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). | 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). | ||
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 [[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]]. | ||
== Interaufo == | == Interaufo == | ||
{{see also| High badness temperaments #Untriton }} | {{see also| High badness 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 | Subgroup: 2.3.5.7 | ||
Revision as of 02:51, 29 January 2022
Triwellismic rank-two temperaments temper out the triwellisma, [1 -1 -7 6⟩ = 235298/234375, the difference seven 7/5s and septimal triple-compound sub-fourth (21/16 plus three octaves).
Temperaments discussed elsewhere include ammonite, chromat, decal, fourfives, hemiwürschmidt, nusecond, semidimfourth, sextilischis, tritonic, triwell and undecental.
Interaufo
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]]
Wedgie: ⟨⟨ 27 -57 -62 -153 -174 16 ]]
POTE generator: ~10125/7168 = 596.247
Badness: 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]]
POTE generator: ~512/363 = 596.248
Optimal GPV sequence: Template:Val list
Badness: 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]]
POTE generator: ~55/39 = 596.248
Optimal GPV sequence: Template:Val list
Badness: 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]]
POTE generator: ~24/17 = 596.250
Optimal GPV sequence: Template:Val list
Badness: 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]]
POTE generator: ~24/17 = 596.250
Optimal GPV sequence: Template:Val list
Badness: 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]]
POTE generator: ~24/17 = 596.249
Optimal GPV sequence: Template:Val list
Badness: 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]]
POTE generator: ~24/17 = 596.249
Optimal GPV sequence: Template:Val list
Badness: 0.014267