Triwellismic temperaments: Difference between revisions

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~~Triwellismic~~ rank-~~two~~ temperaments temper out the triwellisma, {{monzo|1 -1 -7 6}} = 235298/234375, the difference seven [[7/5]]s and septimal ~~triple-compound sub-fourth~~ ([[21/16]] ~~plus three octaves~~).

This page collects miscellaneous [[rank-2 temperament|rank-2]] '''triwellismic temperaments''', which [[tempering out|temper out]] the [[triwellisma]] ({{monzo|legend=1| 1 -1 -7 6 }}, [[ratio]]: 235298/234375), the difference between seven [[7/5]]'s [[octave reduction|octave reduced]] and a septimal subfourth ([[21/16]]).

Temperaments discussed elsewhere include [[~~Porcupine family~~ #~~Ammonite~~|~~ammonite~~]], [[~~Hemimage~~ temperaments #~~Chromat~~|~~chromat~~]], [[~~Pental~~ family #~~Decal~~|~~decal~~]], [[~~Fifive~~ family #~~Fourfives~~|~~fourfives~~]], [[~~Würschmidt~~ family #~~Hemiwürschmidt~~|~~hemiwürschmidt~~]], [[Starling temperaments #Nusecond|~~nusecond~~]], [[~~Ragismic microtemperaments~~ #~~Semidimfourth~~|~~semidimfourth~~]], [[~~Schismatic family~~ #~~sextilififths~~|~~sextilischis~~]], [[~~Marvel temperaments~~ #~~Tritonic~~|~~tritonic~~]], [[Semicomma family #Triwell|~~triwell~~]] ~~and~~ [[~~Hemifamity temperaments~~ #~~Undecental~~|~~undecental~~]].

Temperaments discussed elsewhere include:

* ''[[Undecental]]'' (+5120/5103) → [[Hemifamity temperaments #Undecental|Hemifamity temperaments]]

* ''[[Tritonic]]'' (+225/224) → [[Marvel temperaments #Tritonic|Marvel temperaments]]

* ''[[Sextilifourths]]'' (+32805/32768) → [[Schismatic family #Sextilifourths|Schismatic family]]

* ''[[Ammonite]]'' (+250/243) → [[Porcupine family #Ammonite|Porcupine family]]

* ''[[Decile]]'' (+321489/320000) → [[Quintile family #Decile|Quintile family]]

* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]

* ''[[Chromat]]'' (+10976/10935) → [[Amity family #Chromat|Amity family]]

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

* ''[[Fourfives]]'' (+245/243) → [[Fifive family #Fourfives|Fifive family]]

* ''[[Triwell]]'' (+1029/1024) → [[Semicomma family #Triwell|Semicomma family]]

* ''[[Semidimfourth]]'' (+4375/4374) → [[Ragismic microtemperaments #Semidimfourth|Ragismic microtemperaments]]

== Interaufo ==

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

The name "interaufo" means an augmented fourth interval between [[45/32]] (classic diatonic tritone) and [[729/512]] (Pythagorean tritone). The ''interaufo'' temperament (159&~~amp;~~161) tempers out the same 5-limit comma as the untriton, but has a generator as [[24/17]]-wide tritone, three generators makes 45/32 and five of them makes 7/5 with octave reduction.

The name ''interaufo'' means an augmented fourth interval between [[45/32]] (classic diatonic tritone) and [[729/512]] (Pythagorean tritone). The interaufo temperament (159 & 161) tempers out the same 5-limit comma as the untriton, but has a generator as [[24/17]]-wide tritone, three generators makes 45/32 and five of them makes 7/5 with octave reduction.

Subgroup: 2.3.5.7

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 235298/234375, 33756345/33554432

~~[[Mapping]]: [~~{{~~val~~|1 15 -26 -28~~}}, {{val~~|0 -27 57 62}}]

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10125/7168 = 596.247{{c}}

~~[[POTE generator]]: ~10125/7168~~ = ~~596.247~~

{{Optimal ET sequence|legend=1| 2, …, 157cd, 159, 320, 799dd, 1119bdd }}

~~{{Val list|legend=1| 2, …, 157cd, 159, 320, 799dd, 1119bdd }}~~

[[Badness]] (Smith): 0.342890

[[Badness]]: 0.342890

=== 11-limit ===

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Comma list: 441/440, 102487/102400, 235298/234375

Mapping: [{{~~val~~|1 15 -26 -28 -3~~}}, {{val~~|0 -27 57 62 13}}]

