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


[[Badness]]: 0.342890
[[Badness]] (Smith): 0.342890


=== 11-limit ===
=== 11-limit ===
Line 27: Line 37:
Comma list: 441/440, 102487/102400, 235298/234375
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 ET sequence|legend=1| 2, …, 157cd, 159, 320 }}
{{Optimal ET sequence|legend=0| 2, …, 157cd, 159, 320 }}


Badness: 0.080833
Badness (Smith): 0.080833


=== 13-limit ===
=== 13-limit ===
Line 40: Line 50:
Comma list: 441/440, 1001/1000, 6656/6655, 26411/26364
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 ET sequence|legend=1| 2f, …, 157cdf, 159, 320 }}
{{Optimal ET sequence|legend=0| 2f, …, 157cdf, 159, 320 }}


Badness: 0.041506
Badness (Smith): 0.041506


=== 17-limit ===
=== 17-limit ===
Line 53: Line 63:
Comma list: 441/440, 833/832, 1001/1000, 1089/1088, 8624/8619
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 ET sequence|legend=1| 2f, …, 157cdf, 159, 320 }}
{{Optimal ET sequence|legend=0| 2f, …, 157cdf, 159, 320 }}


Badness: 0.028653
Badness (Smith): 0.028653


=== 19-limit ===
=== 19-limit ===
Line 66: Line 76:
Comma list: 441/440, 513/512, 833/832, 969/968, 1001/1000, 1521/1520
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 ET sequence|legend=1| 2fh, …, 157cdfh, 159, 161, 320 }}
{{Optimal ET sequence|legend=0| 2fh, …, 157cdfh, 159, 161, 320 }}


Badness: 0.020593
Badness (Smith): 0.020593


=== 23-limit ===
=== 23-limit ===
Line 79: Line 89:
Comma list: 441/440, 513/512, 529/528, 833/832, 897/896, 969/968, 1001/1000
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 ET sequence|legend=1| 2fh, …, 157cdfh, 159, 320i }}
{{Optimal ET sequence|legend=0| 2fh, …, 157cdfh, 159, 320i }}


Badness: 0.017242
Badness (Smith): 0.017242


=== 29-limit ===
=== 29-limit ===
Line 92: Line 102:
Comma list: 441/440, 513/512, 529/528, 551/550, 609/608, 783/782, 833/832, 969/968
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 ET sequence|legend=1| 2fh, …, 157cdfhj, 159, 320ij }}
{{Optimal ET sequence|legend=0| 2fh, …, 157cdfhj, 159, 320ij }}


Badness: 0.014267
Badness (Smith): 0.014267


[[Category:Triwellismic temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Triwellismic]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Triwellismic_temperaments"