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 {{Technical data page}}
-This is a collection of [[rank-2 temperament|rank-2]] '''horwell temperaments''', which temper out the [[horwell comma]] ({{monzo|legend=1| -16 1 5 1 }}, [[ratio]]: 65625/65536).
+This page collects [[rank-2 temperament|rank-2]] '''horwell temperaments''', which [[tempering out|temper out]] the [[horwell comma]] ({{monzo|legend=1| -16 1 5 1 }}, [[ratio]]: 65625/65536). For the family of [[rank-3 temperament]]s, see [[Horwell family]].
 Temperaments discussed elsewhere are
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 * ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped|Escapade family]]
 * ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
+* ''[[Counterpara]]'' (+100442349/100000000) → [[Parakleismic family #Counterpara|Parakleismic family]]
 * ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]
 * [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds|Hemimean clan]]
@@ Line 18: / Line 19: @@
 * [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
 * ''[[Eris]]'' (+16875/16807) → [[Canopic clan #Eris|Canopic clan]]
-* ''[[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]].
+Considered below are fifthplus, mutt, oquatonic, emkay, kastro, soviet ferris wheel, 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]].
+Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }} temperament. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator.
+Fifthplus may be extended to the 2.3.5.7.17 subgroup, called prime archagall, derived from the fact that it is an extension of [[archagall]] to a prime subgroup. In either the 7-limit or the add-17 subgroup, [[171edo]] is exceptionally efficient and accurate, but in the latter case 171edo is the tuning where [[2401/2400]] (S49), [[2500/2499]] (S50), and [[1701/1700]] (S18/S20) all vanish, which is natural because this temperament tempers out [[1225/1224]] (S35, S49⋅S50) and [[5832/5831]] ((S18/S20)/S49) while not tempering out any of above individually.
 [[Subgroup]]: 2.3.5.7
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 : error map: {{val| 0.000 -0.126 -0.391 -0.268 }}
-{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}
+{{Optimal ET sequence|legend=1| 22, 105d, 127d, 149, 171, 1903c, 2074c, …, 3613ccd }}
 [[Badness]] (Sintel): 0.654
+=== Prime archagall ===
+{{ See also | MVP archagall }}
+{{ See also | 24576/24565 }}
+Subgroup: 2.3.5.7.17
+Comma list: 1225/1224, 24576/24565, 57375/57344
+{{Mapping|legend=2| 1 -12 10 -22 -3 | 0 23 -13 42 12 }}
+Optimal tunings:
+* WE: ~2 = 1200.0516{{c}}, ~128/85 = 708.8057{{cent}}
+* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 708.7758{{cent}}
+{{Optimal ET sequence|legend=0| 22, 105d, 127d, 149, 171, 1219, 1390 }}
+Badness (Sintel): 0.421
 == Mutt ==
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 Comma list: 441/440, 4375/4356, 16384/16335
-Mapping: {{mapping| 3 -2 6 20 21 | 0 7 1 -12 -11 }}
+{{Mapping|legend=0| 3 -2 6 20 21 | 0 7 1 -12 -11 }}
 Optimal tunings:
@@ Line 87: / Line 107: @@
 Comma list: 364/363, 441/440, 625/624, 2200/2197
-Mapping: {{mapping| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}
+{{Mapping|legend=0| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}
 Optimal tunings:
@@ Line 126: / Line 146: @@
 Comma list: 1375/1372, 6250/6237, 65625/65536
-Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}
+{{Mapping|legend=0| 28 0 65 123 230 | 0 1 0 -1 -3 }}
 Optimal tunings:
@@ Line 141: / Line 161: @@
 Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
-Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
+{{Mapping|legend=0| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
 Optimal tunings:
@@ Line 180: / Line 200: @@
 Comma list: 3025/3024, 4000/3993, 65625/65536
-Mapping: {{mapping| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}
+{{Mapping|legend=0| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}
 Optimal tunings:
@@ Line 195: / Line 215: @@
 Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197
-Mapping: {{mapping| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}
+{{Mapping|legend=0| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}
 Optimal tunings:
@@ Line 232: / Line 252: @@
 Comma list: 385/384, 3388/3375, 12005/11979
-Mapping: {{mapping| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}
+{{Mapping|legend=0| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}
 Optimal tunings:
@@ Line 247: / Line 267: @@
 Comma list: 169/168, 364/363, 385/384, 3388/3375
-Mapping: {{mapping| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}
+{{Mapping|legend=0| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}
 Optimal tunings:
@@ Line 256: / Line 276: @@
 Badness (Sintel): 1.93
+== Soviet ferris wheel ==
+The soviet-ferris-wheel temperament may be described as 320 & 460. It was named by [[Eliora]] in 2022 because it has a period of 1/20 octave, and there are 20 cabins on a standard ferris wheel found throughout most of Eastern Europe and Central Asia (as in abandoned Pripyat wheel, for example).
