Porwell temperaments: Difference between revisions

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* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]

* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]

* ''[[Twothirdtonic]]'' (+686/675) → [[Sengic temperaments #Twothirdtonic|Sengic temperaments]]

* ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]

* ''[[Quinkee]]'' (+1029/1000) → [[Keegic temperaments #Quinkee|Keegic temperaments]]

* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]

* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]

* ''[[Semaja]]'' (+3125/3087) → [[Gariboh clan #Semaja|Gariboh clan]]

* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]

* ''[[Countermiracle]]'' (+823543/819200) → [[Quince clan #Countermiracle|Quince clan]]

* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]

Considered below are hendecatonic, nessafof, grendel~~, twothirdtonic~~, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].

Considered below are hendecatonic, nessafof, grendel, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].

== Hendecatonic ==

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Badness (Sintel): 1.12

~~== Twothirdtonic ==~~

Twothirdtonic tempers out 686/675, the [[senga]], in addition to the porwell comma, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 686/675, 6144/6125~~

~~{{Mapping|legend=1| 1 -10 5 -7 | 0 13 -3 11 }}~~

~~: mapping generators: ~2, ~28/15~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}

~~: [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}

~~: error map: {{val| 0.000 +2.515 +4.962 -3.505 }}~~

~~{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}~~

~~[[Badness]] (Sintel): 2.52~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 121/120, 176/175, 686/675~~

~~Mapping: {{mapping| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}

~~{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}~~

~~Badness (Sintel): 1.35~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 91/90, 121/120, 169/168, 176/175~~

~~Mapping: {{mapping| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}

~~{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}~~

~~Badness (Sintel): 1.07~~

~~== Semaja ==~~

~~{{See also| Llywelynsmic clan }}~~

Cryptically named by [[Petr Pařízek]] in 2011, semaja adds the [[gariboh comma]] to the comma list, and may be described as the {{nowrap| 37 & 53 }} temperament. Its [[ploidacot]] is gamma-19-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). The name actually refers to the fact that two of its ~[[8/7]] generator steps reach a ~[[13/10]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 3125/3087, 6144/6125~~

~~{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}~~

~~: mapping generators: ~2, ~8/7~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1199.4860{{c}}, ~8/7 = 226.3864{{c}}

~~: [[error map]]: {{val| -0.514 +0.415 -2.123 +3.246 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.4697{{c}}

~~: error map: {{val| 0.000 +0.970 -1.026 +4.704 }}~~

~~{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}~~

~~[[Badness]] (Sintel): 2.71~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 121/120, 176/175, 3125/3087~~

~~Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1199.9818{{c}}, ~8/7 = 226.4821{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4851{{c}}

~~{{Optimal ET sequence|legend=0| 16, 37, 53 }}~~

~~Badness (Sintel): 1.98~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 121/120, 169/168, 176/175, 275/273~~

~~Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.1020{{c}}, ~8/7 = 226.4987{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4822{{c}}

~~{{Optimal ET sequence|legend=0| 16, 37, 53 }}~~

~~Badness (Sintel): 1.35~~

== Aufo ==

:''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''

Also named by [[Petr Pařízek]] in 2011, ''aufo'' refers to the augmented fourth, which is a generator of this temperament<ref name="petr's long post"/>. The functional generator however is the [[64/45]] diminished fifth, and like its [[untriton]] variant, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.

Also named by [[Petr Pařízek]] in 2011, ''aufo'' refers to the augmented fourth, which is a generator of this temperament<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>. The functional generator however is the [[64/45]] diminished fifth, and like its [[untriton]] variant, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.

[[Subgroup]]: 2.3.5.7

@@ Line 16: / Line 16: @@
 * ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]
 * ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
+* ''[[Twothirdtonic]]'' (+686/675) → [[Sengic temperaments #Twothirdtonic|Sengic temperaments]]
 * ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]
 * ''[[Quinkee]]'' (+1029/1000) → [[Keegic temperaments #Quinkee|Keegic temperaments]]
 * ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
 * ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
+* ''[[Semaja]]'' (+3125/3087) → [[Gariboh clan #Semaja|Gariboh clan]]
 * ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
 * ''[[Countermiracle]]'' (+823543/819200) → [[Quince clan #Countermiracle|Quince clan]]
 * ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]
-Considered below are hendecatonic, nessafof, grendel, twothirdtonic, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].
+Considered below are hendecatonic, nessafof, grendel, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].
 == Hendecatonic ==
@@ Line 302: / Line 304: @@
 Badness (Sintel): 1.12
-== Twothirdtonic ==
-Twothirdtonic tempers out 686/675, the [[senga]], in addition to the porwell comma, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 686/675, 6144/6125
-{{Mapping|legend=1| 1 -10 5 -7 | 0 13 -3 11 }}
-: mapping generators: ~2, ~28/15
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}
-: [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}
-: error map: {{val| 0.000 +2.515 +4.962 -3.505 }}
-{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
-[[Badness]] (Sintel): 2.52
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 176/175, 686/675
-Mapping: {{mapping| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}
-Optimal tunings:
-* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}
-{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
-Badness (Sintel): 1.35
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 121/120, 169/168, 176/175
-Mapping: {{mapping| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}
-Optimal tunings:
-* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}
-{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
-Badness (Sintel): 1.07
-== Semaja ==
-{{See also| Llywelynsmic clan }}
-Cryptically named by [[Petr Pařízek]] in 2011, semaja adds the [[gariboh comma]] to the comma list, and may be described as the {{nowrap| 37 & 53 }} temperament. Its [[ploidacot]] is gamma-19-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). The name actually refers to the fact that two of its ~[[8/7]] generator steps reach a ~[[13/10]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 3125/3087, 6144/6125
-{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}
-: mapping generators: ~2, ~8/7
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.4860{{c}}, ~8/7 = 226.3864{{c}}
-: [[error map]]: {{val| -0.514 +0.415 -2.123 +3.246 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.4697{{c}}
-: error map: {{val| 0.000 +0.970 -1.026 +4.704 }}
-{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}
-[[Badness]] (Sintel): 2.71
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 176/175, 3125/3087
-Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}
-Optimal tunings:
-* WE: ~2 = 1199.9818{{c}}, ~8/7 = 226.4821{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4851{{c}}
-{{Optimal ET sequence|legend=0| 16, 37, 53 }}
-Badness (Sintel): 1.98
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 169/168, 176/175, 275/273
-Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}
-Optimal tunings:
-* WE: ~2 = 1200.1020{{c}}, ~8/7 = 226.4987{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4822{{c}}
-{{Optimal ET sequence|legend=0| 16, 37, 53 }}
-Badness (Sintel): 1.35
 == Aufo ==
 :''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
-Also named by [[Petr Pařízek]] in 2011, ''aufo'' refers to the augmented fourth, which is a generator of this temperament<ref name="petr's long post"/>. The functional generator however is the [[64/45]] diminished fifth, and like its [[untriton]] variant, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.
+Also named by [[Petr Pařízek]] in 2011, ''aufo'' refers to the augmented fourth, which is a generator of this temperament<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>. The functional generator however is the [[64/45]] diminished fifth, and like its [[untriton]] variant, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.
 [[Subgroup]]: 2.3.5.7