Porwell temperaments: Difference between revisions

This is a collection of [[regular temperament|temperaments]] that [[tempering out|tempers out]] the porwell comma, {{monzo| 11 1 -3 -2 }} ([[6144/6125]]).  

* [[Shrutar]] (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]

* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]

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

: ''For the 5-limit version, see [[11th-octave temperaments #Hendecapent]].''  

=== 11-limit ===

Mapping: {{mapping| 11 0 43 -4 38 | 0 1 -1 2 0 }}

Mapping: {{mapping| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}

Mapping: {{mapping| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}

Mapping: {{mapping| 11 0 43 -4 73 | 0 1 -1 2 -2 }}

Mapping: {{mapping| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}

Mapping: {{mapping| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}

Mapping: {{mapping| 22 0 86 -8 111 | 0 1 -1 2 -1 }}

Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}

Mapping: {{mapping| 3 2 5 10 10 6 | 0 7 5 -4 1 13 }}

{{Optimal ET sequence|legend=0| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}

== 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.  

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

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

: mapping generators: ~2, ~28/15

* [[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

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

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

* 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

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

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

* 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 ==

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. 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

 

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

 

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

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

 

 

 

=== 11-limit ===

 

 

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

* 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 }}

* 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

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.  

Mapping: {{mapping| 1 -3 12 -14 6 | 0 9 -19 33 -5 }}

Mapping: {{mapping| 1 -3 12 -14 6 20 | 0 9 -19 33 -5 -32 }}

Mapping: {{mapping| 1 -3 12 -14 33 | 0 9 -19 33 -58 }}

Mapping: {{mapping| 1 -3 12 -14 33 20 | 0 9 -19 33 -58 -32 }}

Mapping: {{mapping| 7 0 -17 64 124 | 0 1 3 -4 -9 }}

Mapping: {{mapping| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}

Mapping: {{mapping| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}

Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}

Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}

Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}

Polypyth (46 & 121) tempers out the same 5-limit comma as the [[leapday]] temperament (29 & 46), but with the porwell (6144/6125) rather than the hemifamity (5120/5103) tempered out.

Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}

Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }}

Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}

Also named by [[Petr Pařízek]] in 2011, ''whoops'' is a relatively simple extension to the otherwise very accurate microtemperament known as ''whoosh''<ref name="petr's long post"/>.  

Mapping: {{mapping| 1 -16 -11 14 -4 | 0 33 25 -21 14 }}

Mapping: {{mapping| 1 -23 -4 0 14 | 0 35 9 4 -15 }}

Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}

The icositritonic temperament (46 & 161) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].

Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}

Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}

Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}

Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}

Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}

[[Category:Porwell temperaments| ]] <!-- main article -->

[[Category:Rank 2]]