Triwellismic temperaments: Difference between revisions
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{{Technical data page}} | {{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 reduced and a septimal subfourth ([[21/16]]). | 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 [[ | 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 == | ||
Latest revision as of 15:49, 3 June 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
This page collects miscellaneous rank-2 triwellismic temperaments, which temper out the triwellisma (monzo: [1 -1 -7 6⟩, ratio: 235298/234375), the difference between seven 7/5's octave reduced and a septimal subfourth (21/16).
Temperaments discussed elsewhere include:
- Undecental (+5120/5103) → Hemifamity temperaments
- Tritonic (+225/224) → Marvel temperaments
- Sextilifourths (+32805/32768) → Schismatic family
- Ammonite (+250/243) → Porcupine family
- Decile (+321489/320000) → Quintile family
- Nusecond (+126/125) → Starling temperaments
- Chromat (+10976/10935) → Amity family
- Hemiwürschmidt (+2401/2400) → Hemimean clan
- Fourfives (+245/243) → Fifive family
- Triwell (+1029/1024) → Semicomma family
- Semidimfourth (+4375/4374) → 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 & 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
Comma list: 235298/234375, 33756345/33554432
Mapping: [⟨1 15 -26 -28], ⟨0 -27 57 62]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~10125/7168 = 596.247 ¢
Optimal ET sequence: 2, …, 157cd, 159, 320, 799dd, 1119bdd
Badness (Smith): 0.342890
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 102487/102400, 235298/234375
Mapping: [⟨1 15 -26 -28 -3], ⟨0 -27 57 62 13]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~512/363 = 596.248 ¢
Optimal ET sequence: 2, …, 157cd, 159, 320
Badness (Smith): 0.080833
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 1001/1000, 6656/6655, 26411/26364
Mapping: [⟨1 15 -26 -28 -3 -44], ⟨0 -27 57 62 13 96]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~55/39 = 596.248 ¢
Optimal ET sequence: 2f, …, 157cdf, 159, 320
Badness (Smith): 0.041506
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 441/440, 833/832, 1001/1000, 1089/1088, 8624/8619
Mapping: [⟨1 15 -26 -28 -3 -44 18], ⟨0 -27 57 62 13 96 -28]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 596.250 ¢
Optimal ET sequence: 2f, …, 157cdf, 159, 320
Badness (Smith): 0.028653
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 441/440, 513/512, 833/832, 969/968, 1001/1000, 1521/1520
Mapping: [⟨1 15 -26 -28 -3 -44 18 -36], ⟨0 -27 57 62 13 96 -28 81]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 596.250 ¢
Optimal ET sequence: 2fh, …, 157cdfh, 159, 161, 320
Badness (Smith): 0.020593
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 441/440, 513/512, 529/528, 833/832, 897/896, 969/968, 1001/1000
Mapping: [⟨1 15 -26 -28 -3 -44 18 -36 8], ⟨0 -27 57 62 13 96 -28 81 -7]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 596.249 ¢
Optimal ET sequence: 2fh, …, 157cdfh, 159, 320i
Badness (Smith): 0.017242
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 441/440, 513/512, 529/528, 551/550, 609/608, 783/782, 833/832, 969/968
Mapping: [⟨1 15 -26 -28 -3 -44 18 -36 8 -18], ⟨0 -27 57 62 13 96 -28 81 -7 46]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 596.249 ¢
Optimal ET sequence: 2fh, …, 157cdfhj, 159, 320ij
Badness (Smith): 0.014267