+Restricting the temperament to the 5-limit tempers out the comma which is produced by closing twenty [[375/256]]'s at 11 octaves, tempering this interval to 11\20.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 65625/65536, 1977326743/1968300000
+{{Mapping|legend=1| 20 0 57 35 | 0 3 -1 2 }}
+: mapping generators: ~16807/16200, ~3456/2401
+[[Optimal tuning]]s:
+* [[WE]]: ~16807/16200 = 60.0061{{c}}, ~3456/2401 = 634.0252{{c}} (~686/675 = 26.0415{{c}})
+: [[error map]]: {{val| +0.121 +0.121 +0.007 -0.563 }}
+* [[CWE]]: ~16807/16200 = 60.0000{{c}}, ~3456/2401 = 633.9770{{c}} (~686/675 = 26.0230{{c}})
+: error map: {{val| 0.000 -0.024 -0.291 -0.872 }}
+{{Optimal ET sequence|legend=1| 140, 320, 460, 600, 1060d }}
+[[Badness]] (Sintel): 5.25
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 9801/9800, 65625/65536, 184877/184320
+{{Mapping|legend=0| 20 0 57 35 122 | 0 3 -1 2 5 }}
+Optimal tunings:
+* WE: ~512/495 = 60.0085{{c}}, ~231/160 = 633.9990{{c}} (~625/616 = 26.0943{{c}})
+* CWE: ~512/495 = 60.0000{{c}}, ~231/160 = 633.9127{{c}} (~625/616 = 26.0873{{c}})
+{{Optimal ET sequence|legend=0| 140, 320, 460 }}
+Badness (Sintel): 2.92
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 1001/1000, 4225/4224, 4459/4455, 26411/26364
+{{Mapping|legend=0| 20 0 57 35 122 74 | 0 3 -1 2 5 0 }}
+Optimal tunings:
+* WE: ~512/495 = 60.0082{{c}}, ~75/52 = 633.9971{{c}} (~65/64 = 26.2937{{c}})
+* CWE: ~512/495 = 60.0000{{c}}, ~75/52 = 633.9127{{c}} (~65/64 = 26.0873{{c}})
+{{Optimal ET sequence|legend=0| 140, 320, 460 }}
+Badness (Sintel): 1.48
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 833/832, 1001/1000, 1225/1224, 4225/4224, 4459/4455
+{{Mapping|legend=0| 20 0 57 35 122 74 124 | 0 3 -1 2 5 0 -4 }}
+Optimal tunings:
+* WE: ~88/85 = 60.0081{{c}}, ~75/52 = 633.9982{{c}} (~65/64 = 26.0911{{c}})
+* CWE: ~88/85 = 60.0000{{c}}, ~75/52 = 633.9117{{c}} (~65/64 = 26.0883{{c}})
+{{Optimal ET sequence|legend=0| 140, 320, 460 }}
+Badness (Sintel): 1.14
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 833/832, 1001/1000, 1225/1224, 1729/1728, 2926/2925, 4459/4455
+{{Mapping|legend=0| 20 0 57 35 122 74 124 11 | 0 3 -1 2 5 0 -4 7 }}
+Optimal tunings:
+* WE: ~88/85 = 60.0075{{c}}, ~75/52 = 633.9713{{c}} (~65/64 = 26.1108{{c}})
+* CWE: ~88/85 = 60.0000{{c}}, ~75/52 = 633.8948{{c}} (~65/64 = 26.1052{{c}})
+{{Optimal ET sequence|legend=0| 140, 320, 460 }}
+Badness (Sintel): 1.16
 == Bezique ==
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 Comma list: 9801/9800, 46656/46585, 65625/65536
-Mapping: {{mapping| 32 0 125 -113 60 | 0 1 -1 4 1 }}
+{{Mapping|legend=0| 32 0 125 -113 60 | 0 1 -1 4 1 }}
 Optimal tunings:
@@ Line 297: / Line 399: @@
 Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655
-Mapping: {{mapping| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}
+{{Mapping|legend=0| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}
 Optimal tunings:
@@ Line 309: / Line 411: @@
 == References ==
+[[Category:Horwell temperaments| ]] <!-- main article -->
+[[Category:Horwell| ]] <!-- key article -->
 [[Category:Temperament collections]]
-[[Category:Horwell temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]

Horwell temperaments: Difference between revisions

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