No-twos subgroup temperaments: Difference between revisions
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No-twos temperaments exclude the second harmonic, thus meaning the next lowest prime, 3, is the best remaining option for an equave. If you then also remove the 3rd harmonic, the best remaining option for the equave would be [[5/1]] (also known as the pentane). Of course you can keep removing even more primes making 7th, 11th, etc. harmonics as the equave, but it will probably be impractical. | No-twos temperaments exclude the second harmonic, thus meaning the next lowest prime, 3, is the best remaining option for an equave. If you then also remove the 3rd harmonic, the best remaining option for the equave would be [[5/1]] (also known as the pentane). Of course you can keep removing even more primes making 7th, 11th, etc. harmonics as the equave, but it will probably be impractical. | ||
== Pentadacus == | == Pentadacus == | ||
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Badness (Sintel): 0.116 | Badness (Sintel): 0.116 | ||
== Catfish == | |||
[[File:Salmon7.wav|thumb|Catfish[7] scale and 11:13:17 chord on A in 20ed11/5 tuning]] | |||
Named by {{u|CompactStar}} in 2023, catfish tempers out 54925/54043 and 1795625/1771561 in the 5.11.13.17 subgroup. | |||
CompactStar considers it useful to treat this temperament as [[11/5]]-repeating, which implies a generator approximating [[17/11]] or its inverse 121/85. It can be viewed as the no-2's no-3's analog to [[meantone family #Meantone|meantone]] with 11:13:17 being the substitute for [[4:5:6]], and possesses mos scales of the families {{MOS scalesig|2L 3s<11/5>}}, {{MOS scalesig|2L 5s<11/5>}}, {{MOS scalesig|2L 7s<11/5>}}, and {{MOS scalesig|9L 2s<11/5>}}. [[20ed11/5]] is one of the best tunings for this temperament, although it does noticeably flatten the [[13/11]] in the 11:13:17 chord. | |||
[[Subgroup]]: 5.11.13 | |||
[[Comma list]]: {{monzo| 14 -19 9 }} | |||
{{Mapping|legend=3| 1 -4 -10 | 0 9 19 }} | |||
: mapping generators: ~5, ~325/121 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~5 = 2785.1693{{c}}, ~325/121 = 1699.4922{{c}} | |||
: [[error map]]: {{val| -1.144 +3.435 -1.869 }} | |||
* [[CWE]]: ~5 = 2786.3137{{c}}, ~325/121 = 1700.1381{{c}} | |||
: error map: {{val| 0.000 +4.670 -1.042 }} | |||
[[Optimal ET sequence]]: [[18ed5|c18]], [[41ed5|c41]], [[59ed5|c59]], [[372ed5|c372e]], [[431ed5|c431e]], [[490ed5|c490e]] | |||
Badness (Sintel): 10.3 | |||
=== 5.11.13.17 subgroup === | |||
Subgroup: 5.11.13.17 | |||
Comma list: 54925/54043, 1795625/1771561 | |||
{{Mapping|legend=2| 1 -4 -10 -8 | 0 9 19 16 }} | |||
: mapping generators: ~5, ~325/121 | |||
Optimal tunings: | |||
* WE: ~5 = 2785.0096{{c}}, ~325/121 = 1699.3010{{c}} | |||
* CWE: ~5 = 2786.3137{{c}}, ~325/121 = 1700.0352{{c}} | |||
Optimal ET sequence: [[18ed5|c18]], [[41ed5|c41]], [[59ed5|c59]] | |||
Badness (Sintel): 0.905 | |||
; Scales | |||
* [[Catfish7]] | |||
* [[Catfish9]] | |||
= Graphs = | = Graphs = | ||