80th-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|80}} | |||
This page describes 80th-octave temperaments. 80edo is extremely accurate for the 17th harmonic, a relationship which is seen in | This page describes 80th-octave temperaments. [[80edo]] is extremely accurate for the [[17/1|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 | Temperaments discussed elsewhere include | ||
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== Tetraicosic == | == Tetraicosic == | ||
Tetraicosic is described as | Tetraicosic is described as 1600 & 2320, and named after the fact that 4 × 20 = 80. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Badness: 0.0741 | 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: [{{val| 80 1 343 -27 434 13 327}}, {{val| 0 4 -5 8 -5 9 0}}] | |||
Optimal tuning (CTE): ~130/99 = 471.7... | |||
{{Optimal ET sequence|legend=1| 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: [{{val| 80 1 343 -27 434 13 327 -69}}, {{val| 0 4 -5 8 -5 9 0 13}}] | |||
Optimal tuning (CTE): ~67473/51376 = 471.732 | |||
{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }} | |||
== Mercury == | == Mercury == | ||
Named after the 80th element, defined as the 320 & 2000 temperament | Named after the 80th element, defined as the 320 & 2000 temperament. | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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{{mapping|legend=1| 80 3 161 -23 252 197 | 0 5 1 10 1 4 }} | {{mapping|legend=1| 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 tuning (CTE): ~2304/1859 = 371.385 | ||
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{{mapping|legend=1| 80 3 161 -23 252 197 327 | 0 5 1 10 1 4 0 }} | {{mapping|legend=1| 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 tuning (CTE): ~2275/1836 = 371.386 | ||
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Mapping: [{{val|80 3 161 -23 252 197 327 117}}, {{val|0 5 1 10 1 4 0 9}}] | Mapping: [{{val|80 3 161 -23 252 197 327 117}}, {{val|0 5 1 10 1 4 0 9}}] | ||
: mapping generators: ~459/455, ~2275/1836 | |||
Optimal tuning (CTE): ~2275/1836 = 371.386 | Optimal tuning (CTE): ~2275/1836 = 371.386 | ||
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Optimal tuning (CTE): ~920/741 = 374.387 | Optimal tuning (CTE): ~920/741 = 374.387 | ||
[[Category:80edo] | {{Navbox fractional-octave}} | ||
[[Category:80edo]]] | |||