91st-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|91}} | |||
91st-octave temperaments occur naturally through an edo join between any two equal divisions whose greatest common divisor is 91. | |||
Of multiples of 91, [[364edo]] and [[1547edo]] are notable for having high consistency limits. Therefore protactinium, is considered below. | |||
Protactinium is described as the 364 & 1547 temperament and named after the 91st element. | == Protactinium == | ||
Protactinium is described as the 364 & 1547 temperament and named after the 91st element. Despite being consistent only up to 11-limit, [[1911edo]] is also a strong tuning. | |||
Coincidentally, it reaches [[91/64]] in 1 generator. | |||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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[[Support]]ing [[ET]]s: {{EDOs|364, 819, 1183, 1547, 1911, 2730, 3094, 3913, 4277}} | [[Support]]ing [[ET]]s: {{EDOs|364, 819, 1183, 1547, 1911, 2730, 3094, 3913, 4277}} | ||
====11-limit==== | |||
==== 11-limit ==== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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[[Support]]ing [[ET]]s: {{EDOs|364, 819e, 1183, 1547, 1911, 2275, 2730e, 3458}} | [[Support]]ing [[ET]]s: {{EDOs|364, 819e, 1183, 1547, 1911, 2275, 2730e, 3458}} | ||
====13-limit==== | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.0777 | Badness: 0.0777 | ||
====17-limit==== | |||
==== 17-limit ==== | |||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 47: | Line 55: | ||
Badness: 0.0582 | Badness: 0.0582 | ||
{{Navbox fractional-octave}} | |||
Latest revision as of 22:56, 2 September 2026
91st-octave temperaments occur naturally through an edo join between any two equal divisions whose greatest common divisor is 91.
Of multiples of 91, 364edo and 1547edo are notable for having high consistency limits. Therefore protactinium, is considered below.
Protactinium
Protactinium is described as the 364 & 1547 temperament and named after the 91st element. Despite being consistent only up to 11-limit, 1911edo is also a strong tuning.
Coincidentally, it reaches 91/64 in 1 generator.
Subgroup: 2.3.5.7
Comma list: [47 -7 -7 -7⟩, [-2 -25 1 14⟩
Mapping: [⟨91 0 644 -33 1036], ⟨0 1 -3 -2 -5]]
- mapping generators: ~1728/1715, ~3
Optimal tuning (CTE): ~3/2 = 701.991
Supporting ETs: 364, 819, 1183, 1547, 1911, 2730, 3094, 3913, 4277
11-limit
Subgroup: 2.3.5.7.11
Comma list: 234375/234256, 26214400/26198073, 514714375/514434888
Mapping: [⟨91 0 644 -33 1036], ⟨0 1 -3 -2 -5]]
- mapping generators: ~1728/1715, ~3
Optimal tuning (CTE): ~3/2 = 702.015
Supporting ETs: 364, 819e, 1183, 1547, 1911, 2275, 2730e, 3458
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 4096/4095, 91125/91091, 369754/369603, 2912000/2910897
Mapping: [⟨91 0 644 -33 1036 481], ⟨0 1 -3 -2 -5 -1]]
- mapping generators: ~1728/1715, ~3
Optimal tuning (CTE): ~3/2 = 702.0195
Optimal ET sequence: 364, 819e, 1183, 1547
Badness: 0.0777
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 4096/4095, 14400/14399, 42500/42471, 75735/75712, 2100875/2100384
Mapping: [⟨91 0 644 -33 1036 481 -205], ⟨0 1 -3 -2 -5 -1 4]]
Optimal tuning (CTE): ~3/2 = 702.0269
Optimal ET sequence: 364, 1183, 1547, 1911
Badness: 0.0582