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


The minimal generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning.  
The generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 1728/1715, 3136/3125
[[Comma list]]: 1728/1715, 3136/3125


{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}
{{Mapping|legend=1| 1 -5 0 -3 | 0 17 6 15 }}
: mapping generators: ~2, ~98/75


: mapping generators: ~2, ~75/49
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.2968{{c}}, ~98/75 = 464.5722{{c}}
: [[error map]]: {{val| -0.703 -0.712 +1.119 +1.867 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98/75 = 464.8120{{c}}
: error map: {{val| 0.000 -0.150 +2.559 +3.355 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
{{Optimal ET sequence|legend=1| 31, 80, 111, 364cdd, 475ccddd }}


{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}
[[Badness]] (Sintel): 1.28
 
[[Badness]]: 0.050472


=== 11-limit ===
=== 11-limit ===
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Comma list: 176/175, 540/539, 1331/1323
Comma list: 176/175, 540/539, 1331/1323


{{Mapping|legend=0| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 | 0 17 6 15 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125
Optimal tunings:
* WE: ~2 = 1199.3073{{c}}, ~72/55 = 464.6063{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 464.8423{{c}}


{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.022476
Badness (Sintel): 0.743


==== 13-limit ====
==== 13-limit ====
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Comma list: 176/175, 351/350, 540/539, 1375/1372
Comma list: 176/175, 351/350, 540/539, 1375/1372


{{Mapping|legend=0| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 | 0 17 6 15 27 -24 }}


Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126
Optimal tunings:
* WE: ~2 = 1199.3145{{c}}, ~72/55 = 464.6080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 464.8636{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.025204
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
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Comma list: 176/175, 256/255, 351/350, 640/637, 715/714
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714


{{Mapping|legend=0| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 13 | 0 17 6 15 27 -24 -23 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125
Optimal tunings:
* WE: ~2 = 1199.3067{{c}}, ~17/13 = 464.6061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 464.8699{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.019919
Badness (Sintel): 1.01


===== 19-limit =====
===== 19-limit =====
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Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323


{{Mapping|legend=0| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 13 12 | 0 17 6 15 27 -24 -23 -20 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116
Optimal tunings:
* WE: ~2 = 1199.2346{{c}}, ~17/13 = 464.5878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 464.8816{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.016301
Badness (Sintel): 0.992


===== 23-limit =====
===== 23-limit =====
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Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459


{{Mapping|legend=0| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 13 12 -4 | 0 17 6 15 27 -24 -23 -20 22 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106
Optimal tunings:
* WE: ~2 = 1199.3227{{c}}, ~17/13 = 464.6320{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 464.8912{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}
{{Optimal ET sequence|legend=0| 31, 80, 111, 191cdh }}


Badness: 0.014957
Badness (Sintel): 1.07


==== Semishly ====
==== Semishly ====
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Comma list: 144/143, 176/175, 196/195, 275/273
Comma list: 144/143, 176/175, 196/195, 275/273


{{Mapping|legend=0| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 1 | 0 17 6 15 27 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980
Optimal tunings:
* WE: ~2 = 1198.3103{{c}}, ~13/10 = 464.3248{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 464.9110{{c}}


{{Optimal ET sequence|legend=1| 31, 49f, 80f }}
{{Optimal ET sequence|legend=0| 31, 49f, 80f }}


Badness: 0.028408
Badness (Sintel): 1.17


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


Emka tempers out {{monzo| -50 -8 27 }} in the 5-limit. This temperament can be described as 37 & 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87 & 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.
Emka tempers out {{monzo| -50 -8 27 }} in the 5-limit. This temperament can be described as the 37 & 50 temperament, which tempers out the hemimean comma and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87 & 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 3136/3125, 84035/82944
[[Comma list]]: 3136/3125, 84035/82944


{{Mapping|legend=1| 1 14 6 12 | 0 -27 -8 -20 }}
{{Mapping|legend=1| 1 -13 -2 -8 | 0 27 8 20 }}
: mapping generators: ~2, ~35/24


