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Breedsmic temperaments: Difference between revisions

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* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]


== Hemififths ==
Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, newt, unthirds, septidiasemi, subneutral, maviloid, lockerbie, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  
{{Main| Hemififths }}
 
Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.
 
By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 5120/5103
 
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
: mapping generators: ~2, ~49/40
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7412{{c}}, ~49/40 = 351.4016{{c}}
: [[error map]]: {{val| -0.259 +0.590 +0.021 -0.346 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4671{{c}}
: error map: {{val| 0.000 +0.979 +0.364 +0.246 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 
[[Algebraic generator]]: (2 + sqrt(2))/2
 
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338 }}
 
[[Badness]] (Sintel): 0.563
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 896/891
 
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.2845{{c}}, ~11/9 = 351.3110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4956{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
 
Badness (Sintel): 0.777
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 243/242, 364/363
 
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1198.8875{{c}}, ~11/9 = 351.2475{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.5438{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
 
Badness (Sintel): 0.789
 
=== Semihemi ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3388/3375, 5120/5103
 
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
: mapping generators: ~99/70, ~400/231
 
Optimal tunings:
* WE: ~99/70 = 599.8556{{c}}, ~400/231 = 951.2757{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~400/231 = 951.4939{{c}}
 
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
 
Badness (Sintel): 1.40
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 676/675, 847/845, 1716/1715
 
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8513{{c}}, ~26/15 = 951.2662{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/15 = 951.4905{{c}}
 
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
 
Badness (Sintel): 0.876
 
=== Quadrafifths ===
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5120/5103
 
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
: mapping generators: ~2, ~243/220
 
Optimal tunings:
* WE: ~2 = 1199.7520{{c}}, ~243/220 = 175.7015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7360{{c}}
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
 
Badness (Sintel): 1.33
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
 
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
 
Optimal tunings:
* WE: ~2 = 1199.6502{{c}}, ~72/65 = 175.6957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7461{{c}}
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}
 
Badness (Sintel): 1.29


== Tertiaseptal ==
== Tertiaseptal ==
Line 460: Line 337:
Badness (Sintel): 0.853
Badness (Sintel): 0.853


== Quasiorwell ==
== Emmthird ==
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 & 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.
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
 
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.


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


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


{{Mapping|legend=1| 1 -7 3 1 | 0 38 -3 8 }}
{{Mapping|legend=1| 1 -3 -17 -8 | 0 14 59 33 }}
: mapping generators: ~2, ~1024/875
: mapping generators: ~2, ~2744/2187


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9403{{c}}, ~1024/875 = 271.0935{{c}}
* [[WE]]: ~2 = 1200.0435{{c}}, ~2744/2187 = 393.0021{{c}}
: [[error map]]: {{val| -0.060 +0.018 +0.226 -0.137 }}
: [[error map]]: {{val| +0.043 -0.057 +0.069 -0.106 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1024/875 = 271.1064{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2744/2187 = 392.9887{{c}}
: error map: {{val| 0.000 +0.087 +0.367 +0.025 }}
: error map: {{val| 0.000 -0.113 +0.022 -0.197 }}


{{Optimal ET sequence|legend=1| 31, …, 177, 208, 239, 270, 571, 841, 1111 }}
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}


[[Badness]] (Sintel): 0.907
[[Badness]] (Sintel): 0.424


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


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


Mapping: {{mapping| 1 -7 3 1 -11 | 0 38 -3 8 64 }}
Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9484{{c}}, ~90/77 = 271.0989{{c}}
* WE: ~2 = 1199.8090{{c}}, ~1372/1089 = 392.9286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1099{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1372/1089 = 392.9870{{c}}


{{Optimal ET sequence|legend=0| 31, …, 177e, 208, 239, 270 }}
{{Optimal ET sequence|legend=0| 58, 113, 171 }}


Badness (Sintel): 0.580
Badness (Sintel): 1.73


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


Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
Comma list: 243/242, 364/363, 441/440, 2200/2197
 
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}
 
Optimal tunings:
* WE: ~2 = 1199.7756{{c}}, ~180/143 = 392.9154{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~180/143 = 392.9840{{c}}
 
