No-twos subgroup temperaments: Difference between revisions
Complete data for suhail and erigone |
Complete data for mizar and remus |
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===== Hemigone ===== | ===== Hemigone ===== | ||
By tempering out [[3971/3969]], erigone's generator ([[~]][[99/49]]) is split into two [[27/19]]'s. Meanwhile, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]). [[2277/2275]] may be used in the same way to extend erigone for prime 23. | |||
By tempering out [[3971/3969]], erigone's generator ([[~]][[99/49]]) is split into two [[27/19]]'s. Meanwhile, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]). | |||
[[2277/2275]] may be used in the same way to extend erigone for prime 23. | |||
Subgroup: 3.5.7.11.13.17 | Subgroup: 3.5.7.11.13.17 | ||
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=== Mizar === | === Mizar === | ||
Mizar exploits the sirius tuning of the 25/21 generator being close to [[13/11]] (in order to split 7/5 evenly); additionally this tempers out [[459/455]], equating [[17/13]] to [[35/27]]. | Mizar exploits the sirius tuning of the 25/21 generator being close to [[13/11]] (in order to split 7/5 evenly); additionally this tempers out [[459/455]], equating [[17/13]] to [[35/27]]. | ||
The mapping for prime 17 is similar to what [[dubhe]] does: tempering out [[2025/2023]] to split the 7-limit generator in half; in this case, 25/7 is split into two intervals of [[17/9]], which turns out to occupy the position of a [[macrodiatonic]] fifth, specifically a macro-[[flattone]] fifth. | The mapping for prime 17 is similar to what [[dubhe]] does: tempering out [[2025/2023]] to split the 7-limit generator in half; in this case, 25/7 is split into two intervals of [[17/9]], which turns out to occupy the position of a [[macrodiatonic]] fifth, specifically a macro-[[flattone]] fifth. | ||
Subgroup: 3.5.7.11 | |||
Comma list: 3125/3087, 295245/290521 | |||
{{Mapping|legend=2| 1 -2 -4 12 | 0 6 10 -17 }} | |||
: mapping generators: ~3, ~729/385 | |||
Optimal tunings: | |||
* WE: ~3 = 1901.0672{{c}}, ~729/385 = 1097.6627{{c}} | |||
* CWE: ~3 = 1901.9550{{c}}, ~729/385 = 1098.1722{{c}} | |||
Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]], [[168edt|b168]] | |||
Badness (Sintel): 1.82 | |||
==== 3.5.7.11.13 subgroup | |||
Subgroup: 3.5.7.11.13 | |||
Comma list: 275/273, 847/845, 59049/57967 | |||
{{Mapping|legend=2| 1 -2 -4 12 11 | 0 6 10 -17 -15 }} | |||
Optimal tunings: | |||
* WE: ~3 = 1900.6176{{c}}, ~729/385 = 1097.5046{{c}} | |||
* CWE: ~3 = 1901.9550{{c}}, ~729/385 = 1098.2837{{c}} | |||
Optimal ET sequence: [[19edt|b19]], [[26edt|b26]], [[45edt|b45]], [[71edt|b71]] | |||
Badness (Sintel): 1.12 | |||
==== 3.5.7.11.13.17 subgroup ==== | ==== 3.5.7.11.13.17 subgroup ==== | ||
Subgroup: 3.5.7.11.13.17 | Subgroup: 3.5.7.11.13.17 | ||
Comma list: 275/273, 459/455, | Comma list: 275/273, 459/455, 847/845, 2025/2023 | ||
{{Mapping|legend=2| 1 -2 -4 12 11 2 | 0 6 10 -17 -15 1 }} | {{Mapping|legend=2| 1 -2 -4 12 11 2 | 0 6 10 -17 -15 1 }} | ||
Optimal tunings: | Optimal tunings: | ||
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=== Remus === | === Remus === | ||
By splitting the generator of sirius into three, remus efficiently represents the no-2's 13-limit with [[mos scale]]s of 18, 25, 32, or 39 steps. | By splitting the generator of sirius into three, remus efficiently represents the no-2's 13-limit with [[mos scale]]s of 18, 25, 32, or 39 steps. | ||
This is essentially [[electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament. | This is essentially [[electra]] but with prime 7, or more accurately, electra is the no-7's [[restriction]] of this temperament. | ||
Subgroup: 3.5.7.11 | |||
Comma list: 1331/1323, 3125/3087 | |||
{{Mapping|legend=2| 1 -5 -9 -5 | 0 9 15 10 }} | |||
: mapping generators: ~3, ~11/5 | |||
Optimal tunings: | |||
* WE: ~3 = 1902.6279{{c}}, ~11/5 = 1366.3374{{c}} | |||
* CWE: ~3 = 1901.9550{{c}}, ~11/5 = 1365.8996{{c}} | |||
Optimal ET sequence: [[7edt|b7]], [[25edt|b25d]], [[32edt|b32]], [[39edt|b39]], [[71edt|b71]], [[110edt|b110]], [[149edt|b149]] | |||
Badness (Sintel): 0.538 | |||
==== 3.5.7.11.13 subgroup ==== | ==== 3.5.7.11.13 subgroup ==== | ||
Subgroup: 3.5.7.11.13 | Subgroup: 3.5.7.11.13 | ||
Comma list: 275/273, | Comma list: 275/273, 847/845, 1331/1323 | ||
{{Mapping|legend=2| 1 -5 -9 -5 -7 | 0 9 15 10 13 }} | {{Mapping|legend=2| 1 -5 -9 -5 -7 | 0 9 15 10 13 }} | ||
Optimal tunings: | Optimal tunings: | ||