15th-octave temperaments: Difference between revisions
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{{ | {{Infobox fractional-octave|15}} | ||
[[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 [[Cloudy clan|pentadecal]]. | Temperaments discussed elsewhere include [[Cloudy clan #Pentadecal|pentadecal]] and [[Landscape microtemperaments#Slendscape|slendscape]]. | ||
== | == Pentadecoid == | ||
Subgroup: 2.3.7.11 | 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]] ({{nowrap| 255 & 270 }}). | ||
[[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 | ||
[[Optimal tuning]] ([[CTE]]): ~22/21 = | [[Optimal tuning]] ([[CTE]]): ~22/21 = 80.0000, ~8/7 = 231.0071 (~1029/1024 = 8.9929) | ||
{{Optimal ET sequence|legend=1| 120, 135, 660, 795, 930, 1065, 1200 }} | {{Optimal ET sequence|legend=1| 120, 135, 660, 795, 930, 1065, 1200, 1335, 2535e }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.951 | ||
== Phosphorus == | == Phosphorus == | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 250047/250000, {{monzo|60 -96 1 32}} | [[Comma list]]: 250047/250000, {{monzo| 60 -96 1 32 }} | ||
[[Mapping]]: {{mapping| 15 8 -4 -4 | 0 13 32 38 }} | [[Mapping]]: {{mapping| 15 8 -4 -4 | 0 13 32 38 }} | ||
: | : mapping generators: ~{{monzo| -21 34 4 -15 }}, ~210827008/199290375 | ||
[[Optimal tuning]] ([[CTE]]): ~{{monzo|-21 34 4 -15}} = | [[Optimal tuning]] ([[CTE]]): ~{{monzo|-21 34 4 -15}} = 80.0000, ~28/27 = 62.9267 | ||
{{Optimal ET sequence|legend=1| 1125, 1335, 2460 }} | {{Optimal ET sequence|legend=1| 1125, 1335, 2460 }} | ||
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[[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }} | [[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }} | ||
: | : mapping generators: ~22/21, ~13 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~22/21 = | * [[CTE]]: ~22/21 = 80.000, ~13/8 = 840.528 (~40/39 = 39.472) | ||
* [[POTE]]: ~22/21 = | * [[POTE]]: ~22/21 = 80.000, ~13/8 = 852.924 (~40/39 = 27.076) | ||
{{Optimal ET sequence|legend=1| 15, 30 }} | {{Optimal ET sequence|legend=1| 15, 30 }} | ||
Latest revision as of 12:36, 11 June 2025
15edo has some close approximations including 22/21 to one step and 77/64 to four steps.
Temperaments discussed elsewhere include pentadecal and slendscape.
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
Phosphorus
Subgroup: 2.3.5.7
Comma list: 250047/250000, [60 -96 1 32⟩
Mapping: [⟨15 8 -4 -4], ⟨0 13 32 38]]
- mapping generators: ~[-21 34 4 -15⟩, ~210827008/199290375
Optimal tuning (CTE): ~[-21 34 4 -15⟩ = 80.0000, ~28/27 = 62.9267
Optimal ET sequence: 1125, 1335, 2460
Badness: 0.381587
Quindecic
Quindecic preserves the 11-limit structure of 15edo, with an independent generator for harmonic 13.
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 49/48, 55/54, 77/75
Mapping: [⟨15 24 35 42 52 0], ⟨0 0 0 0 0 1]]
- mapping generators: ~22/21, ~13
- CTE: ~22/21 = 80.000, ~13/8 = 840.528 (~40/39 = 39.472)
- POTE: ~22/21 = 80.000, ~13/8 = 852.924 (~40/39 = 27.076)
Badness: 0.028944