{{Optimal ET sequence|legend=0| 58, 113, 171 }}
 
Badness (Sintel): 1.11
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197


Mapping: {{mapping| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9916{{c}}, ~90/77 = 271.1051{{c}}
* WE: ~2 = 1199.8396{{c}}, ~64/51 = 392.9322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1070{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/51 = 392.9826{{c}}
 
{{Optimal ET sequence|legend=0| 58, 113, 171 }}
 
Badness (Sintel): 1.18


{{Optimal ET sequence|legend=0| 31, 239, 270, 571, 841, 1111 }}
== Hemififths ==
{{Main| Hemififths }}


Badness (Sintel): 0.741
Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.


== Neominor ==
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.
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, 5120/5103


{{Mapping|legend=1| 1 -3 -29 -14 | 0 6 41 22 }}
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
: mapping generators: ~2, ~320/189
: mapping generators: ~2, ~49/40


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4276{{c}}, ~320/189 = 917.0471{{c}}
* [[WE]]: ~2 = 1199.7412{{c}}, ~49/40 = 351.4016{{c}}
: [[error map]]: {{val| +0.428 -0.955 +0.216 +0.224 }}
: [[error map]]: {{val| -0.259 +0.590 +0.021 -0.346 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/189 = 916.7320{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4671{{c}}
: error map: {{val| 0.000 -1.563 -0.301 -0.722 }}
: error map: {{val| 0.000 +0.979 +0.364 +0.246 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 
[[Algebraic generator]]: (2 + sqrt(2))/2


{{Optimal ET sequence|legend=1| 17c, 55c, 72, 161, 233, 305 }}
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338 }}


[[Badness]] (Sintel): 2.23
[[Badness]] (Sintel): 0.563


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


Comma list: 243/242, 441/440, 35937/35840
Comma list: 243/242, 441/440, 896/891


Mapping: {{mapping| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.3466{{c}}, ~56/33 = 916.9889{{c}}
* WE: ~2 = 1199.2845{{c}}, ~11/9 = 351.3110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/33 = 916.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4956{{c}}


{{Optimal ET sequence|legend=0| 17c, 55c, 72, 161, 233, 305 }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}


Badness (Sintel): 0.924
Badness (Sintel): 0.777


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


Comma list: 169/168, 243/242, 364/363, 441/440
Comma list: 144/143, 196/195, 243/242, 364/363


Mapping: {{mapping| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.6874{{c}}, ~22/13 = 917.2313{{c}}
* WE: ~2 = 1198.8875{{c}}, ~11/9 = 351.2475{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 916.7228{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.5438{{c}}


{{Optimal ET sequence|legend=0| 17c, 55cf, 72 }}
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}


Badness (Sintel): 1.11
Badness (Sintel): 0.789


=== 17-limit ===
=== Semihemi ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11


Comma list: 169/168, 221/220, 243/242, 273/272, 364/363
Comma list: 2401/2400, 3388/3375, 5120/5103


Mapping: {{mapping| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
: mapping generators: ~99/70, ~400/231


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.6905{{c}}, ~17/10 = 917.2356{{c}}
* WE: ~99/70 = 599.8556{{c}}, ~400/231 = 951.2757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 916.7252{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~400/231 = 951.4939{{c}}


{{Optimal ET sequence|legend=0| 17cg, 55cfg, 72 }}
{{Optimal ET sequence|legend=0| 58, 140, 198 }}


Badness (Sintel): 0.918
Badness (Sintel): 1.40


== Emmthird ==
==== 13-limit ====
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
Subgroup: 2.3.5.7.11.13


[[Subgroup]]: 2.3.5.7
Comma list: 352/351, 676/675, 847/845, 1716/1715


[[Comma list]]: 2401/2400, 14348907/14336000
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


{{Mapping|legend=1| 1 -3 -17 -8 | 0 14 59 33 }}
Optimal tunings:
: mapping generators: ~2, ~2744/2187
* WE: ~99/70 = 599.8513{{c}}, ~26/15 = 951.2662{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/15 = 951.4905{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
* [[WE]]: ~2 = 1200.0435{{c}}, ~2744/2187 = 393.0021{{c}}
: [[error map]]: {{val| +0.043 -0.057 +0.069 -0.106 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2744/2187 = 392.9887{{c}}
: error map: {{val| 0.000 -0.113 +0.022 -0.197 }}


{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
Badness (Sintel): 0.876


[[Badness]] (Sintel): 0.424
=== Quadrafifths ===
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.  


