line 919:
Comma list: 243/242, 351/350, 441/440, 847/845
Comma list: 243/242, 351/350, 441/440, 847/845


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


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


{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586bf }}
{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586bf }}
Line 932: Line 932:
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845


Mapping: [{{val| 1 5 9 7 12 11 3 }}, {{val| 0 -22 -43 -27 -55 -47 7 }}]
Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}


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


{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586bf }}
{{Optimal ET sequence|legend=1| 58, 103, 161, 425b, 586bf }}
Line 941: Line 941:


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 13 17 13 }}, {{val| 0 -28 -36 -25 }}]
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}


{{Multival|legend=1| 28 36 25 -8 -39 -43 }}
{{Multival|legend=1| 28 36 25 -8 -39 -43 }}


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


{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
Line 962: Line 962:
Comma list: 243/242, 441/440, 78408/78125
Comma list: 243/242, 441/440, 78408/78125


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


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


{{Optimal ET sequence|legend=1| 103, 130, 233, 363, 493e, 856be }}
{{Optimal ET sequence|legend=1| 103, 130, 233, 363, 493e, 856be }}
Line 975: Line 975:
Comma list: 243/242, 351/350, 441/440, 10985/10976
Comma list: 243/242, 351/350, 441/440, 10985/10976


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


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


{{Optimal ET sequence|legend=1| 103, 130, 233, 363 }}
{{Optimal ET sequence|legend=1| 103, 130, 233, 363 }}
Line 984: Line 984:


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


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


[[Mapping]]: [{{val| 1 -13 -4 -4 }}, {{val| 0 30 13 14 }}]
{{Mapping|legend=1| 1 -13 -4 -4 | 0 30 13 14 }}


{{Multival|legend=1|30 13 14 -49 -62 -4}}
{{Multival|legend=1| 30 13 14 -49 -62 -4 }}


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


{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
Line 1,003: Line 1,003:
Comma list: 385/384, 1375/1372, 4000/3993
Comma list: 385/384, 1375/1372, 4000/3993


Mapping: [{{val| 1 -13 -4 -4 2 }}, {{val| 0 30 13 14 3 }}]
Mapping: {{mapping| 1 -13 -4 -4 2 | 0 30 13 14 3 }}


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


{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
Line 1,016: Line 1,016:
Comma list: 169/168, 364/363, 385/384, 625/624
Comma list: 169/168, 364/363, 385/384, 625/624


Mapping: [{{val| 1 -13 -4 -4 2 -7 }}, {{val| 0 30 13 14 3 22 }}]
Mapping: {{mapping| 1 -13 -4 -4 2 -7 | 0 30 13 14 3 22 }}


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


{{Optimal ET sequence|legend=1| 37, 72, 109, 181f }}
{{Optimal ET sequence|legend=1| 37, 72, 109, 181f }}
Line 1,027: Line 1,027:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{Val|1 1 9 6}}, {{Val|0 2 -23 -11}}]
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}


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


{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
Line 1,044: Line 1,044:
Comma list: 243/242, 441/440, 1815/1792
Comma list: 243/242, 441/440, 1815/1792


Mapping: [{{Val|1 1 9 6 2}}, {{Val|0 2 -23 -11 5}}]
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}


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


{{Optimal ET sequence|legend=1| 31, 86ce, 117ce, 148bce }}
{{Optimal ET sequence|legend=1| 31, 86ce, 117ce, 148bce }}
Line 1,059: Line 1,059:
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}


[[Mapping]]: {{val| 1 43 -74 -25 }}, {{val| 0 -70 129 47 }}
{{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }}


Mapping generators: ~2, ~675/448
: mapping generators: ~2, ~675/448


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


{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}
Line 1,074: Line 1,074:
Comma list: 2401/2400, 820125/819896, 2097152/2096325
Comma list: 2401/2400, 820125/819896, 2097152/2096325


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


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


{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 1795 }}
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 1795 }}
Line 1,087: Line 1,087:
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167


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


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


{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 1795f }}
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 1795f }}
Line 1,100: Line 1,100:
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: {{val| 1 43 -74 -25 36 25 -103 }}, {{val| 0 -70 129 47 -55 -36 181 }}
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 | 0 -70 129 47 -55 -36 181 }}


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


{{Optimal ET sequence|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
{{Optimal ET sequence|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
Line 1,113: Line 1,113:
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: {{val| 1 43 -74 -25 36 25 -103 -49 }}, {{val| 0 -70 129 47 -55 -36 181 90 }}
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 -49 | 0 -70 129 47 -55 -36 181 90 }}


