91st-octave temperaments

Fractional-octave temperaments

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← 90th-octave • 91st-octave • 92nd-octave →

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