No-twos subgroup temperaments: Difference between revisions
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Erigone splits the (tritave-augmented) generator of [[No-twos subgroup temperaments#Procyon|procyon]] into three, allowing for an accurate representation of 11/9 at -19 generators and 13/9 at -13 generators. | Erigone splits the (tritave-augmented) generator of [[No-twos subgroup temperaments#Procyon|procyon]] into three, allowing for an accurate representation of 11/9 at -19 generators and 13/9 at -13 generators. | ||
[[Subgroup]]: 3.5.7.11.13 | [[Subgroup]]: [[3.5.7.11.13_subgroup|3.5.7.11.13]] | ||
[[Comma list]]: 1575/1573, 4459/4455 | [[Comma list]]: 847/845, 1575/1573, 4459/4455 | ||
{{Mapping|legend=2|1 9 5 9 7|0 -21 -9 -19 -13}} | {{Mapping|legend=2|1 9 5 9 7|0 -21 -9 -19 -13}} | ||
: sval mapping generators: ~3, ~49/33 | : [[Transversal_generators|sval mapping generators]]: ~[[3/1|3]], ~[[49/33]] | ||
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~49/33 = 682. | [[Optimal tuning]] ([[WE]]): ~3 = 1901.9699, ~49/33 = 682.4486 <!-- IDK why the second rounded value disagrees with Sintel's calculator, I got it from FloraC's --> | ||
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~49/33 = 682.4427 | |||
[[Optimal ET sequence]]: [[25edt|b25ce]], [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[301edt|b301]], [[471edt|b471]] | [[Optimal ET sequence]]: [[25edt|b25ce]], [[39edt|b39]], [[92edt|b92]], [[131edt|b131]], [[170edt|b170]], [[301edt|b301]], [[471edt|b471]] | ||
Badness (Dirichlet): 0. | [[Badness]] (Smith): 0.00271 | ||
[[Badness]] (Dirichlet): 0.21396 | |||
=== Hemigone === | |||
By tempering out [[3971/3969]], erigone's tritave-augmented generator ([[49/11]]) is split into two [[19/9]]s. Then, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]). | |||
[[Subgroup]]: [[3.5.7.11.13.17.19_subgroup|3.5.7.11.13.17.19]] | |||
[[Comma list]]: 665/663, 847/845, 1575/1573, 1617/1615, 4459/4455 | |||
{{Mapping|legend=2|1 30 14 28 20 25 2|0 -42 -18 -38 -26 -33 1}} | |||
: [[Transversal_generators|sval mapping generators]]: ~[[3/1|3]], ~[[19/9]] | |||
[[Optimal tuning]] ([[WE]]): ~3 = 1902.0918, ~19/9 = 1292.3032 | |||
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~19/9 = 1292.2083 | |||
[[Optimal ET sequence]]: [[25edt|b25ce]], [[53edt|b53]], [[78edt|b78]], [[131edt|b131]], [[209edt|b209]], [[340edt|b340]] | |||
[[Badness]] (Smith): 0.00501 | |||
[[Badness]] (Dirichlet): 0.45479 | |||
=== <small><small>(no-2s)</small> </small>23-limit === | |||
[[2277/2275]] may be used in the same way to extend the simpler [[#Erigone|erigone]] to the 3.5.7.11.13.23 subgroup. | |||
[[Subgroup]]: [[3.5.7.11.13.17.19.23_subgroup|3.5.7.11.13.17.19.23]] | |||
[[Comma list]]: 665/663, 847/845, 1575/1573, 1617/1615, 2277/2275, 4459/4455 | |||
{{Mapping|legend=2|1 30 14 28 20 25 2 64|0 -42 -18 -38 -26 -33 1 -90}} | |||
: [[Transversal_generators|sval mapping generators]]: ~[[3/1|3]], ~[[19/9]] | |||
[[Optimal tuning]] ([[WE]]): ~3 = 1902.0149, ~19/9 = 1292.2401 | |||
[[Optimal tuning]] ([[CWE]]): ~3 = 1\1edt, ~19/9 = 1292.1988 | |||
[[Optimal ET sequence]]: [[53edt|b53i]], [[78edt|b78i]], [[131edt|b131]], [[340edt|b340]], [[471edt|b471]] | |||
[[Badness]] (Smith): 0.00539 | |||
[[Badness]] (Dirichlet): 0.54174 | |||
== Sirius == | == Sirius == | ||