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


{{Optimal ET sequence|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
{{Optimal ET sequence|legend=1| 120g, 191g, 311, 431, 742, 1795f }}
Line 1,123: Line 1,123:
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 03:04, 18 September 2023

This page discusses miscellaneous rank-2 temperaments tempering out the breedsma, [-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)2 = 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:

Hemififths

Main article: Hemififths

Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with 99edo and 140edo providing good tunings, and 239edo an even better one; and other possible tunings are 160(1/25), giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14(1/13), giving just 7's. It may be called the 41 & 58 temperament. 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[clarification needed].

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: [⟨1 1 -5 -1], ⟨0 2 25 13]]

Wedgie: ⟨⟨ 2 25 13 35 15 -40 ]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.477

Minimax tuning:

[[1 0 0 0⟩, [7/5 0 2/25 0⟩, [0 0 1 0⟩, [8/5 0 13/25 0⟩]
Eigenmonzo (unchanged-interval) basis: 2.5

Algebraic generator: (2 + sqrt(2))/2

Optimal ET sequence: 41, 58, 99, 239, 338

Badness: 0.022243

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 896/891

Mapping: [⟨1 1 -5 -1 2], ⟨0 2 25 13 5]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.521

Optimal ET sequence: 17c, 41, 58, 99e

Badness: 0.023498

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242, 364/363

Mapping: [⟨1 1 -5 -1 2 4], ⟨0 2 25 13 5 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.573

Optimal ET sequence: 17c, 41, 58, 99ef, 157eff

Badness: 0.019090

Semihemi

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3388/3375, 5120/5103

Mapping: [⟨2 0 -35 -15 -47], ⟨0 2 25 13 34]]

Optimal tuning (POTE): ~99/70 = 1\2, ~49/40 = 351.505

Optimal ET sequence: 58, 140, 198

Badness: 0.042487

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 1716/1715

Mapping: [⟨2 0 -35 -15 -47 -37], ⟨0 2 25 13 34 28]]

Optimal tuning (POTE): ~99/70 = 1\2, ~49/40 = 351.502

Optimal ET sequence: 58, 140, 198, 536f

Badness: 0.021188

Quadrafifths

This has been logged as semihemififths in Graham Breed's temperament finder, but quadrafifths arguably makes more sense.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 5120/5103

Mapping: [⟨1 1 -5 -1 8], ⟨0 4 50 26 -31]]

Optimal tuning (POTE): ~2 = 1\1, ~243/220 = 175.7378

Optimal ET sequence: 41, 157, 198, 239, 676b, 915be

Badness: 0.040170

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 2401/2400, 3025/3024

Mapping: [⟨1 1 -5 -1 8 10], ⟨0 4 50 26 -31 -43]]

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.7470

Optimal ET sequence: 41, 157, 198, 437f, 635bcff

Badness: 0.031144

Tertiaseptal

Main article: 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 & 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. 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: [⟨1 3 2 3], ⟨0 -22 5 -3]]

Wedgie: ⟨⟨ 22 -5 3 -59 -57 21 ]]

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

Optimal ET sequence: 31, 109, 140, 171

Badness: 0.012995

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 2 3 7], ⟨0 -22 5 -3 -55]]

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

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: [⟨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: 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: [⟨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: 31, 109eg, 140e, 171

Badness: 0.027398

Tertia

Subgroup:2.3.5.7.11

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

Mapping: [⟨1 3 2 3 5], ⟨0 -22 5 -3 -24]]

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

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: [⟨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: 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: [⟨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: 31, 109g, 140, 311e, 451ee

Badness: 0.022416

Tertiaseptia

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 2 3 -4], ⟨0 -22 5 -3 116]]

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

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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 140, 171, 311

Badness: 0.009825

Hemitert

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 2 3 6], ⟨0 -44 10 -6 -79]]

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

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: [⟨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: 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: [⟨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: 31, 280, 311, 653f, 964f

Badness: 0.025298

Semitert

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 6 4 6 1], ⟨0 -22 5 -3 46]]

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

Optimal ET sequence: 62e, 140, 202, 342

Badness: 0.025790

Quasiorwell

In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = [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 & 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)1/8, giving just 7's, or 3841/38, giving pure fifths.