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


Comma list: 243/242, 441/440, 1792000/1771561
Comma list: 2401/2400, 3025/3024, 5120/5103


Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
: mapping generators: ~2, ~243/220


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8090{{c}}, ~1372/1089 = 392.9286{{c}}
* WE: ~2 = 1199.7520{{c}}, ~243/220 = 175.7015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1372/1089 = 392.9870{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7360{{c}}


{{Optimal ET sequence|legend=0| 58, 113, 171 }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}


Badness (Sintel): 1.73
Badness (Sintel): 1.33


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


Comma list: 243/242, 364/363, 441/440, 2200/2197
Comma list: 352/351, 847/845, 2401/2400, 3025/3024


Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7756{{c}}, ~180/143 = 392.9154{{c}}
* WE: ~2 = 1199.6502{{c}}, ~72/65 = 175.6957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~180/143 = 392.9840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7461{{c}}


{{Optimal ET sequence|legend=0| 58, 113, 171 }}
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}


Badness (Sintel): 1.11
Badness (Sintel): 1.29


=== 17-limit ===
== Osiris ==
Subgroup: 2.3.5.7.11.13.17
{{See also| Metric microtemperaments #Geb }}
 
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
 
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}
 
Optimal tunings:
* WE: ~2 = 1199.8396{{c}}, ~64/51 = 392.9322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/51 = 392.9826{{c}}
 
{{Optimal ET sequence|legend=0| 58, 113, 171 }}
 
Badness (Sintel): 1.18
 
== Quinmite ==
The generator for quinmite is quasi-tempered minor third [[25/21]], flatter than 6/5 by the starling comma, [[126/125]]. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.


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


[[Comma list]]: 2401/2400, 1959552/1953125
[[Comma list]]: 2401/2400, 31381059609/31360000000


{{Mapping|legend=1| 1 -7 -5 -3 | 0 34 29 23 }}
{{Mapping|legend=1| 1 13 33 21 | 0 32 86 51 }}
: mapping generators: ~2, ~25/21
: mapping generators: ~2, ~2187/1400


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9361{{c}}, ~25/21 = 302.9808{{c}}
* [[WE]]: ~2 = 1200.0285{{c}}, ~2187/1400 = 771.9522{{c}}
: [[error map]]: {{val| -0.064 -0.162 +0.448 -0.077 }}
: [[error map]]: {{val| +0.028 -0.025 +0.068 -0.117 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 302.9953{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2187/1400 = 771.9343{{c}}
: error map: {{val| 0.000 -0.116 +0.549 +0.065 }}
: error map: {{val| 0.000 -0.056 +0.039 -0.175 }}


{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c }}
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}


[[Badness]] (Sintel): 0.945
[[Badness]] (Sintel): 0.716


== Unthirds ==
== Quasiorwell ==
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.
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 & 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.


The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.


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


[[Comma list]]: 2401/2400, 68359375/68024448
[[Comma list]]: 2401/2400, 29360128/29296875


{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
{{Mapping|legend=1| 1 -7 3 1 | 0 38 -3 8 }}
: mapping generators: ~2, ~3969/3125
: mapping generators: ~2, ~1024/875


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0859{{c}}, ~3969/3125 = 416.7465{{c}}
* [[WE]]: ~2 = 1199.9403{{c}}, ~1024/875 = 271.0935{{c}}
: [[error map]]: {{val| +0.086 +0.281 -0.431 -0.218 }}
: [[error map]]: {{val| -0.060 +0.018 +0.226 -0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3969/3125 = 416.7184{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1024/875 = 271.1064{{c}}
: error map: {{val| 0.000 +0.220 -0.547 -0.399 }}
: error map: {{val| 0.000 +0.087 +0.367 +0.025 }}


{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
{{Optimal ET sequence|legend=1| 31, …, 177, 208, 239, 270, 571, 841, 1111 }}


