No-twos subgroup temperaments: Difference between revisions
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Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]]. | Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]]. | ||
* Arcturus, antares, aldebaran and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals. | * Arcturus, antares, aldebaran, delta centauri and polaris have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals. | ||
* BPS has a ~9/7 generator, two of which give the ~5/3. | * BPS has a ~9/7 generator, two of which give the ~5/3. | ||
* Sirius has a ~25/21 generator, three of which give the ~5/3. | * Sirius has a ~25/21 generator, three of which give the ~5/3. | ||
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=== Polturus === | === Polturus === | ||
This extension of arcturus adds [[polaris]]'s mapping for [[11/9]], mapping it to 5 generators down. | This extension of arcturus adds [[#Polaris|polaris]]'s mapping for [[11/9]], mapping it to 5 generators down. | ||
Subgroup: 3.5.7.11 | Subgroup: 3.5.7.11 | ||
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Badness (Sintel): 0.177 | Badness (Sintel): 0.177 | ||
== Vega == | |||
[[Subgroup]]: 3.5.7 | |||
[[Comma list]]: 6561/6125 | |||
{{Mapping|legend=3| 1 0 4 | 0 2 -3 }} | |||
: mapping generators: ~3, ~81/35 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~3 = 1891.9431{{c}}, ~81/35 = 1397.0975{{c}} | |||
: [[error map]]: {{val| -10.012 +7.881 +7.654 }} | |||
* [[CWE]]: ~3 = 1901.9550{{c}}, ~81/35 = 1404.6664{{c}} | |||
: error map: {{val| 0.000 +23.019 +24.995 }} | |||
[[Optimal ET sequence]]: [[4edt|b4]], [[11edt|b11d]], [[15edt|b15]], [[19edt|b19]], [[23edt|b23]], [[42edt|b42d]], [[65edt|b65cd]] | |||
[[Badness]] (Sintel): 1.34 | |||
== Sirius == | == Sirius == | ||
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== Miaplacidus == | == Miaplacidus == | ||
Miaplacidus tempers out [[5859375/5764801]], the amount by which eight [[7/5]]'s fall short of [[15/1]]. It thus tunes the chord 7:9:15 accurately. Having a generator of ~7/5, it possesses [[mos scale]]s of the families {{mos scalesig|link= | Miaplacidus tempers out [[5859375/5764801]], the amount by which eight [[7/5]]'s fall short of [[15/1]]. It thus tunes the chord 7:9:15 accurately. Having a generator of ~7/5, it possesses [[mos scale]]s of the families {{mos scalesig|link=1|3L 4s<3/1>}}, {{mos scalesig|link=1|3L 7s<3/1>}}, {{mos scalesig|link=1|3L 10s<3/1>}}, and {{mos scalesig|link=1|13L 3s<3/1>}}. Good tunings for this temperament are [[13edt]], [[16edt]], and [[29edt]]. | ||
[[Subgroup]]: 3.5.7 | [[Subgroup]]: 3.5.7 | ||
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= 3.5.11-subgroup temperaments = | = 3.5.11-subgroup temperaments = | ||
== Polaris == | == Polaris == | ||
Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]]. It contains [[mos scale]]s of the families {{mos scalesig|link=1|2L 5s<3/1>}}, {{mos scalesig|link=1|2L 7s<3/1>}}, and {{mos scalesig|link=1|2L 9s<3/1>}}. The hendecatonic scale is generally the most usable of these, with the heptatonic and enneatonic scales usually being lopsided. Good tunings for polaris include [[15edt]], [[17edt]] and [[32edt]]. | |||
Polaris tempers out the comma 177147/171875, and thus equates seven [[5/3]]'s with [[15/11]], or equivalently seven [[9/5]]'s with [[11/9]]. | |||
[[Subgroup]]: 3.5.11 | [[Subgroup]]: 3.5.11 | ||
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[[Badness]] (Sintel): 1.01 | [[Badness]] (Sintel): 1.01 | ||
; Scales | |||
* [[Polaris9]] | |||
* [[Polaris11]] | |||
== Delta centauri == | == Delta centauri == | ||
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Named by {{u|CompactStar}} in 2023, catfish tempers out 54925/54043 and 1795625/1771561 in the 5.11.13.17 subgroup. | 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 {{ | 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 | [[Subgroup]]: 5.11.13 | ||
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* [[Catfish7]] | * [[Catfish7]] | ||
* [[Catfish9]] | * [[Catfish9]] | ||
[[Category:Catalogs of subgroup temperaments]] | [[Category:Catalogs of subgroup temperaments]] | ||
[[Category:Catalogs of rank-2 temperaments]] | [[Category:Catalogs of rank-2 temperaments]] | ||