Adding 3025/3024 extends to the 11-limit and gives ⟨⟨ 38 -3 8 64 … ]] for the initial wedgie, and as expected, 270 remains an excellent tuning.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 29360128/29296875

Mapping: [⟨1 31 0 9], ⟨0 -38 3 -8]]

Wedgie: ⟨⟨ 38 -3 8 -93 -94 27 ]]

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

Optimal ET sequence: 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: [⟨1 31 0 9 53], ⟨0 -38 3 -8 -64]]

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

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: [⟨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: 31, 239, 270, 571, 841, 1111

Badness: 0.017921

Decoid

See also: Quintosec family § Decoid

Decoid tempers out 2401/2400 and 67108864/66976875, as well as the linus comma, [11 -10 -10 10⟩. Either 8/7 or 16/15 can be used its generator. It may be described as the 130 & 270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the quintosec temperament.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 67108864/66976875

Mapping: [⟨10 0 47 36], ⟨0 2 -3 -1]]

mapping generators: ~15/14, ~8192/4725

Wedgie: ⟨⟨ 20 -30 -10 -94 -72 61 ]]

Optimal tuning (POTE): ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099)

Optimal ET sequence: 10, 120, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc

Badness: 0.033902

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 5632/5625, 9801/9800

Mapping: [⟨10 0 47 36 98], ⟨0 2 -3 -1 -8]]

Optimal tuning (POTE): ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070)

Optimal ET sequence: 10e, 130, 270, 670, 940, 1210, 2150c

Badness: 0.018735

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095

Mapping: [⟨10 0 47 36 98 37], ⟨0 2 -3 -1 -8 0]]

Optimal tuning (POTE): ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083)

Optimal ET sequence: 10e, 130, 270, 940, 1210f, 1480cf

Badness: 0.013475

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: [⟨1 3 12 8], ⟨0 -6 -41 -22]]

Wedgie: ⟨⟨ 6 41 22 51 18 -64 ]]

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

Optimal ET sequence: 72, 161, 233, 305

Badness: 0.088221

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 12 8 7], ⟨0 -6 -41 -22 -15]]

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

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: [⟨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: 72, 161f, 233f

Badness: 0.026942

Emmthird

The generator for emmthird temperament 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: [⟨1 -3 -17 -8], ⟨0 14 59 33]]

Wedgie: ⟨⟨ 14 59 33 61 13 -89 ]]

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

Optimal ET sequence: 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: [⟨1 -3 -17 -8 -8], ⟨0 14 59 33 35]]

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

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: [⟨1 -3 -17 -8 -8 -13], ⟨0 14 59 33 35 51]]

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

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: [⟨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: 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[1][2].

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1959552/1953125

Mapping: [⟨1 -7 -5 -3], ⟨0 34 29 23]]

Wedgie: ⟨⟨ 34 29 23 -33 -59 -28 ]]

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

Optimal ET sequence: 95, 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc

Badness: 0.037322

Unthirds

The generator for unthirds temperament is undecimal major third, 14/11.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 68359375/68024448

Mapping: [⟨1 -13 -14 -9], ⟨0 42 47 34]]

Wedgie: ⟨⟨ 42 47 34 -23 -64 -53 ]]

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

Optimal ET sequence: 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: [⟨1 -13 -14 -9 -8], ⟨0 42 47 34 33]]

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

Optimal ET sequence: 72, 167, 239, 311, 1316c

Badness: 0.022926

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -13 -14 -9 -9 -47], ⟨0 42 47 34 33 146]]

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

Optimal ET sequence: 72, 311, 694, 1005c, 1699cd

Badness: 0.020888

Newt

This temperament has a generator of neutral third (0.2 cents flat of 49/40) and tempers out the garischisma.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 33554432/33480783

Mapping: [⟨1 1 19 11], ⟨0 2 -57 -28]]

Wedgie: ⟨⟨ 2 -57 -28 -95 -50 95 ]]

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

Optimal ET sequence: 41, 188, 229, 270, 1121, 1391, 1661, 1931, 2201, 6333bbcc

Badness: 0.041878

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 19 11 -10], ⟨0 2 -57 -28 46]]

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

Optimal ET sequence: 41, 188, 229, 270, 581, 851, 1121, 1972, 3093b, 4214b

Badness: 0.019461

13-limit

Subgroup: 2.3.5.7.11.13

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

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: 41, 229, 270, 581, 851, 2283b, 3134b

Badness: 0.013830

Septidiasemi

Main article: 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: [⟨1 -1 6 4], ⟨0 26 -37 -12]]

Wedgie: ⟨⟨ 26 -37 -12 -119 -92 76 ]]

