25th-octave temperaments
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Manganese
Manganese is the simplest 5-limit 25th-octave temperament by TE simple badness at 0.1 cent error [1].
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