80th-octave temperaments: Difference between revisions
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{{Fractional-octave navigation|80}} | {{Fractional-octave navigation|80}} | ||
This page describes 80th-octave temperaments. | This page describes 80th-octave temperaments. 80edo is extremely accurate for the 17th harmonic, a relationship which is seen in some of these temperaments. In addition, it is the first equal division to be consistent in the 19-limit. | ||
Temperaments discussed elsewhere include | |||
* ''Octogintic'', → [[Parkleiness temperaments#Octogintic|Parkleiness temperaments]] | |||
'' | |||
== Tetraicosic == | == Tetraicosic == | ||
Revision as of 23:23, 7 July 2023
Template:Fractional-octave navigation
This page describes 80th-octave temperaments. 80edo is extremely accurate for the 17th harmonic, a relationship which is seen in some of these temperaments. 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 720 & 1600, 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
Mercury
Named after the 80th element, defined as the 320 & 2000 temperament. 7 periods are mapped to ~17/16, 11 periods are mapped to ~11/10.
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