[[Badness]] (Sintel): 1.90
[[Badness]] (Sintel): 0.907


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


Comma list: 2401/2400, 3025/3024, 4000/3993
Comma list: 2401/2400, 3025/3024, 5632/5625


Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}
Mapping: {{mapping| 1 -7 3 1 -11 | 0 38 -3 8 64 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0246{{c}}, ~14/11 = 416.7270{{c}}
* WE: ~2 = 1199.9484{{c}}, ~90/77 = 271.0989{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7190{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1099{{c}}


{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}
{{Optimal ET sequence|legend=0| 31, …, 177e, 208, 239, 270 }}


Badness (Sintel): 0.758
Badness (Sintel): 0.580


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


Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095


Mapping: {{mapping| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}
Mapping: {{mapping| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0536{{c}}, ~14/11 = 416.7343{{c}}
* WE: ~2 = 1199.9916{{c}}, ~90/77 = 271.1051{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1070{{c}}
 
{{Optimal ET sequence|legend=0| 31, 239, 270, 571, 841, 1111 }}
 
Badness (Sintel): 0.741
 
== Quinmite ==
The generator for quinmite is quasi-tempered minor third [[25/21]], flatter than 6/5 by the starling comma, [[126/125]]. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1959552/1953125
 
{{Mapping|legend=1| 1 -7 -5 -3 | 0 34 29 23 }}
: mapping generators: ~2, ~25/21
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9361{{c}}, ~25/21 = 302.9808{{c}}
: [[error map]]: {{val| -0.064 -0.162 +0.448 -0.077 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 302.9953{{c}}
: error map: {{val| 0.000 -0.116 +0.549 +0.065 }}


{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c }}


Badness (Sintel): 0.863
[[Badness]] (Sintel): 0.945


== Newt ==
== Newt ==
Line 778: Line 683:


Badness (Sintel): 0.438
Badness (Sintel): 0.438
== Unthirds ==
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.
The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 68359375/68024448
{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
: mapping generators: ~2, ~3969/3125
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0859{{c}}, ~3969/3125 = 416.7465{{c}}
: [[error map]]: {{val| +0.086 +0.281 -0.431 -0.218 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3969/3125 = 416.7184{{c}}
: error map: {{val| 0.000 +0.220 -0.547 -0.399 }}
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
[[Badness]] (Sintel): 1.90
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 4000/3993
Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}
Optimal tunings:
* WE: ~2 = 1200.0246{{c}}, ~14/11 = 416.7270{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7190{{c}}
{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}
Badness (Sintel): 0.758
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Mapping: {{mapping| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}
Optimal tunings:
* WE: ~2 = 1200.0536{{c}}, ~14/11 = 416.7343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7164{{c}}
{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}
Badness (Sintel): 0.863


== Septidiasemi ==
== Septidiasemi ==
Line 847: Line 804:


Badness (Sintel): 1.39
Badness (Sintel): 1.39
== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 1224440064/1220703125
{{Mapping|legend=1| 1 -21 -22 -15 | 0 52 56 41 }}
: mapping generators: ~2, ~875/648
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9863{{c}}, ~875/648 = 521.1837{{c}}
: [[error map]]: {{val| -0.014 -0.115 +0.274 -0.089 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~875/648 = 521.1894{{c}}
: error map: {{val| 0.000 -0.106 +0.293 -0.060 }}
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
[[Badness]] (Sintel): 1.46


== Subneutral ==
== Subneutral ==
Line 888: Line 825:
[[Badness]] (Sintel): 1.16
[[Badness]] (Sintel): 1.16


== Osiris ==
== Maviloid ==
{{See also| Metric microtemperaments #Geb }}
{{See also| Ragismic microtemperaments #Parakleismic }}


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


[[Comma list]]: 2401/2400, 31381059609/31360000000
[[Comma list]]: 2401/2400, 1224440064/1220703125


{{Mapping|legend=1| 1 13 33 21 | 0 32 86 51 }}
{{Mapping|legend=1| 1 -21 -22 -15 | 0 52 56 41 }}
: mapping generators: ~2, ~2187/1400
: mapping generators: ~2, ~875/648


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0285{{c}}, ~2187/1400 = 771.9522{{c}}
* [[WE]]: ~2 = 1199.9863{{c}}, ~875/648 = 521.1837{{c}}
: [[error map]]: {{val| +0.028 -0.025 +0.068 -0.117 }}
: [[error map]]: {{val| -0.014 -0.115 +0.274 -0.089 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2187/1400 = 771.9343{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~875/648 = 521.1894{{c}}
: error map: {{val| 0.000 -0.056 +0.039 -0.175 }}
: error map: {{val| 0.000 -0.106 +0.293 -0.060 }}
 
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
 
[[Badness]] (Sintel): 1.46
 
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''
 
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.


