25th-octave temperaments

Fractional-octave temperaments

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← 24th-octave • 25th-octave • 26th-octave →

Manganese

Manganese is the simplest 5-limit 25th-octave temperament by TE simple badness at 0.1 cent error .

Subgroup: 2.3.5

Comma list: [211 50 -125⟩

Mapping: [⟨25 2 43], ⟨0 5 2]]

mapping generators: ~[76 18 -45⟩ = 1\25, ~[54 13 -32⟩ = 361.189

Optimal tuning (CTE): ~[54 13 -32⟩ = 361.189

Supporting ETs: 525, 1000, ...

Hemimanganese

While manganese is a rather bulky temperament, splitting the generator in half lends itself to a high-limit interpretaton, which can be described as the 525 & 2000 temperament. Hemimanganese maps 17/14 into 7\25.

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 67392/67375, 28600000/28588707, 66997216/66976875

Mapping: [⟨25 2 43 89 15 164], ⟨0 10 4 -5 19 -19]]

mapping generators: ~147/143 = 1\25, ~10000/9009 = 180.598

Optimal tuning (CTE): ~10000/9009 = 180.598

Supporting ETs: 525, 1475, 2000, 2525, 3475

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 4096/4095, 14400/14399, 67392/67375, 361250/361179, 983178/983125

Mapping: [⟨25 2 43 89 15 164 121], ⟨0 10 4 -5 19 -19 -5]]

mapping generators: ~147/143 = 1\25, ~272/245 = 180.599

Optimal tuning (CTE): ~272/245 = 180.599

Supporting ETs: 525, 1475, 2000, 2525, 3475

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 4096/4095, 6175/6174, 14400/14399, 23409/23408, 67392/67375, 89376/89375

Mapping: [⟨25 2 43 89 15 164 121 46], ⟨0 10 4 -5 19 -19 -5 16]]

mapping generators: ~1573/1530 = 1\25, ~3773/3400 = 180.598

Optimal tuning (CTE): ~3773/3400 = 180.598

Supporting ETs: 525, 1475, 2000, 2525, 3475

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 4096/4095, 6175/6174, 8625/8624, 10626/10625, 14400/14399, 67392/67375, 89376/89375

Mapping: [⟨25 2 43 89 15 164 121 46 162], ⟨0 10 4 -5 19 -19 -5 16 -13]]

mapping generators: ~3213/3125 = 1\25, ~272/245 = 180.598

Optimal tuning (CTE): ~272/245 = 180.598

Supporting ETs: 525, 1475, 2000, 2525, 3475