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.

== Quindecic ==

Temperaments discussed elsewhere include [[blackwood family #Quindecic|quindecic]], [[keemic temperaments #Pentadecal|pentadecal]], [[Landscape microtemperaments #Slendscape|slendscape]], and [[Landscape microtemperaments #Phosphorus|phosphorus]].

~~Quindecic preserves the~~ [[~~11-limit~~]] ~~structure of~~ [[~~15edo~~]], ~~with an independent generator for~~ [[~~13/1~~|~~harmonic 13~~]].

[[~~Subgroup~~]]~~: 2~~.3.~~5.7.11~~.13

== 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 }}).

[[~~Comma list~~]]: ~~28/27, 49/48, 55/54, 77/75~~

[[Subgroup]]: 2.3.7.11

[[~~Mapping~~]]: ~~{{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}~~

[[Comma list]]: 1362944/1361367, 1771561/1769472

~~: mapping generators: ~22/21, ~13~~

~~[[Optimal tuning]]s~~:

: mapping generators: ~22/21, ~7

* [[CTE]]: ~22/21 ~~= 1\15~~, ~~~13/8 = 840.528 (~40/39 = 39.472)~~

* [[POTE]]: ~22/21 = 1\15, ~13/8 = 852.924 (~40/39 = 27.076)

{{Optimal ~~ET sequence|legend~~=~~1| 15~~, ~~30 }}~~

[[Optimal tuning]] ([[CTE]]): ~22/21 = 80.0000, ~8/7 = 231.0071 (~1029/1024 = 8.9929)

[[Badness]] (~~Smith~~): 0.~~028944~~

{{Optimal ET sequence|legend=1| 120, 135, 660, 795, 930, 1065, 1200, 1335, 2535e }}

[[Badness]] (Sintel): 0.951

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-{{Fractional-octave navigation|15}}
+{{Infobox fractional-octave|15}}
+[[15edo]] has some close approximations including [[22/21]] to one step and [[77/64]] to four steps.
-== Quindecic ==
+Temperaments discussed elsewhere include [[blackwood family #Quindecic|quindecic]], [[keemic temperaments #Pentadecal|pentadecal]], [[Landscape microtemperaments #Slendscape|slendscape]], and [[Landscape microtemperaments #Phosphorus|phosphorus]].
-Quindecic preserves the [[11-limit]] structure of [[15edo]], with an independent generator for [[13/1|harmonic 13]].
-[[Subgroup]]: 2.3.5.7.11.13
+== 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 }}).
-[[Comma list]]: 28/27, 49/48, 55/54, 77/75
+[[Subgroup]]: 2.3.7.11
-[[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}
+[[Comma list]]: 1362944/1361367, 1771561/1769472
-: mapping generators: ~22/21, ~13
+{{Mapping|legend=1| 15 108 0 94 | 0 -2 1 -1 }}
-[[Optimal tuning]]s:
+: mapping generators: ~22/21, ~7
-* [[CTE]]: ~22/21 = 1\15, ~13/8 = 840.528 (~40/39 = 39.472)
-* [[POTE]]: ~22/21 = 1\15, ~13/8 = 852.924 (~40/39 = 27.076)
-{{Optimal ET sequence|legend=1| 15, 30 }}
+[[Optimal tuning]] ([[CTE]]): ~22/21 = 80.0000, ~8/7 = 231.0071 (~1029/1024 = 8.9929)
-[[Badness]] (Smith): 0.028944
+{{Optimal ET sequence|legend=1| 120, 135, 660, 795, 930, 1065, 1200, 1335, 2535e }}
+[[Badness]] (Sintel): 0.951
 {{Fractional-octave footer}}