44th-octave temperaments

Fractional-octave temperaments

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← 43rd-octave • 44th-octave • 45th-octave →

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