56th-octave temperaments

Fractional-octave temperaments

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← 55th-octave • 56th-octave • 57th-octave →

Barium

One step of 56edo is close to a syntonic comma. Named after the 56th element, barium tempers out the barium comma ([-225 224 -56⟩), which sets 56 syntonic commas equal to the octave. It can be expressed as the 224 & 2072 temperament.

Subgroup: 2.3.5

Comma list: [-225 224 -56⟩

Mapping: [⟨56 0 -225], ⟨0 1 4]]

Mapping generators: ~81/80, ~3

Optimal tuning (CTE): ~3/2 = 701.9379

Optimal ET sequence: 224, 1176, 1400, 1624, 1848, 2072, 5992, 8064, 26264, 34328b, 42392b

Badness: 4.70

7-limit

Subgroup: 2.3.5.7

Comma list: [-12 29 -11 -3⟩, [47 -7 -7 -7⟩

Mapping: [⟨56 0 -225 601], ⟨0 1 4 -5]]

Optimal tuning (CTE): ~3/2 = 701.9433

Optimal ET sequence: 224, 1176, 1400, 1624, 1848, 2072, 5768, 7616, 17080, 24696cd

Badness: 0.227

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 1019215872/1019046875, 14765025303/14763950080

Mapping: [⟨56 0 -225 601 460], ⟨0 1 4 -5 -3]]

Optimal tuning (CTE): ~3/2 = 701.9431

Optimal ET sequence: 224, 1176, 1400, 1624, 1848, 3920, 5768, 7616, 21000cd, 28616cd

Badness: 0.0345

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 9801/9800, 67392/67375, 26802913280/26795786661

Mapping: [⟨56 0 -225 601 460 651], ⟨0 1 4 -5 -3 -5]]

Optimal tuning (CTE): ~3/2 = 701.9431

Optimal ET sequence: 224, 1848, 2072