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.

Temperaments discussed elsewhere include [[~~Cloudy clan~~ #Pentadecal|pentadecal]] ~~and~~ [[Landscape microtemperaments#Slendscape|slendscape]].

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

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

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[[Badness]] (Sintel): 0.951

~~== Phosphorus ==~~

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 250047/250000, {{monzo| 60 -96 1 32 }}~~

~~[[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}} = 80.0000, ~28/27 = 62.9267~~

~~{{Optimal ET sequence|legend=1| 1125, 1335, 2460 }}~~

~~[[Badness]]: 0.381587~~

~~== Quindecic ==~~

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

~~[[Subgroup]]: 2.3.5.7.11.13~~

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

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

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

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

* [[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)

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

~~[[Badness]]: 0.028944~~

@@ Line 2: / Line 2: @@
 [[15edo]] has some close approximations including [[22/21]] to one step and [[77/64]] to four steps.
-Temperaments discussed elsewhere include [[Cloudy clan #Pentadecal|pentadecal]] and [[Landscape microtemperaments#Slendscape|slendscape]].
+Temperaments discussed elsewhere include [[blackwood family #Quindecic|quindecic]], [[keemic temperaments #Pentadecal|pentadecal]], [[Landscape microtemperaments #Slendscape|slendscape]], and [[Landscape microtemperaments #Phosphorus|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]] ({{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 }}).
+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
@@ Line 20: / Line 20: @@
 [[Badness]] (Sintel): 0.951
-== Phosphorus ==
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 250047/250000, {{monzo| 60 -96 1 32 }}
-[[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}} = 80.0000, ~28/27 = 62.9267
-{{Optimal ET sequence|legend=1| 1125, 1335, 2460 }}
-[[Badness]]: 0.381587
-== Quindecic ==
-Quindecic preserves the [[11-limit]] structure of [[15edo]], with an independent generator for [[13/1|harmonic 13]].
-[[Subgroup]]: 2.3.5.7.11.13
-[[Comma list]]: 28/27, 49/48, 55/54, 77/75
-[[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}
-: mapping generators: ~22/21, ~13
-[[Optimal tuning]]s:
-* [[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)
-{{Optimal ET sequence|legend=1| 15, 30 }}
-[[Badness]]: 0.028944
 {{Fractional-octave footer}}