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Switch to Sintel's badness, WE & CWE tunings (1/)
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= 2.5.7 subgroup =
= 2.5.7 subgroup =
== Didacus ==
== Didacus ==
{{main|Didacus}}
{{Main| Didacus }}


See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].
See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].
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{{Mapping|legend=3| 1 0 -3 | 0 2 5 }}
{{Mapping|legend=3| 1 0 -3 | 0 2 5 }}
: sval mapping generators: ~2, ~56/25


{{Mapping|legend=5| 1 0 0 -3 | 0 0 2 5 }}
{{Mapping|legend=5| 1 0 0 -3 | 0 0 2 5 }}
: mapping generators: ~2, ~56/25


: [[gencom]]: [2 56/25; 3136/3125]
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.8194{{c}}, ~28/25 = 193.7425{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772
: [[error map]]: {{val| -0.181 +0.810 -0.474 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 193.7167{{c}}
: error map: {{val| 0.000 +1.120 -0.243 }}


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}
{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
[[Badness]] (Sintel): 0.0908
 
[[Badness]] (Sintel): 0.091


= Strong extensions =
= Strong extensions =
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| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
|}
|}
<nowiki />* Defined by intersection with other documented extensions
<nowiki/>* Defined by intersection with other documented extensions


== Hemiwürschmidt ==
== Hemiwürschmidt ==
{{See also| Würschmidt family }}
{{See also| Würschmidt family }}


'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.
Hemiwürschmidt (sometimes spelled ''hemiwuerschmidt'') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 2401/2400, 3136/3125
[[Comma list]]: 2401/2400, 3136/3125


{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}
{{Mapping|legend=1| 1 -1 2 2 | 0 16 2 5 }}
: mapping generators: ~2, ~28/25


Mapping generators: ~2, ~25/14
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.7153{{c}}, ~28/25 = 193.8522{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
: [[error map]]: {{val| -0.285 -0.035 +0.821 -0.134 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 193.8883{{c}}
: error map: {{val| 0.000 +0.258 +1.463 +0.616 }}


{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}


[[Badness]]: 0.020307
[[Badness]] (Sintel): 0.514


=== 2.3.5.7.23 subgroup ===
=== 11-limit ===
As described at the page for [[würschmidt]], there is an extension to prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7.23
Comma list: 243/242, 441/440, 3136/3125


[[Comma list]]: 576/575, 736/735, 1127/1125
{{Mapping|legend=0| 1 -1 2 2 -3 | 0 16 2 5 40 }}


{{Mapping|legend=1| 1 15 4 7 28 | 0 -16 -2 -5 -28 }}
Optimal tunings:
* WE: ~2 = 1199.7863{{c}}, ~28/25 = 193.8058{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.8342{{c}}


Mapping generators: ~2, ~25/14
{{Optimal ET sequence|legend=0| 31, 68e, 99e, 130 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.901
Badness (Sintel): 0.697


{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness (Sintel): 0.304
Comma list: 243/242, 351/350, 441/440, 3584/3575


=== 11-limit ===
{{Mapping|legend=0| 1 -1 2 2 -3 10 | 0 16 2 5 40 -39 }}
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 3136/3125
Optimal tunings:  
* WE: ~2 = 1199.8626{{c}}, ~28/25 = 193.8063{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.8277{{c}}


{{Mapping|legend=0| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}
{{Optimal ET sequence|legend=0| 31, 99e, 130, 291, 421e, 551ce }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840
Badness (Sintel): 0.953


{{Optimal ET sequence|legend=1| 31, 99e, 130, 811ce }}
==== 2.3.5.7.11.13.23 subgroup ====
As described at the page for [[würschmidt]], there is an extension for prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).


Badness: 0.021069
Subgroup: 2.3.5.7.11.13.23


==== 13-limit ====
Comma list: 243/242, 276/275, 351/350, 441/440, 3584/3575
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 351/350, 441/440, 3584/3575
{{Mapping|legend=0| 1 -1 2 2 -3 10 0 | 0 16 2 5 40 -39 28 }}
 
{{Mapping|legend=0| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829
Optimal tunings:
* WE: ~2 = 1199.9052{{c}}, ~28/25 = 193.8170{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.8316{{c}}


{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}
{{Optimal ET sequence|legend=0| 31, 99e, 130 }}


Badness: 0.023074
Badness (Sintel): 0.839


==== Hemithir ====
==== Hemithir ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 144/143, 196/195, 243/242, 625/624


{{Mapping|legend=0| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}
{{Mapping|legend=0| 1 -1 2 2 -3 5 | 0 16 2 5 40 -8 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918
Optimal tunings:
* WE: ~2 = 1199.1395{{c}}, ~28/25 = 193.7788{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.9023{{c}}


{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
{{Optimal ET sequence|legend=0| 31, 68e, 99ef }}


Badness: 0.031199
Badness (Sintel): 1.29


=== Hemiwur ===
=== Hemiwur ===
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Comma list: 121/120, 176/175, 1375/1372
Comma list: 121/120, 176/175, 1375/1372


{{Mapping|legend=0| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}
{{Mapping|legend=0| 1 -1 2 2 2 | 0 16 2 5 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884
Optimal tunings:
* WE: ~2 = 1200.1742{{c}}, ~28/25 = 193.9117{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.8896{{c}}


{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
{{Optimal ET sequence|legend=0| 31, 68, 99, 130e }}


Badness: 0.029270
Badness (Sintel): 0.968


==== 13-limit ====
==== 13-limit ====
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Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 121/120, 176/175, 196/195, 275/273


{{Mapping|legend=0| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}
{{Mapping|legend=0| 1 -1 2 2 2 5 | 0 16 2 5 9 -8 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004
Optimal tunings:
* WE: ~2 = 1199.4972{{c}}, ~28/25 = 193.9231{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.9942{{c}}


{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
{{Optimal ET sequence|legend=0| 31, 68 }}


Badness: 0.028432
Badness (Sintel): 1.17


==== Hemiwar ====
==== Hemiwar ====
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Comma list: 66/65, 105/104, 121/120, 1375/1372
Comma list: 66/65, 105/104, 121/120, 1375/1372


{{Mapping|legend=0| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}
{{Mapping|legend=0| 1 -1 2 2 2 0 | 0 16 2 5 9 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698
Optimal tunings:
* WE: ~2 = 1199.9465{{c}}, ~28/25 = 193.6889{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.6956{{c}}


{{Optimal ET sequence|legend=1| 6f, 31 }}
{{Optimal ET sequence|legend=0| 6f, 25bf, 31 }}


Badness: 0.044886
Badness (Sintel): 1.85


=== Quadrawürschmidt ===
=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  
This extension has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense because it splits the generator in four.  


The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].
The generator of quadrawürschmidt is essentially a [[meantone family #Meantone|meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 whole tones up. The small comma between the generator and 3/2 represents [[441/440]][[~]][[540/539]].


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Comma list: 2401/2400, 3025/3024, 3136/3125
Comma list: 2401/2400, 3025/3024, 3136/3125


{{Mapping|legend=0| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}
{{Mapping|legend=0| 1 -17 0 -3 -25 | 0 32 4 10 49 }}
 
: mapping generators: ~2, ~220/147
: mapping generators: ~2, ~147/110


Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404
Optimal tunings:
* WE: ~2 = 1199.7360{{c}}, ~220/147 = 696.8062{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~220/147 = 696.9548{{c}}


{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}


Badness: 0.034814
Badness (Sintel): 1.15


=== Semihemiwür ===
=== Semihemiwür ===
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