Mapping: {{mapping| 1 15 -26 -28 -3 | 0 -27 57 62 13 }}

POTE ~~generator~~: ~512/363 = 596.248

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~512/363 = 596.248{{c}}

Optimal ~~GPV~~ sequence~~: {{Val list~~| 2, …, 157cd, 159, 320 }}

Badness: 0.080833

Badness (Smith): 0.080833

=== 13-limit ===

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Comma list: 441/440, 1001/1000, 6656/6655, 26411/26364

Mapping: [{{~~val~~|1 15 -26 -28 -3 -44~~}}, {{val~~|0 -27 57 62 13 96}}]

Mapping: {{mapping| 1 15 -26 -28 -3 -44 | 0 -27 57 62 13 96 }}

POTE ~~generator~~: ~55/39 = 596.248

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~55/39 = 596.248{{c}}

Optimal ~~GPV~~ sequence~~: {{Val list~~| 2f, …, 157cdf, 159, 320 }}

Badness: 0.041506

Badness (Smith): 0.041506

=== 17-limit ===

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Comma list: 441/440, 833/832, 1001/1000, 1089/1088, 8624/8619

Mapping: [{{~~val~~|1 15 -26 -28 -3 -44 18~~}}, {{val~~|0 -27 57 62 13 96 -28}}]

Mapping: {{mapping| 1 15 -26 -28 -3 -44 18 | 0 -27 57 62 13 96 -28 }}

POTE ~~generator~~: ~24/17 = 596.250

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.250{{c}}

Optimal ~~GPV~~ sequence~~: {{Val list~~| 2f, …, 157cdf, 159, 320 }}

Badness: 0.028653

Badness (Smith): 0.028653

=== 19-limit ===

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Comma list: 441/440, 513/512, 833/832, 969/968, 1001/1000, 1521/1520

Mapping: [{{~~val~~|1 15 -26 -28 -3 -44 18 -36~~}}, {{val~~|0 -27 57 62 13 96 -28 81}}]

Mapping: {{mapping| 1 15 -26 -28 -3 -44 18 -36 | 0 -27 57 62 13 96 -28 81 }}

POTE ~~generator~~: ~24/17 = 596.250

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.250{{c}}

Optimal ~~GPV~~ sequence~~: {{Val list~~| 2fh, …, 157cdfh, 159, 161, 320 }}

{{Optimal ET sequence|legend=0| 2fh, …, 157cdfh, 159, 161, 320 }}

Badness: 0.020593

Badness (Smith): 0.020593

=== 23-limit ===

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Comma list: 441/440, 513/512, 529/528, 833/832, 897/896, 969/968, 1001/1000

Mapping: [{{~~val~~|1 15 -26 -28 -3 -44 18 -36 8~~}}, {{val~~|0 -27 57 62 13 96 -28 81 -7}}]

Mapping: {{mapping| 1 15 -26 -28 -3 -44 18 -36 8 | 0 -27 57 62 13 96 -28 81 -7 }}

POTE ~~generator~~: ~24/17 = 596.249

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.249{{c}}

Optimal ~~GPV~~ sequence~~: {{Val list~~| 2fh, …, 157cdfh, 159, 320i }}

Badness: 0.017242

Badness (Smith): 0.017242

=== 29-limit ===

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Comma list: 441/440, 513/512, 529/528, 551/550, 609/608, 783/782, 833/832, 969/968

Mapping: [{{~~val~~|1 15 -26 -28 -3 -44 18 -36 8 -18~~}}, {{val~~|0 -27 57 62 13 96 -28 81 -7 46}}]

Mapping: {{mapping| 1 15 -26 -28 -3 -44 18 -36 8 -18 | 0 -27 57 62 13 96 -28 81 -7 46 }}

POTE ~~generator~~: ~24/17 = 596.249

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.249{{c}}

Optimal ~~GPV~~ sequence~~: {{Val list~~| 2fh, …, 157cdfhj, 159, 320ij }}

Badness: 0.014267

Badness (Smith): 0.014267

[[Category:~~Temperament collection~~]]

[[Category:Triwellismic temperaments| ]]

[[Category:~~Triwellismic~~]]

