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]].

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

~~[[Mapping]]:~~ {{~~mapping~~| 15 108 0 94 | 0 -2 1 -1 }}

: ~~Mapping~~ generators: ~22/21, ~7

: mapping generators: ~22/21, ~7

[[Optimal tuning]] ([[CTE]]): ~22/21 = ~~1＼15~~, ~8/7 = 231.0071

[[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.~~025231~~

[[Badness]] (Sintel): 0.951

== 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~~ generators: ~{{monzo|-21 34 4 -15}}, ~210827008/199290375

: mapping generators: ~{{monzo| -21 34 4 -15 }}, ~210827008/199290375

[[Optimal tuning]] ([[CTE]]): ~{{monzo|-21 34 4 -15}} = ~~1＼15~~, ~28/27 = 62.9267

[[Optimal tuning]] ([[CTE]]): ~{{monzo|-21 34 4 -15}} = 80.0000, ~28/27 = 62.9267

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

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

: mapping generators: ~22/21, ~13

[[Optimal tuning]]s:

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

* [[CTE]]: ~22/21 = 80.000, ~13/8 = 840.528 (~40/39 = 39.472)

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

* [[POTE]]: ~22/21 = 80.000, ~13/8 = 852.924 (~40/39 = 27.076)

@@ Line 1: / Line 1: @@
-{{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.
-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
-[[Mapping]]: {{mapping| 15 108 0 94 | 0 -2 1 -1 }}
+{{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 = 1＼15, ~8/7 = 231.0071
+[[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]]: 0.025231
+[[Badness]] (Sintel): 0.951
 == 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 generators: ~{{monzo|-21 34 4 -15}}, ~210827008/199290375
+: mapping generators: ~{{monzo| -21 34 4 -15 }}, ~210827008/199290375
-[[Optimal tuning]] ([[CTE]]): ~{{monzo|-21 34 4 -15}} = 1＼15, ~28/27 = 62.9267
+[[Optimal tuning]] ([[CTE]]): ~{{monzo|-21 34 4 -15}} = 80.0000, ~28/27 = 62.9267
 {{Optimal ET sequence|legend=1| 1125, 1335, 2460 }}
@@ Line 43: / Line 45: @@
 [[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}
-: Mapping generators: ~22/21, ~13
+: mapping generators: ~22/21, ~13
 [[Optimal tuning]]s:
-* [[CTE]]: ~22/21 = 1＼15, ~13/8 = 840.528 (~40/39 = 39.472)
+* [[CTE]]: ~22/21 = 80.000, ~13/8 = 840.528 (~40/39 = 39.472)
-* [[POTE]]: ~22/21 = 1＼15, ~13/8 = 852.924 (~40/39 = 27.076)
+* [[POTE]]: ~22/21 = 80.000, ~13/8 = 852.924 (~40/39 = 27.076)
 {{Optimal ET sequence|legend=1| 15, 30 }}