: mapping generators: ~2, ~48/35
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.4701{{c}}, ~35/24 = 648.4717{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
: [[error map]]: {{val| +0.470 +0.669 +0.520 -3.153 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 648.2307{{c}}
: error map: {{val| 0.000 +0.274 -0.468 -4.212 }}


{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}


[[Badness]]: 0.144338
[[Badness]] (Sintel): 3.65


=== 11-limit ===
=== 11-limit ===
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Comma list: 385/384, 2401/2376, 3136/3125
Comma list: 385/384, 2401/2376, 3136/3125


{{Mapping|legend=0| 1 14 6 12 3 | 0 -27 -8 -20 1 }}
{{Mapping|legend=0| 1 -13 -2 -8 4 | 0 27 8 20 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765
Optimal tunings:
* WE: ~2 = 1200.2552{{c}}, ~16/11 = 648.3727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2399{{c}}


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}
{{Optimal ET sequence|legend=0| 37, 50, 87 }}


Badness: 0.054744
Badness (Sintel): 1.81


=== 13-limit ===
=== 13-limit ===
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Comma list: 196/195, 364/363, 385/384, 625/624
Comma list: 196/195, 364/363, 385/384, 625/624


{{Mapping|legend=0| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}
{{Mapping|legend=0| 1 -13 -2 -8 4 1 | 0 27 8 20 -1 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758
Optimal tunings:
* WE: ~2 = 1200.1506{{c}}, ~16/11 = 648.3230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2442{{c}}


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}


Badness: 0.029741
Badness (Sintel): 1.23


== Decipentic ==
== Decipentic ==
The generator for the decipentic temperament (43 &amp; 56) is the tenth root of the [[5/1|5th harmonic (5/1)]], 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.
Decipentic may be described as the 43 & 56 temperament. The minimal generator for this temperament is the 10th root of the [[5/1|5th harmonic]], 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 3136/3125, 2097152/2083725
[[Comma list]]: 3136/3125, 2097152/2083725


{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
{{Mapping|legend=1| 1 -13 10 22 | 0 19 -10 -25 }}
: mapping generators: ~2, ~128/75


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5804{{c}}, ~128/75 = 920.8777{{c}} (~75/64 = 278.7027{{c}})
: [[error map]]: {{val| -0.420 +0.175 +0.714 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/75 = 921.2002{{c}} (~75/64 = 278.7998{{c}})
: error map: {{val| 0.000 +0.849 +1.684 +1.169 }}


{{Optimal ET sequence|legend=1| 13, 43, 56, 99 }}
{{Optimal ET sequence|legend=1| 43, 56, 99, 538cd, 637cd, …, 1033bccdd }}


[[Badness]]: 0.087325
[[Badness]] (Sintel): 2.21


=== 11-limit ===
=== 11-limit ===
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{{Mapping|legend=0| 1 6 0 -3 3 | 0 -19 10 25 2 }}
{{Mapping|legend=0| 1 6 0 -3 3 | 0 -19 10 25 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799
Optimal tunings:
* WE: ~2 = 1199.3002{{c}}, ~128/75 = 920.6639{{c}} (~75/64 = 278.6364{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~128/75 = 921.2019{{c}} (~75/64 = 278.7981{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 43, 56, 99e }}


Badness: 0.061413
Badness (Sintel): 2.03


==== 13-limit ====
==== 13-limit ====
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Comma list: 169/168, 441/440, 832/825, 975/968
Comma list: 169/168, 441/440, 832/825, 975/968


{{Mapping|legend=0| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}
{{Mapping|legend=0| 1 -13 10 22 5 6 | 0 19 -10 -25 -2 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802
Optimal tunings:
* WE: ~2 = 1199.5986{{c}}, ~22/13 = 920.8900{{c}} (~13/11 = 278.7085{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 921.1988{{c}} (~13/11 = 278.7981{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=0| 43, 56, 99e }}


Badness: 0.047611
Badness (Sintel): 1.97


==== 17-limit ====
==== 17-limit ====
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Comma list: 169/168, 221/220, 256/255, 273/272, 375/374
Comma list: 169/168, 221/220, 256/255, 273/272, 375/374


