44th-octave temperaments
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One step of 44edo is extremely close to the septimal comma, 64/63. The relationship is preserved even up tens of thousands of edos.
Septiruthenic
- Main article: Septiruthenia
Subgroup: 2.3.5.7
Comma list: [-263 88 0 44⟩
Mapping: [⟨44 0 0 263], ⟨0 1 0 -2], ⟨0 0 1 0]]
Mapping generators: ~64/63, ~3, ~5
Optimal ET sequence: 176, 308, 352, 484, 528, 836, 1848, 2684, 4532, 7744, 8580, 9592, 10428, 12276, 14960, 27236, 42196, 99352, …
Badness: 116 × 10-3
Septiruthenian
Subgroup: 2.3.7
Comma list: [-263 88 44⟩
Mapping: [⟨44 0 263], ⟨0 1 -2]]
Mapping generators: ~64/63, ~3
Ruthenium
Ruthenium is named after the 44th element, prescribes the additional mapping for 5, and can be expressed as the 1848 & 2684 temperament.
Subgroup: 2.3.5.7
Comma list: [-8 23 -5 -6⟩, [51 -13 -1 -10⟩
Mapping: [⟨44 0 -386 263], ⟨0 1 7 -2]]
Mapping generators: ~64/63, ~3
Optimal tuning (CTE): ~3/2 = 701.9420
Optimal ET sequence: 176, 660, 836, 1848, 2684, 4532, 19976, 24508, 29040, 33572
Badness: 0.111
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 1771561/1771470, 67110351/67108864
Mapping: [⟨44 0 -386 263 -57], ⟨0 1 7 -2 3]]
Optimal tuning (CTE): ~3/2 = 701.9429
Optiml GPV sequence: 176, 660, 836, 1848, 2684, 4532, 15444, 19976e
Badness: 0.0209
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 196625/196608, 823680/823543, 1771561/1771470
Mapping: [⟨44 0 -386 263 -57 1976], ⟨0 1 7 -2 3 -26]]
Optimal tuning (CTE): ~3/2 = 701.939
Optiml GPV sequence: 836, 1848, 2684, 7216, 9900, 12584
Badness: 0.0396