{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.


[[Badness]] (Sintel): 0.716
Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.


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


[[Comma list]]: 2401/2400, 28672/28125
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}


{{Mapping|legend=1| 1 -13 8 2 | 0 18 -7 1 }}
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
: mapping generators: ~2, ~7/4
: mapping generators: ~2, ~3828125/2985984


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.5503{{c}}, ~7/4 = 971.3132{{c}} (~8/7 = 227.2371{{c}})
* [[WE]]: ~2 = 1199.9950{{c}}, ~3828125/2985984 = 431.1055{{c}}
: [[error map]]: {{val| -1.450 +0.528 +2.896 -0.412 }}
: [[error map]]: {{val| -0.005 -0.024 +0.146 -0.120 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 972.4675{{c}} (~8/7 = 227.5325{{c}})
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3828125/2985984 = 431.1072{{c}}
: error map: {{val| 0.000 +2.460 +6.414 +3.642 }}
: error map: {{val| 0.0000 -0.020 +0.155 -0.108 }}


{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913, 1183 }}


[[Badness]] (Sintel): 4.01
[[Badness]] (Sintel): 1.51


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


Comma list: 176/175, 2401/2400, 2560/2541
Comma list: 2401/2400, 3025/3024, 766656/765625


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.4615{{c}}, ~7/4 = 971.2535{{c}} (~8/7 = 227.2079{{c}})
* WE: ~2 = 1200.0199{{c}}, ~77/60 = 431.1147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.4918{{c}} (~8/7 = 227.5082{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1078{{c}}


{{Optimal ET sequence|legend=0| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}


Badness (Sintel): 1.96
Badness (Sintel): 0.865


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


Comma list: 176/175, 196/195, 364/363, 512/507
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.4012{{c}}, ~7/4 = 971.2110{{c}} (~8/7 = 227.1903{{c}})
* WE: ~2 = 1200.0707{{c}}, ~77/60 = 431.1316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.5030{{c}} (~8/7 = 227.4970{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1069{{c}}


{{Optimal ET sequence|legend=0| 21, 37, 58, 153bcef, 211bccdeeff }}
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}


Badness (Sintel): 1.33
Badness (Sintel): 0.662


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


[[Comma list]]: 2401/2400, 341796875/339738624
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224


{{Mapping|legend=1| 1 -27 -7 -9 | 0 46 15 19 }}
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
: mapping generators: ~2, ~192/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2050{{c}}, ~192/125 = 745.8170{{c}}
: [[error map]]: {{val| +0.205 +0.094 -0.493 -0.147 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/125 = 745.6927{{c}}
: error map: {{val| 0.000 -0.092 -0.924 -0.665 }}
 
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
 
Badness (Sintel): 2.54
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 43923/43750
 
Mapping: {{mapping| 1 -27 -7 -9 -4 | 0 46 15 19 12 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4064{{c}}, ~77/50 = 745.9349{{c}}
* WE: ~2 = 1199.9639{{c}}, ~77/60 = 431.0957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/50 = 745.6876{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1083{{c}}


{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}
{{Optimal ET sequence|legend=0| 103, 167, 270 }}


Badness (Sintel): 1.87
Badness (Sintel): 1.07


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


Comma list: 385/384, 625/624, 847/845, 1375/1372
Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224


Mapping: {{mapping| 1 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.3728{{c}}, ~20/13 = 745.9152{{c}}
* WE: ~2 = 1199.8693{{c}}, ~41/32 = 431.0650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 745.6879{{c}}
* CWE: ~2 = 1200.000{{c}}, ~41/32 = 431.1109{{c}}