[[Category:Temperament collections]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-Triwellismic rank-two temperaments temper out the triwellisma, {{monzo|1 -1 -7 6}} = 235298/234375, the difference seven [[7/5]]s and septimal triple-compound sub-fourth ([[21/16]] plus three octaves).
+{{Technical data page}}
+This page collects miscellaneous [[rank-2 temperament|rank-2]] '''triwellismic temperaments''', which [[tempering out|temper out]] the [[triwellisma]] ({{monzo|legend=1| 1 -1 -7 6 }}, [[ratio]]: 235298/234375), the difference between seven [[7/5]]'s [[octave reduction|octave reduced]] and a septimal subfourth ([[21/16]]).
-Temperaments discussed elsewhere include [[Porcupine family #Ammonite|ammonite]], [[Hemimage temperaments #Chromat|chromat]], [[Pental family #Decal|decal]], [[Fifive family #Fourfives|fourfives]], [[Würschmidt family #Hemiwürschmidt|hemiwürschmidt]], [[Starling temperaments #Nusecond|nusecond]], [[Ragismic microtemperaments #Semidimfourth|semidimfourth]], [[Schismatic family #sextilififths|sextilischis]], [[Marvel temperaments #Tritonic|tritonic]], [[Semicomma family #Triwell|triwell]] and [[Hemifamity temperaments #Undecental|undecental]].
+Temperaments discussed elsewhere include:
+* ''[[Undecental]]'' (+5120/5103) → [[Hemifamity temperaments #Undecental|Hemifamity temperaments]]
+* ''[[Tritonic]]'' (+225/224) → [[Marvel temperaments #Tritonic|Marvel temperaments]]
+* ''[[Sextilifourths]]'' (+32805/32768) → [[Schismatic family #Sextilifourths|Schismatic family]]
+* ''[[Ammonite]]'' (+250/243) → [[Porcupine family #Ammonite|Porcupine family]]
+* ''[[Decile]]'' (+321489/320000) → [[Quintile family #Decile|Quintile family]]
+* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
+* ''[[Chromat]]'' (+10976/10935) → [[Amity family #Chromat|Amity family]]
+* ''[[Hemiwürschmidt]]'' (+2401/2400) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
+* ''[[Fourfives]]'' (+245/243) → [[Fifive family #Fourfives|Fifive family]]
+* ''[[Triwell]]'' (+1029/1024) → [[Semicomma family #Triwell|Semicomma family]]
+* ''[[Semidimfourth]]'' (+4375/4374) → [[Ragismic microtemperaments #Semidimfourth|Ragismic microtemperaments]]
 == Interaufo ==
-{{see also| High badness temperaments #Untriton }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
-The name "interaufo" means an augmented fourth interval between [[45/32]] (classic diatonic tritone) and [[729/512]] (Pythagorean tritone). The ''interaufo'' temperament (159&amp;161) tempers out the same 5-limit comma as the untriton, but has a generator as [[24/17]]-wide tritone, three generators makes 45/32 and five of them makes 7/5 with octave reduction.
+The name ''interaufo'' means an augmented fourth interval between [[45/32]] (classic diatonic tritone) and [[729/512]] (Pythagorean tritone). The interaufo temperament (159 & 161) tempers out the same 5-limit comma as the untriton, but has a generator as [[24/17]]-wide tritone, three generators makes 45/32 and five of them makes 7/5 with octave reduction.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 235298/234375, 33756345/33554432
-[[Mapping]]: [{{val|1 15 -26 -28}}, {{val|0 -27 57 62}}]
+{{Mapping|legend=1| 1 15 -26 -28 | 0 -27 57 62 }}
-{{Multival|legend=1| 27 -57 -62 -153 -174 16 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10125/7168 = 596.247{{c}}
-[[POTE generator]]: ~10125/7168 = 596.247
+{{Optimal ET sequence|legend=1| 2, …, 157cd, 159, 320, 799dd, 1119bdd }}