{{Mapping|legend=0| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}
{{Mapping|legend=0| 1 -13 10 22 5 6 11 | 0 19 -10 -25 -2 -3 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798
Optimal tunings:
* WE: ~2 = 1199.5089{{c}}, ~17/10 = 920.8252{{c}} (~13/11 = 278.6837{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 921.2034{{c}} (~13/11 = 278.7966{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=0| 43, 56, 99e }}


Badness: 0.031191
Badness (Sintel): 1.59


==== 19-limit ====
==== 19-limit ====
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Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285
Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285


{{Mapping|legend=0| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}
{{Mapping|legend=0| 1 -13 10 22 5 6 11 15 | 0 19 -10 -25 -2 -3 -9 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.790
Optimal tunings:
* WE: ~2 = 1199.4107{{c}}, ~17/10 = 920.7575{{c}} (~13/11 = 278.6532{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 921.2020{{c}} (~13/11 = 278.7980{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=0| 43, 56, 99e }}


Badness: 0.023899
Badness (Sintel): 1.45


=== Quasijerome ===
=== Quasijerome ===
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Comma list: 3136/3125, 15488/15435, 16384/16335
Comma list: 3136/3125, 15488/15435, 16384/16335


{{Mapping|legend=0| 1 6 0 -3 3 | 0 -38 20 50 47 }}
{{Mapping|legend=0| 1 -32 20 47 45 | 0 38 -20 -50 -47 }}
: mapping generators: ~2, ~825/448


Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403
Optimal tunings:
* WE: ~2 = 1199.6276{{c}}, ~825/448 = 1060.2861{{c}} (~896/825 = 139.3595{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~825/448 = 1060.5982{{c}} (~896/825 = 139.4016{{c}})


{{Optimal ET sequence|legend=1| 43, 112, 155, 198, 439cd, 637cd }}
{{Optimal ET sequence|legend=0| 43, 112, 155, 198, 439cd, 637cd }}


Badness: 0.092996
Badness (Sintel): 3.07


==== 13-limit ====
==== 13-limit ====
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Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435


{{Mapping|legend=0| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}
{{Mapping|legend=0| 1 -32 20 47 45 -29 | 0 38 -20 -50 -47 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403
Optimal tunings:
* WE: ~2 = 1199.6184{{c}}, ~24/13 = 1060.2595{{c}} (~13/12 = 139.3589{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~24/13 = 1060.5971{{c}} (~13/12 = 139.4029{{c}})


{{Optimal ET sequence|legend=1| 43, 155, 198, 439cdf, 637cdf }}
{{Optimal ET sequence|legend=0| 43, 155, 198, 439cdf, 637cdf }}


Badness: 0.044328
Badness (Sintel): 1.83


== Sengagen ==
== Sengagen ==
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== Mowglic ==
== Mowglic ==
The mowglic temperament (19 &amp; 161) is an extension of the [[Syntonic–kleismic equivalence continuum #Mowgli|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.
The mowglic temperament (19 &amp; 161) is an extension of the [[Syntonic–kleismic equivalence continuum #Mowgli|mowgli]] temperament which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Tremka ==
== Tremka ==
The name ''tremka'' was initially used for the [[No-sevens subgroup temperaments|no-sevens version]] of 50 &amp; 111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.
The name ''tremka'' was initially used for the [[No-sevens subgroup temperaments|no-sevens version]] of 50 & 111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.


=== 7-limit ===
=== 7-limit ===
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: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''


The undetrita temperament (111 &amp; 118) tempers out the hemimean comma (3136/3125) and [[scheme comma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt]], and the name ''undetrita'' is a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111 &amp; 118.
The undetrita temperament (111 & 118) tempers out the hemimean comma (3136/3125) and [[scheme comma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt]], and the name ''undetrita'' is a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111 & 118.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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= Subgroup extensions =
= Subgroup extensions =
== Undecimal didacus ==
== Undecimal didacus ==
In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])<sup>3</sup>. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.
In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])<sup>3</sup>. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.
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