91st-octave temperaments

Fractional-octave temperaments

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

Protactinium

Protactinium is described as the 364 & 1547 temperament and named after the 91st element. 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