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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
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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 }}
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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 }}
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{{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]]
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"