15th-octave temperaments: Difference between revisions
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[[15edo]] has some close approximations including [[22/21]] to one step and [[77/64]] to four steps. | [[15edo]] has some close approximations including [[22/21]] to one step and [[77/64]] to four steps. | ||
Temperaments discussed elsewhere include [[ | Temperaments discussed elsewhere include [[blackwood family #Quindecic|quindecic]], [[keemic temperaments #Pentadecal|pentadecal]], [[Landscape microtemperaments #Slendscape|slendscape]], and [[Landscape microtemperaments #Phosphorus|phosphorus]]. | ||
== Pentadecoid == | == Pentadecoid == | ||
In pentadecoid, the period (1\15) is given an interpretation of [[22/21]], and four of them represent the keenanismic minor third, [[77/64]] = ([[147/128]])⋅([[22/21]]). The temperament is named for an analog of its relating temperaments, [[decoid]] ({{nowrap| 130 & 140 }}) and [[quintosec family #Triacontoid|triacontoid]] ({{nowrap| 120 & 150 }}). Triacontoid is a weak extension of pentadecoid, as well as [[Landscape microtemperaments #Slendscape|slendscape]] ({{nowrap| 255 & 270 }}). | |||
[[Subgroup]]: 2.3.7.11 | [[Subgroup]]: 2.3.7.11 | ||
[[Comma list]]: 1362944/1361367, 1771561/1769472 | [[Comma list]]: 1362944/1361367, 1771561/1769472 | ||
{{Mapping|legend=1| 15 108 0 94 | 0 -2 1 -1 }} | |||
: mapping generators: ~22/21, ~7 | : mapping generators: ~22/21, ~7 | ||
[[Optimal tuning]] ([[CTE]]): ~22/21 = 80.0000, ~8/7 = 231.0071 | [[Optimal tuning]] ([[CTE]]): ~22/21 = 80.0000, ~8/7 = 231.0071 (~1029/1024 = 8.9929) | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 120, 135, 660, 795, 930, 1065, 1200, 1335, 2535e }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.951 | ||
{{Fractional-octave footer}} | {{Fractional-octave footer}} | ||
Latest revision as of 10:57, 28 June 2026
15edo has some close approximations including 22/21 to one step and 77/64 to four steps.
Temperaments discussed elsewhere include quindecic, pentadecal, slendscape, and phosphorus.
Pentadecoid
In pentadecoid, the period (1\15) is given an interpretation of 22/21, and four of them represent the keenanismic minor third, 77/64 = (147/128)⋅(22/21). The temperament is named for an analog of its relating temperaments, decoid (130 & 140) and triacontoid (120 & 150). Triacontoid is a weak extension of pentadecoid, as well as slendscape (255 & 270).
Subgroup: 2.3.7.11
Comma list: 1362944/1361367, 1771561/1769472
Mapping: [⟨15 108 0 94], ⟨0 -2 1 -1]]
- mapping generators: ~22/21, ~7
Optimal tuning (CTE): ~22/21 = 80.0000, ~8/7 = 231.0071 (~1029/1024 = 8.9929)
Optimal ET sequence: 120, 135, 660, 795, 930, 1065, 1200, 1335, 2535e
Badness (Sintel): 0.951