Horwell temperaments: Difference between revisions

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 * ''[[Kaboom]]'' (+4802000/4782969) → [[Vavoom family #Kaboom|Vavoom family]]
 * ''[[Soviet ferris wheel]]'' (+{{monzo| -5 -9 -5 11 }}) → [[20th-octave temperaments #Soviet ferris wheel|20th-octave temperaments]]
+Considered below are fifthplus, mutt, oquatonic, emkay, kastro, and bezique, in the order of increasing [[badness]].
+== Fifthplus ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sesesix]].''
+Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }}. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator. It is a restriction of [[24576/24565 #2.3.5.7.17 subgroup (prime archagall)|prime archagall]].
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 65625/65536, 420175/419904
+{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
+: mapping generators: ~2, ~5488/3645
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0934{{c}}, ~5488/3645 = 708.8291{{c}}
+: [[error map]]: {{val| +0.093 -0.007 -0.158 -0.059 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5488/3645 = 708.7752{{c}}
+: error map: {{val| 0.000 -0.126 -0.391 -0.268 }}
+{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}
+[[Badness]] (Sintel): 0.654
 == Mutt ==
@@ Line 73: / Line 97: @@
 Badness (Sintel): 1.20
-== Fifthplus ==
+== Oquatonic ==
-: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sesesix]].''
+: ''For the 5-limit version, see [[28th-octave temperaments #Oquatonic (5-limit)]].''
+Oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the [[dimcomp comma]] (390625/388962). In this temperament, the [[5/4]] major third is mapped to 9\28.
-Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }}. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator. It is a restriction of [[24576/24565 #2.3.5.7.17 subgroup (prime archagall)|prime archagall]].
+The name ''oquatonic'' was given by [[Petr Pařízek]] in 2011 as an abbreviation of the Italian [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref name="petr's long post"/>.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 65625/65536, 420175/419904
+[[Comma list]]: 65625/65536, 390625/388962
-{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
+{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
-: mapping generators: ~2, ~5488/3645
+: mapping generators: ~128/125, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.0934{{c}}, ~5488/3645 = 708.8291{{c}}
+* [[WE]]: ~128/125 = 42.8570{{c}}, ~3/2 = 702.1112{{c}}
-: [[error map]]: {{val| +0.093 -0.007 -0.158 -0.059 }}
+: [[error map]]: {{val| -0.004 +0.152 -0.609 +0.477 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~5488/3645 = 708.7752{{c}}
+* [[CWE]]: ~128/125 = 42.8571{{c}}, ~3/2 = 702.1132{{c}}
-: error map: {{val| 0.000 -0.126 -0.391 -0.268 }}
+: error map: {{val| 0.000 +0.158 -0.599 +0.489 }}
+{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
+[[Badness]] (Sintel): 2.23
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 1375/1372, 6250/6237, 65625/65536
+Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}
+Optimal tunings:
+* WE: ~128/125 = 42.8577{{c}}, ~3/2 = 702.0275{{c}}
+* CWE: ~128/125 = 42.8571{{c}}, ~3/2 = 702.0174{{c}}
+{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
+Badness (Sintel): 1.58
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
+Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
+Optimal tunings:
+* WE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0289{{c}}
+* CWE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}
-{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}
+{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
-[[Badness]] (Sintel): 0.654
+Badness (Sintel): 0.908
 == Emkay ==
@@ Line 200: / Line 256: @@
 Badness (Sintel): 1.93
-== Oquatonic ==
-: ''For the 5-limit version, see [[28th-octave temperaments #Oquatonic (5-limit)]].''
-Oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the [[dimcomp comma]] (390625/388962). In this temperament, the [[5/4]] major third is mapped to 9\28.
-The name ''oquatonic'' was given by [[Petr Pařízek]] in 2011 as an abbreviation of the Italian [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref name="petr's long post"/>.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 65625/65536, 390625/388962
-{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
-: mapping generators: ~128/125, ~3
-[[Optimal tuning]]s:
-* [[WE]]: ~128/125 = 42.8570{{c}}, ~3/2 = 702.1112{{c}}
-: [[error map]]: {{val| -0.004 +0.152 -0.609 +0.477 }}
-* [[CWE]]: ~128/125 = 42.8571{{c}}, ~3/2 = 702.1132{{c}}
-: error map: {{val| 0.000 +0.158 -0.599 +0.489 }}
-{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
-[[Badness]] (Sintel): 2.23
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 1375/1372, 6250/6237, 65625/65536
-Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}
-Optimal tunings:
-* WE: ~128/125 = 42.8577{{c}}, ~3/2 = 702.0275{{c}}
-* CWE: ~128/125 = 42.8571{{c}}, ~3/2 = 702.0174{{c}}
-{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
-Badness (Sintel): 1.58
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
-Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
-Optimal tunings:
-* WE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0289{{c}}
-* CWE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}
-{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
-Badness (Sintel): 0.908
 == Bezique ==