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Triwellismic temperaments: Difference between revisions

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Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Undecental]]'' (+5120/5103) → [[Hemifamity temperaments #Undecental|Hemifamity temperaments]]
* ''[[Undecental]]'' (+5120/5103) → [[Aberschismic temperaments #Undecental|Aberschismic temperaments]]
* ''[[Tritonic]]'' (+225/224) → [[Marvel temperaments #Tritonic|Marvel temperaments]]
* ''[[Tritonic]]'' (+225/224) → [[Marvel temperaments #Tritonic|Marvel temperaments]]
* ''[[Sextilifourths]]'' (+32805/32768) → [[Schismatic family #Sextilifourths|Schismatic family]]
* ''[[Sextilifourths]]'' (+32805/32768) → [[Schismatic family #Sextilifourths|Schismatic family]]
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: ''For the 5-limit version, see [[Miscellaneous 5-limit 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.
Interaufo may be described as the {{nowrap| 159 & 161 }} temperament. It tempers out the same 5-limit comma as the untriton, but has a generator as a [[~]][[17/12]] tritone, three generators make [[64/45]] and five make [[10/7]] with [[octave reduction]]. It was named by [[Xenllium]] in 2022, referring to the augmented fourth between [[45/32]] (ptolemaic augmented fourth) and [[729/512]] (Pythagorean augmented fourth).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 235298/234375, 33756345/33554432
[[Comma list]]: 235298/234375, 33756345/33554432


{{Mapping|legend=1| 1 15 -26 -28 | 0 -27 57 62 }}
{{Mapping|legend=1| 1 -12 31 34 | 0 27 -57 -62 }}
: mapping generators: ~2, ~14336/10125


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10125/7168 = 596.247{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2580{{c}}, ~14336/10125 = 603.8868{{c}}
: [[error map]]: {[val| +0.258 -0.216 +0.366 -0.787 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14336/10125 = 603.7517{{c}}
: error map: {[val| 0.000 -0.659 -0.161 -1.432 }}


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


[[Badness]] (Smith): 0.342890
[[Badness]] (Sintel): 8.68


=== 11-limit ===
=== 11-limit ===
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Comma list: 441/440, 102487/102400, 235298/234375
Comma list: 441/440, 102487/102400, 235298/234375


Mapping: {{mapping| 1 15 -26 -28 -3 | 0 -27 57 62 13 }}
{{Mapping|legend=0| 1 -12 31 34 10 | 0 27 -57 -62 -13 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~512/363 = 596.248{{c}}
Optimal tunings:
* WE: ~2 = 1200.2088{{c}}, ~363/256 = 603.8574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~363/256 = 603.7515{{c}}


{{Optimal ET sequence|legend=0| 2, …, 157cd, 159, 320 }}
{{Optimal ET sequence|legend=0| 159, 320 }}


Badness (Smith): 0.080833
Badness (Sintel): 2.67


=== 13-limit ===
=== 13-limit ===
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Comma list: 441/440, 1001/1000, 6656/6655, 26411/26364
Comma list: 441/440, 1001/1000, 6656/6655, 26411/26364


Mapping: {{mapping| 1 15 -26 -28 -3 -44 | 0 -27 57 62 13 96 }}
{{Mapping|legend=0| 1 -12 31 34 10 52 | 0 27 -57 -62 -13 -96 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~55/39 = 596.248{{c}}
Optimal tunings:
* WE: ~2 = 1200.2085{{c}}, ~78/55 = 603.8574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/55 = 603.7513{{c}}


{{Optimal ET sequence|legend=0| 2f, …, 157cdf, 159, 320 }}
{{Optimal ET sequence|legend=0| 159, 320 }}


Badness (Smith): 0.041506
Badness (Sintel): 1.72


=== 17-limit ===
=== 17-limit ===
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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: {{mapping| 1 15 -26 -28 -3 -44 18 | 0 -27 57 62 13 96 -28 }}
{{Mapping|legend=0| 1 -12 31 34 10 52 -10 | 0 27 -57 -62 -13 -96 28 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.250{{c}}
Optimal tunings:
* WE: ~2 = 1200.1599{{c}}, ~17/12 = 603.8310{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 603.7499{{c}}


{{Optimal ET sequence|legend=0| 2f, …, 157cdf, 159, 320 }}
{{Optimal ET sequence|legend=0| 159, 320 }}


Badness (Smith): 0.028653
Badness (Sintel): 1.46


=== 19-limit ===
=== 19-limit ===
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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: {{mapping| 1 15 -26 -28 -3 -44 18 -36 | 0 -27 57 62 13 96 -28 81 }}
{{Mapping|legend=0| 1 -12 31 34 10 52 -10 45 | 0 27 -57 -62 -13 -96 28 -81 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.250{{c}}
Optimal tunings:
* WE: ~2 = 1200.1710{{c}}, ~17/12 = 603.8357{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 603.7490{{c}}


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


Badness (Smith): 0.020593
Badness (Sintel): 1.25


=== 23-limit ===
=== 23-limit ===
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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: {{mapping| 1 15 -26 -28 -3 -44 18 -36 8 | 0 -27 57 62 13 96 -28 81 -7 }}
{{Mapping|legend=0| 1 -12 31 34 10 52 -10 45 1 | 0 27 -57 -62 -13 -96 28 -81 7 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.249{{c}}
Optimal tunings:
* WE: ~2 = 1200.2121{{c}}, ~17/12 = 603.8577{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 603.7503{{c}}


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


Badness (Smith): 0.017242
Badness (Sintel): 1.23


=== 29-limit ===
=== 29-limit ===
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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: {{mapping| 1 15 -26 -28 -3 -44 18 -36 8 -18 | 0 -27 57 62 13 96 -28 81 -7 46 }}
{{Mapping|legend=0| 1 -12 31 34 10 52 -10 45 1 28 | 0 27 -57 -62 -13 -96 28 -81 7 -46 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~24/17 = 596.249{{c}}
Optimal tunings:
* WE: ~2 = 1200.2331{{c}}, ~17/12 = 603.8681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 603.7500{{c}}


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


Badness (Smith): 0.014267
Badness (Sintel): 1.19


[[Category:Triwellismic temperaments| ]] <!-- main article -->
[[Category:Triwellismic temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
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