-{{Val list|legend=1| 2, …, 157cd, 159, 320, 799dd, 1119bdd }}
+[[Badness]] (Smith): 0.342890
-[[Badness]]: 0.342890
 === 11-limit ===
@@ Line 27: / Line 37: @@
 Comma list: 441/440, 102487/102400, 235298/234375
-Mapping: [{{val|1 15 -26 -28 -3}}, {{val|0 -27 57 62 13}}]
+Mapping: {{mapping| 1 15 -26 -28 -3 | 0 -27 57 62 13 }}
-POTE generator: ~512/363 = 596.248
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~512/363 = 596.248{{c}}
-Optimal GPV sequence: {{Val list| 2, …, 157cd, 159, 320 }}
+{{Optimal ET sequence|legend=0| 2, …, 157cd, 159, 320 }}
-Badness: 0.080833
+Badness (Smith): 0.080833
 === 13-limit ===
@@ Line 40: / Line 50: @@
 Comma list: 441/440, 1001/1000, 6656/6655, 26411/26364
-Mapping: [{{val|1 15 -26 -28 -3 -44}}, {{val|0 -27 57 62 13 96}}]
+Mapping: {{mapping| 1 15 -26 -28 -3 -44 | 0 -27 57 62 13 96 }}
-POTE generator: ~55/39 = 596.248
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~55/39 = 596.248{{c}}
-Optimal GPV sequence: {{Val list| 2f, …, 157cdf, 159, 320 }}
+{{Optimal ET sequence|legend=0| 2f, …, 157cdf, 159, 320 }}
-Badness: 0.041506
+Badness (Smith): 0.041506
 === 17-limit ===
@@ Line 53: / Line 63: @@
 Comma list: 441/440, 833/832, 1001/1000, 1089/1088, 8624/8619
-Mapping: [{{val|1 15 -26 -28 -3 -44 18}}, {{val|0 -27 57 62 13 96 -28}}]
+Mapping: {{mapping| 1 15 -26 -28 -3 -44 18 | 0 -27 57 62 13 96 -28 }}
-POTE generator: ~24/17 = 596.250
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.250{{c}}
-Optimal GPV sequence: {{Val list| 2f, …, 157cdf, 159, 320 }}
+{{Optimal ET sequence|legend=0| 2f, …, 157cdf, 159, 320 }}
-Badness: 0.028653
+Badness (Smith): 0.028653
 === 19-limit ===
@@ Line 66: / Line 76: @@
 Comma list: 441/440, 513/512, 833/832, 969/968, 1001/1000, 1521/1520
-Mapping: [{{val|1 15 -26 -28 -3 -44 18 -36}}, {{val|0 -27 57 62 13 96 -28 81}}]
+Mapping: {{mapping| 1 15 -26 -28 -3 -44 18 -36 | 0 -27 57 62 13 96 -28 81 }}
-POTE generator: ~24/17 = 596.250
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.250{{c}}
-Optimal GPV sequence: {{Val list| 2fh, …, 157cdfh, 159, 161, 320 }}
+{{Optimal ET sequence|legend=0| 2fh, …, 157cdfh, 159, 161, 320 }}
-Badness: 0.020593
+Badness (Smith): 0.020593
 === 23-limit ===
@@ Line 79: / Line 89: @@
 Comma list: 441/440, 513/512, 529/528, 833/832, 897/896, 969/968, 1001/1000
-Mapping: [{{val|1 15 -26 -28 -3 -44 18 -36 8}}, {{val|0 -27 57 62 13 96 -28 81 -7}}]
+Mapping: {{mapping| 1 15 -26 -28 -3 -44 18 -36 8 | 0 -27 57 62 13 96 -28 81 -7 }}
-POTE generator: ~24/17 = 596.249
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.249{{c}}
-Optimal GPV sequence: {{Val list| 2fh, …, 157cdfh, 159, 320i }}
+{{Optimal ET sequence|legend=0| 2fh, …, 157cdfh, 159, 320i }}
-Badness: 0.017242
+Badness (Smith): 0.017242
 === 29-limit ===
@@ Line 92: / Line 102: @@
 Comma list: 441/440, 513/512, 529/528, 551/550, 609/608, 783/782, 833/832, 969/968
-Mapping: [{{val|1 15 -26 -28 -3 -44 18 -36 8 -18}}, {{val|0 -27 57 62 13 96 -28 81 -7 46}}]
+Mapping: {{mapping| 1 15 -26 -28 -3 -44 18 -36 8 -18 | 0 -27 57 62 13 96 -28 81 -7 46 }}
-POTE generator: ~24/17 = 596.249
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.249{{c}}
-Optimal GPV sequence: {{Val list| 2fh, …, 157cdfhj, 159, 320ij }}
+{{Optimal ET sequence|legend=0| 2fh, …, 157cdfhj, 159, 320ij }}
-Badness: 0.014267
+Badness (Smith): 0.014267
-[[Category:Temperament collection]]
+[[Category:Triwellismic temperaments| ]] <!-- main article -->
-[[Category:Triwellismic]]
+[[Category:Temperament collections]]
 [[Category:Rank 2]]