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

Optimal ET sequence: 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd

Badness: 0.044115

Sedia

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

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

Mapping: [⟨1 -1 6 4 -3], ⟨0 26 -37 -12 65]]

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

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: [⟨1 -1 6 4 -3 4], ⟨0 26 -37 -12 65 -3]]

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

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: [⟨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 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: [⟨1 31 34 26], ⟨0 -52 -56 -41]]

Wedgie: ⟨⟨ 52 56 41 -32 -81 -62 ]]

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

Optimal ET sequence: 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: [⟨1 19 0 6], ⟨0 -60 8 -11]]

Wedgie: ⟨⟨ 60 -8 11 -152 -151 48 ]]

Optimal tuning (POTE): ~2 = 1\1, ~57344/46875 = 348.301

Optimal ET sequence: 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: [⟨1 13 33 21], ⟨0 -32 -86 -51]]

Wedgie: ⟨⟨ 32 86 51 62 -9 -123 ]]

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

Optimal ET sequence: 157, 171, 1012, 1183, 1354, 1525, 1696, 6955dd

Badness: 0.028307

Gorgik

Subgroup: 2.3.5.7

Comma list: 2401/2400, 28672/28125

Mapping: [⟨1 5 1 3], ⟨0 -18 7 -1]]

Wedgie: ⟨⟨ 18 -7 1 -53 -49 22 ]]

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

Optimal ET sequence: 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: [⟨1 5 1 3 1], ⟨0 -18 7 -1 13]]

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

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: [⟨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: 21, 37, 58, 153bcef, 211bccdeeff

Badness: 0.032205

Fibo

Subgroup: 2.3.5.7

Comma list: 2401/2400, 341796875/339738624

Mapping: [⟨1 19 8 10], ⟨0 -46 -15 -19]]

Wedgie: ⟨⟨ 46 15 19 -83 -99 2 ]]

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

Optimal ET sequence: 37, 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: [⟨1 19 8 10 8], ⟨0 -46 -15 -19 -12]]

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

Optimal ET sequence: 37, 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: [⟨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: 37, 103, 140, 243e

Badness: 0.027429

Mintone

In addition to 2401/2400, mintone tempers out 177147/175000 = [-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 & 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: [⟨1 5 9 7], ⟨0 -22 -43 -27]]

Wedgie: ⟨⟨ 22 43 27 17 -19 -58 ]]

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

Optimal ET sequence: 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: [⟨1 5 9 7 12], ⟨0 -22 -43 -27 -55]]

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

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: [⟨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: 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: [⟨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: 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: [⟨1 13 17 13], ⟨0 -28 -36 -25]]

Wedgie: ⟨⟨ 28 36 25 -8 -39 -43 ]]

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

Optimal ET sequence: 27, 76, 103, 130

Badness: 0.079579

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 13 17 13 32], ⟨0 -28 -36 -25 -70]]

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

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: [⟨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: 103, 130, 233, 363

Badness: 0.021694

Cotritone

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625/387072

Mapping: [⟨1 -13 -4 -4], ⟨0 30 13 14]]

Wedgie: ⟨⟨ 30 13 14 -49 -62 -4 ]]

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

Optimal ET sequence: 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: [⟨1 -13 -4 -4 2], ⟨0 30 13 14 3]]

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

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: [⟨1 -13 -4 -4 2 -7], ⟨0 30 13 14 3 22]]

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

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: [⟨1 1 9 6], ⟨0 2 -23 -11]]

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

Optimal ET sequence: 31, 117c, 148bc, 179bc

Badness: 0.110820

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 9 6 2], ⟨0 2 -23 -11 5]]

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

Optimal ET sequence: 31, 86ce, 117ce, 148bce

Badness: 0.046181

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, [93 -32 -17 -1⟩

Mapping: [⟨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: 120, 191, 311, 742, 1053, 2848, 3901

Badness: 0.202

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 43 -74 -25 36], ⟨0 -70 129 47 -55]]

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

Optimal ET sequence: 120, 191, 311, 742, 1053, 1795

Badness: 0.0523

13-limit

Subgroup: 2.3.5.7.11.13

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

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: 120, 191, 311, 742, 1053, 1795f

Badness: 0.0325

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619

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: 120g, 191g, 311, 431, 742, 1795f

Badness: 0.0325

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: [⟨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: 120g, 191g, 311, 431, 742, 1795f

Badness: 0.0138

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
  2. ↑ Yahoo! Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
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