Horwell temperaments: Difference between revisions

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Temperaments discussed elsewhere are

* ''[[~~Semabila~~]]'' (+49/48) → [[~~Mabila~~ family #~~Septimal mabila~~|~~Mabila~~ family]]

* [[Pontiac]] (+4375/4374) → [[Schismatic family #Pontiac|Schismatic family]]

* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic family]]

* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity family]]

* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]

* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]

* ''[[Worschmidt]]'' (+126/125) → [[Würschmidt family #Worschmidt|Würschmidt family]]

* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped|Escapade family]]

* ''[[~~Maquiloid~~]]'' (+~~686~~/~~675~~) → [[~~Maquila~~ family #~~Maquiloid~~|~~Maquila~~ family]]

* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]

* ''[[~~Keen~~]]'' (+~~875~~/~~864~~) → [[~~Diaschismic~~ family #~~Keen~~|~~Diaschismic~~ 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]]

* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]

* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]

* [[Pontiac]] (+4375/4374) → [[Schismatic family #Pontiac|Schismatic family]]

* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]

* ''[[Bisupermajor]]'' (+10976/10935) → [[Hemimage temperaments #Bisupermajor|Hemimage temperaments]]

* ''[[~~Eris~~]]'' (+~~16875~~/~~16807~~) → [[~~Mirkwai clan #Eris|Mirkwai clan]]~~

* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]

* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]

* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #~~Paramity~~|~~Amity~~ family]]

* ''[[Kaboom]]'' (+4802000/4782969) → [[Vavoom family #Kaboom|Vavoom family]]

* [[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]]

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: ''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 ===

Subgroup: 2.3.5.7.17

Comma list: 1225/1224, 24576/24565, 57375/57344

Subgroup-val mapping: {{mapping| 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 }}

Optimal tunings:

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Comma list: 364/363, 441/440, 625/624, 2200/2197

~~Mapping:~~ {{~~mapping~~| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}

Optimal tunings:

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Comma list: 1375/1372, 6250/6237, 65625/65536

~~Mapping:~~ {{~~mapping~~| 28 0 65 123 230 | 0 1 0 -1 -3 }}

Optimal tunings:

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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:

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Comma list: 3025/3024, 4000/3993, 65625/65536

~~Mapping:~~ {{~~mapping~~| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}

Optimal tunings:

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Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}

Optimal tunings:

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Comma list: 385/384, 3388/3375, 12005/11979

~~Mapping:~~ {{~~mapping~~| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}

Optimal tunings:

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Comma list: 169/168, 364/363, 385/384, 3388/3375

~~Mapping:~~ {{~~mapping~~| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}

Optimal tunings:

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Comma list: 9801/9800, 46656/46585, 65625/65536

~~Mapping:~~ {{~~mapping~~| 32 0 125 -113 60 | 0 1 -1 4 1 }}

Optimal tunings:

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Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655

~~Mapping:~~ {{~~mapping~~| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}

Optimal tunings:

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== References ==

[[Category:Horwell temperaments| ]]

[[Category:Temperament collections]]

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

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

@@ Line 3: / Line 3: @@
 Temperaments discussed elsewhere are
-* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
+* [[Pontiac]] (+4375/4374) → [[Schismatic family #Pontiac|Schismatic family]]
+* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic family]]
+* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity family]]
+* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]
+* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
 * ''[[Worschmidt]]'' (+126/125) → [[Würschmidt family #Worschmidt|Würschmidt family]]
 * ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped|Escapade family]]
-* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]
+* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
-* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic 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]]
-* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
-* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
-* [[Pontiac]] (+4375/4374) → [[Schismatic family #Pontiac|Schismatic family]]
-* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]
 * ''[[Bisupermajor]]'' (+10976/10935) → [[Hemimage temperaments #Bisupermajor|Hemimage temperaments]]
-* ''[[Eris]]'' (+16875/16807) → [[Mirkwai clan #Eris|Mirkwai clan]]
+* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]
-* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]
-* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity family]]
 * ''[[Kaboom]]'' (+4802000/4782969) → [[Vavoom family #Kaboom|Vavoom family]]
+* [[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]]
@@ Line 25: / Line 26: @@
 : ''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
+Subgroup-val mapping: {{mapping| 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 ==
@@ Line 72: / Line 93: @@
 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 108: @@
 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 147: @@
 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 162: @@
 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 201: @@
 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 216: @@
 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 253: @@
 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 268: @@
 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 282: / Line 303: @@
 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 318: @@
 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 330: @@
 == References ==
+[[Category:Horwell temperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Horwell temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]