80th-octave temperaments
This page describes 80th-octave temperaments. 80edo is extremely accurate for the 17th harmonic, a relationship which is so far seen in all documented temperaments on this page. In addition, it is the first equal division to be consistent in the 19-limit.
Temperaments discussed elsewhere include
- Octogintic, → Parkleiness temperaments
Tetraicosic
Tetraicosic is described as 1600 & 2320, and named after the fact that 4 × 20 = 80.
Subgroup: 2.3.5.7
Comma list: [-52 17 12 -1⟩, [25 42 -8 -26⟩
Mapping: [⟨80 1 343 -27], ⟨0 4 -5 8]]
Mapping generators: ~31637227888/31381059609, ~7381125/5619712
Optimal tuning (CTE): ~7381125/5619712 = 471.7339
Optimal ET sequence: 720, 1600, 2320, 3920
Badness: 1.49
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 928760463360/928426965851, [49 -13 -14 -1 2⟩
Mapping: [⟨80 1 343 -27 434], ⟨0 4 -5 8 -5]]
Optimal tuning (CTE): ~2278125/1734656 = 471.7342
Optimal ET sequence: 720, 1600, 2320, 3920
Badness: 0.178
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 4100625/4100096, 14236560/14235529, 143327232/143286143
Mapping: [⟨80 1 343 -27 434 13], ⟨0 4 -5 8 -5 9]]
Optimal tuning (CTE): ~130/99 = 471.7322
Optimal ET sequence: 720, 1600, 2320, 3920
Badness: 0.0741
17-limit
Subgroup 2.3.5.7.11.13.17
Comma list: 9801/9800, 14400/14399, 373527/373490, 1812608/1812525, 4685824/4685625
Mapping: [⟨80 1 343 -27 434 13 327], ⟨0 4 -5 8 -5 9 0]]
Optimal tuning (CTE): ~130/99 = 471.7...
Optimal ET sequence: 720, 1600, 2320, 3920
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 9801/9800, 10830/10839, 12636/12635, 23409/23408, 373527/373490, 32133332/32131125
Mapping: [⟨80 1 343 -27 434 13 327 -69], ⟨0 4 -5 8 -5 9 0 13]]
Optimal tuning (CTE): ~67473/51376 = 471.732
Optimal ET sequence: 720, 1600, 2320, 3920
Mercury
Named after the 80th element, defined as the 320 & 2000 temperament.
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 67392/67375, 1399680/1399489, [14 -8 -8 5 2 -1⟩
Mapping: [⟨80 3 161 -23 252 197], ⟨0 5 1 10 1 4]]
- mapping generators: ~9295/9216, ~2304/1859
Optimal tuning (CTE): ~2304/1859 = 371.385
Optimal ET sequence: 320, 2000, 4320
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 9801/9800, 12376/12375, 67392/67375, 4230144/4229225, 494534656/494515125
Mapping: [⟨80 3 161 -23 252 197 327], ⟨0 5 1 10 1 4 0]]
- mapping generators: ~459/455, ~1859/1500
Optimal tuning (CTE): ~2275/1836 = 371.386
Optimal ET sequence: 320, 2000, 4320
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 9801/9800, 12376/12375, 67392/67375, 392445/392392, 401408/401375, 1549184/1549125
Mapping: [⟨80 3 161 -23 252 197 327 117], ⟨0 5 1 10 1 4 0 9]]
- mapping generators: ~459/455, ~2275/1836
Optimal tuning (CTE): ~2275/1836 = 371.386
Optimal ET sequence: 320, 2000, 4320
Octodeca
Octodeca can be described as the 80 & 1920 temperament.
Subgroup: 2.3.5.7
Comma list: [21 60 -50⟩, 184528125/184473632
Mapping: [⟨80 2 36 -25], ⟨0 5 6 10]]
- mapping generators: ~1071875/1062882 = 1\80, ~7476806640625/6025163444928 = 374.386
Optimal tuning (CTE): ~7476806640625/6025163444928 = 374.386
Supporting ETs: 80, 1920, 2000
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 5832/5831, 9801/9800, 89376/89375, 123201/123200, 392445/392392, 653184/653125
Mapping: [⟨80 2 36 -25 -127 -321 -327 -240], ⟨0 5 6 10 6 -1 0 4]]
- mapping generators: ~12000/12103 = 1\80, ~1625/1309 = 374.386
Optimal tuning (CTE): ~1625/1309 = 374.386
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 5832/5831, 8625/8624, 9801/9800, 10626/10625, 89376/89375, 95013/95000, 123201/123200
Mapping: [⟨80 2 36 -25 -127 -321 -327 -240 -287], ⟨0 5 6 10 6 -1 0 4 3]]
- mapping generators: ~12000/12103 = 1\80, ~920/741 = 374.387
Optimal tuning (CTE): ~920/741 = 374.387
]