{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}
{{Optimal ET sequence|legend=0| 103, 167, 270 }}


Badness (Sintel): 1.13
Badness (Sintel): 1.25


== Mintone ==
== Neominor ==
In addition to 2401/2400, mintone tempers out 177147/175000 ({{monzo| -3 11 -5 -1 }}) in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 & 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.
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, 177147/175000
[[Comma list]]: 2401/2400, 177147/175616


{{Mapping|legend=1| 1 -17 -34 -20 | 0 22 43 27 }}
{{Mapping|legend=1| 1 -3 -29 -14 | 0 6 41 22 }}
: mapping generators: ~2, ~9/5
: mapping generators: ~2, ~320/189


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1458{{c}}, ~9/5 = 1013.7798{{c}}
* [[WE]]: ~2 = 1200.4276{{c}}, ~320/189 = 917.0471{{c}}
: [[error map]]: {{val| +0.146 -1.277 +1.263 +0.314 }}
: [[error map]]: {{val| +0.428 -0.955 +0.216 +0.224 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6611{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/189 = 916.7320{{c}}
: error map: {{val| 0.000 -1.410 +1.116 +0.025 }}
: error map: {{val| 0.000 -1.563 -0.301 -0.722 }}


{{Optimal ET sequence|legend=1| 45, 58, 103, 161 }}
{{Optimal ET sequence|legend=1| 17c, 55c, 72, 161, 233, 305 }}


[[Badness]] (Sintel): 3.18
[[Badness]] (Sintel): 2.23


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


Comma list: 243/242, 441/440, 43923/43750
Comma list: 243/242, 441/440, 35937/35840


Mapping: {{mapping| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}
Mapping: {{mapping| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1491{{c}}, ~9/5 = 1013.7809{{c}}
* WE: ~2 = 1200.3466{{c}}, ~56/33 = 916.9889{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6593{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/33 = 916.7330{{c}}


{{Optimal ET sequence|legend=0| 45e, 58, 103, 161, 425b }}
{{Optimal ET sequence|legend=0| 17c, 55c, 72, 161, 233, 305 }}


Badness (Sintel): 1.32
Badness (Sintel): 0.924


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


Comma list: 243/242, 351/350, 441/440, 847/845
Comma list: 169/168, 243/242, 364/363, 441/440


Mapping: {{mapping| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0928{{c}}, ~9/5 = 1013.7311{{c}}
* WE: ~2 = 1200.6874{{c}}, ~22/13 = 917.2313{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 916.7228{{c}}


{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}
{{Optimal ET sequence|legend=0| 17c, 55cf, 72 }}


Badness (Sintel): 0.903
Badness (Sintel): 1.11


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


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


Mapping: {{mapping| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1085{{c}}, ~9/5 = 1013.7433{{c}}
* WE: ~2 = 1200.6905{{c}}, ~17/10 = 917.2356{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6537{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 916.7252{{c}}


{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}
{{Optimal ET sequence|legend=0| 17cg, 55cfg, 72 }}


Badness (Sintel): 1.03
Badness (Sintel): 0.918


== Catafourth ==
== Catafourth ==
Line 1,166: Line 1,093:


Badness (Sintel): 1.19
Badness (Sintel): 1.19
== Fibo ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 341796875/339738624
{{Mapping|legend=1| 1 -27 -7 -9 | 0 46 15 19 }}
: mapping generators: ~2, ~192/125
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2050{{c}}, ~192/125 = 745.8170{{c}}
: [[error map]]: {{val| +0.205 +0.094 -0.493 -0.147 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/125 = 745.6927{{c}}
: error map: {{val| 0.000 -0.092 -0.924 -0.665 }}
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
Badness (Sintel): 2.54
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 43923/43750
Mapping: {{mapping| 1 -27 -7 -9 -4 | 0 46 15 19 12 }}
Optimal tunings:
* WE: ~2 = 1200.4064{{c}}, ~77/50 = 745.9349{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/50 = 745.6876{{c}}
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}
Badness (Sintel): 1.87
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 385/384, 625/624, 847/845, 1375/1372
Mapping: {{mapping| 1 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}
Optimal tunings:
* WE: ~2 = 1200.3728{{c}}, ~20/13 = 745.9152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 745.6879{{c}}
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}
Badness (Sintel): 1.13


== Quasimoha ==
== Quasimoha ==
Line 1,202: Line 1,177:
Badness (Sintel): 1.53
Badness (Sintel): 1.53


== Lockerbie ==
== Mintone ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''
In addition to 2401/2400, mintone tempers out 177147/175000 ({{monzo| -3 11 -5 -1 }}) in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It 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.
 
