15th-octave temperaments: Difference between revisions

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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 [[Cloudy clan #Pentadecal|pentadecal]] and [[Landscape microtemperaments#Slendscape|slendscape]].
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 }}
== 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 }}).


: mapping generators: ~22/21, ~13
[[Subgroup]]: 2.3.7.11


[[Optimal tuning]]s:
[[Comma list]]: 1362944/1361367, 1771561/1769472
* [[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 }}
{{Mapping|legend=1| 15 108 0 94 | 0 -2 1 -1 }}


[[Badness]] (Smith): 0.028944
: mapping generators: ~22/21, ~7


== Pentadecal ==
[[Optimal tuning]] ([[CTE]]): ~22/21 = 80.0000, ~8/7 = 231.0071 (~1029/1024 = 8.9929)
Subgroup: 2.3.5


[[Comma]]: 30517578125/29386561536
{{Optimal ET sequence|legend=1| 120, 135, 660, 795, 930, 1065, 1200, 1335, 2535e }}


[[Mapping]]: [{{Val| 15 0 11 }}, {{Val| 0 1 1 }}]
[[Badness]] (Sintel): 0.951
 
[[POTE generator]]: ~3/2 = 702.652
 
{{Optimal ET sequence|legend=1| 15, 45, 60, 75, 135, 210c, 345cc }}
 
[[Badness]]: 0.692042
 
=== 7-limit ===
The ''pentadecal'' temperament tempers out the 15-5/3-comma (Quintriyo, pentadecatonic minor thirds comma), {{monzo|-11 -15 15}}. This temperament can be described as 15&60 temperament, tempering out the [[cloudy comma]], 16807/16384 and the [[875/864|keema]], 875/864 in the 7-limit.


== Phosphorus ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 16807/16384
[[Comma list]]: 250047/250000, {{monzo| 60 -96 1 32 }}


{{Mapping|legend=1| 15 0 11 42 | 0 1 1 0 }}
[[Mapping]]: {{mapping| 15 8 -4 -4 | 0 13 32 38 }}  


: mapping generators: ~21/20, ~3
: mapping generators: ~{{monzo| -21 34 4 -15 }}, ~210827008/199290375


[[Optimal tuning]] ([[POTE]]): ~21/20 = 1\15, ~3/2 = 700.223 (~126/125 = 19.777)
[[Optimal tuning]] ([[CTE]]): ~{{monzo|-21 34 4 -15}} = 80.0000, ~28/27 = 62.9267


{{Optimal ET sequence|legend=1| 15, 45, 60 }}
{{Optimal ET sequence|legend=1| 1125, 1335, 2460 }}


[[Badness]]: 0.114833
[[Badness]]: 0.381587


=== 11-limit ===
== Quindecic ==
Subgroup: 2.3.5.7.11
Quindecic preserves the [[11-limit]] structure of [[15edo]], with an independent generator for [[13/1|harmonic 13]].  


Comma list: 100/99, 385/384, 1372/1331
[[Subgroup]]: 2.3.5.7.11.13


Mapping: {{mapping| 15 0 11 42 52 | 0 1 1 0 0 }}
[[Comma list]]: 28/27, 49/48, 55/54, 77/75


: mapping generators: ~21/20, ~3
[[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}


Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 702.733 (~56/55 = 17.267)
: mapping generators: ~22/21, ~13


{{Optimal ET sequence|legend=1| 15, 45, 60, 75de, 135de }}
[[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)


Badness: 0.077420
{{Optimal ET sequence|legend=1| 15, 30 }}
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 144/143, 1372/1331
 
Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}
 
: mapping generators: ~21/20, ~3
 
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.715 (~91/90 = 18.285)
 
{{Optimal ET sequence|legend=1| 15, 45f, 60, 135de, 195cddee }}
 
Badness: 0.051740
 
=== Quindecal ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 441/440, 875/864
 
Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}
 
: mapping generators: ~21/20, ~3
 
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 700.318 (~126/125 = 19.682)
 
{{Optimal ET sequence|legend=1| 15, 45e, 60e }}
 
Badness: 0.044405
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 196/195, 352/351, 875/864
 
Mapping: {{mapping| 15 0 11 42 28 103 | 0 1 1 0 1 -2 }}
 
: mapping generators: ~21/20, ~3
 
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.647 (~65/64 = 18.353)
 
{{Optimal ET sequence|legend=1| 15f, 45e, 60e }}


Badness: 0.055361
[[Badness]]: 0.028944


{{Fractional-octave footer}}
{{Fractional-octave footer}}

Latest revision as of 12:36, 11 June 2025

← 14th-octave • 15th-octave • 16th-octave →

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

Optimal tunings:

  • 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: 15, 30

Badness: 0.028944


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