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.
 
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.
 
Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.


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


[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}
[[Comma list]]: 2401/2400, 177147/175000


{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
{{Mapping|legend=1| 1 -17 -34 -20 | 0 22 43 27 }}
: mapping generators: ~2, ~3828125/2985984
: mapping generators: ~2, ~9/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9950{{c}}, ~3828125/2985984 = 431.1055{{c}}
* [[WE]]: ~2 = 1200.1458{{c}}, ~9/5 = 1013.7798{{c}}
: [[error map]]: {{val| -0.005 -0.024 +0.146 -0.120 }}
: [[error map]]: {{val| +0.146 -1.277 +1.263 +0.314 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3828125/2985984 = 431.1072{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6611{{c}}
: error map: {{val| 0.0000 -0.020 +0.155 -0.108 }}
: error map: {{val| 0.000 -1.410 +1.116 +0.025 }}


{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913, 1183 }}
{{Optimal ET sequence|legend=1| 45, 58, 103, 161 }}


[[Badness]] (Sintel): 1.51
[[Badness]] (Sintel): 3.18


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


Comma list: 2401/2400, 3025/3024, 766656/765625
Comma list: 243/242, 441/440, 43923/43750


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0199{{c}}, ~77/60 = 431.1147{{c}}
* WE: ~2 = 1200.1491{{c}}, ~9/5 = 1013.7809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1078{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6593{{c}}


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


Badness (Sintel): 0.865
Badness (Sintel): 1.32


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


Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224
Comma list: 243/242, 351/350, 441/440, 847/845


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0707{{c}}, ~77/60 = 431.1316{{c}}
* WE: ~2 = 1200.0928{{c}}, ~9/5 = 1013.7311{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1069{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6556{{c}}


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


Badness (Sintel): 0.662
Badness (Sintel): 0.903


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


Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
 
Mapping: {{mapping| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}
 
Optimal tunings:
* WE: ~2 = 1200.1085{{c}}, ~9/5 = 1013.7433{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6537{{c}}
 
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}
 
Badness (Sintel): 1.03
 
== Gorgik ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 28672/28125
 
{{Mapping|legend=1| 1 -13 8 2 | 0 18 -7 1 }}
: mapping generators: ~2, ~7/4
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.5503{{c}}, ~7/4 = 971.3132{{c}} (~8/7 = 227.2371{{c}})
: [[error map]]: {{val| -1.450 +0.528 +2.896 -0.412 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 972.4675{{c}} (~8/7 = 227.5325{{c}})
: error map: {{val| 0.000 +2.460 +6.414 +3.642 }}
 
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
 
[[Badness]] (Sintel): 4.01
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 2401/2400, 2560/2541


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9639{{c}}, ~77/60 = 431.0957{{c}}
* WE: ~2 = 1198.4615{{c}}, ~7/4 = 971.2535{{c}} (~8/7 = 227.2079{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1083{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.4918{{c}} (~8/7 = 227.5082{{c}})


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


Badness (Sintel): 1.07
Badness (Sintel): 1.96


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


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


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8693{{c}}, ~41/32 = 431.0650{{c}}
* WE: ~2 = 1198.4012{{c}}, ~7/4 = 971.2110{{c}} (~8/7 = 227.1903{{c}})
* CWE: ~2 = 1200.000{{c}}, ~41/32 = 431.1109{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.5030{{c}} (~8/7 = 227.4970{{c}})


{{Optimal ET sequence|legend=0| 103, 167, 270 }}
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bcef, 211bccdeeff }}


Badness (Sintel): 1.25
Badness (Sintel): 1.33


== Hemigoldis ==
== Hemigoldis ==
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"