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15edo: Difference between revisions

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A regular keyboard can be designed using this system by placing 7 black keys as Porcupine[7] and 8 whites as Porcupine[8]. In fact, [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] has already done this with his pink Halberstadt keyboard.
A regular keyboard can be designed using this system by placing 7 black keys as Porcupine[7] and 8 whites as Porcupine[8]. In fact, [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] has already done this with his pink Halberstadt keyboard.


==Approximation to JI==
== Approximation to JI ==
[[File:15ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 13-limit intervals]]
[[File:15ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 13-limit intervals]]
15edo offers some minor improvements over 12et in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a [[5L 5s]] [[MOS scale]] wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as the [[blackwood]] temperament, named after [[Easley Blackwood Jr.]], who is the first to document its existence. It has also been written on extensively by [[Igliashon Jones]] in the paper [http://www.cityoftheasleep.com/etc/5nEDOs.pdf ''Five is Not an Odd Number'']. For an in-depth treatment of harmony in 15edo based on this temperament (and its 7- and 11-limit extensions), see [[Blacksmith temperament modal harmony (in 15edo)]].
15edo offers some minor improvements over 12et in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a [[5L 5s]] [[mos scale]] wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as the [[blackwood]] temperament, named after [[Easley Blackwood Jr.]], who is the first to document its existence. It has also been written on extensively by [[Igliashon Jones]] in the paper [http://www.cityoftheasleep.com/etc/5nEDOs.pdf ''Five is Not an Odd Number'']. For an in-depth treatment of harmony in 15edo based on this temperament (and its 7- and 11-limit extensions), see [[Blackwood temperament modal harmony (in 15edo)]].


===15-odd-limit interval mappings===
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals}}


==Regular temperament properties==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
|-
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" |Tuning error
! colspan="2" | Tuning error
|-
|-
!
! [[TE error|Absolute]] (¢)
[[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3.5
| 2.3.5
| 128/125, 250/243
| 128/125, 250/243
|{{mapping| 15 24 35 }}
| {{Mapping| 15 24 35 }}
| −5.75
| −5.75
|4.63
| 4.63
|5.81
| 5.81
|-
|-
|2.3.5.7
| 2.3.5.7
|28/27, 49/48, 126/125
| 28/27, 49/48, 126/125
|
| {{Mapping| 15 24 35 42 }}
{{mapping| 15 24 35 42 }}
| −3.55
|−3.55
| 5.56
|5.56
| 6.97
|6.97
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|28/27, 49/48, 55/54, 77/75
| 28/27, 49/48, 55/54, 77/75
|{{mapping| 15 24 35 42 52 }}
| {{Mapping| 15 24 35 42 52 }}
|−3.34
| −3.34
|4.99
| 4.99
| 6.25
| 6.25
|}
|}
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=== Errors by subgroup ===
=== Errors by subgroup ===
{| class="wikitable center-3 mw-collapsible mw-collapsed"
{| class="wikitable center-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" |Errors by subgroup
|+ style="font-size: 105%; white-space: nowrap;" | Errors by subgroup
|-
|-
! Subgroup
! Subgroup
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|-
|-
| 2.3
| 2.3
| {{mapping| 15 24 }}
| {{Mapping| 15 24 }}
| 8.979801
| 8.979801
|-
|-
| 2.5
| 2.5
| {{mapping| 15 35 }}
| {{Mapping| 15 35 }}
| 6.826357
| 6.826357
|-
|-
| 2.7
| 2.7
| {{mapping| 15 42 }}
| {{Mapping| 15 42 }}
| 4.418738
| 4.418738
|-
|-
| 2.11
| 2.11
| {{mapping| 15 52 }}
| {{Mapping| 15 52 }}
| 4.336492
| 4.336492
|-
|-
| 2.3.5
| 2.3.5
|  
| {{Mapping| 15 24 35 }}
{{mapping| 15 24 35 }}
| 10.742841
| 10.742841
|-
|-
| 2.3.7
| 2.3.7
| {{mapping| 15 24 42 }}
| {{Mapping| 15 24 42 }}
| 17.481581
| 17.481581
|-
|-
| 2.3.11
| 2.3.11
|  
| {{Mapping| 15 24 52 }}
{{mapping| 15 24 52 }}
| 16.831238
| 16.831238
|-
|-
| 2.5.7
| 2.5.7
| {{mapping| 15 35 42 }}
| {{Mapping| 15 35 42 }}
| 10.509269
| 10.509269
|-
|-
| 2.5.11
| 2.5.11
| {{mapping| 15 35 52 }}
| {{Mapping| 15 35 52 }}
| 8.335693
| 8.335693
|-
|-
| 2.7.11
| 2.7.11
| {{mapping| 15 42 52 }}
| {{Mapping| 15 42 52 }}
| 8.002641
| 8.002641
|-
|-
| 2.3.5.7
| 2.3.5.7
| {{mapping| 15 24 35 42 }}
| {{Mapping| 15 24 35 42 }}
| 15.603114
| 15.603114
|-
|-
| 2.3.5.11
| 2.3.5.11
| {{mapping| 15 24 35 52 }}
| {{Mapping| 15 24 35 52 }}
| 14.693746
| 14.693746
|-
|-
| 2.3.7.11
| 2.3.7.11
| {{mapping| 15 24 42 52 }}
| {{Mapping| 15 24 42 52 }}
| 18.660367
| 18.660367
|-
|-
| 2.5.7.11
| 2.5.7.11
| {{mapping| 15 35 42 52 }}
| {{Mapping| 15 35 42 52 }}
| 11.462127
| 11.462127
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| {{mapping| 15 24 35 42 52 }}
| {{Mapping| 15 24 35 42 52 }}
| 17.258371
| 17.258371
|}
|}


===Uniform maps===
=== Uniform maps ===
{{Uniform map|edo=15}}
{{Uniform map|edo=15}}


===Rank-2 temperaments===
===Rank-2 temperaments===
* [[List of 15et rank two temperaments by badness]]
* [[List of 15et rank two temperaments by badness]]
*[[List of edo-distinct 15et rank two temperaments]]
* [[List of edo-distinct 15et rank two temperaments]]


{| class="wikitable center-all left-4 left-5"
{| class="wikitable center-all left-4 left-5"
|-
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
!Generator
! Generator
! Associated<br>ratio
! Associated<br>ratio
!Temperaments
! Temperaments
!Mos scales
! Mos scales
|-
|-
| 1
| 1
|1\15
| 1\15
|21/20
| 21/20
|[[Nautilus]]<br>[[Valentine]]
| [[Nautilus]]<br>[[Valentine]]
|
|
|-
|-
|1
| 1
|2\15
| 2\15
|11/10
| 11/10
|[[Porcupine]] / [[opossum]]
| [[Porcupine]] / [[opossum]]
|[[1L 6s]], [[7L 1s]]
| [[1L 6s]], [[7L 1s]]
|-
|-
|1
| 1
|4\15
| 4\15
|6/5<br>77/64
| 6/5<br>77/64
|[[Cata]] / [[keemun]] / [[catalan]]<br>[[Orgone]] / [[superkleismic]]
| [[Cata]] / [[keemun]] / [[catalan]]<br>[[Orgone]] / [[superkleismic]]
| [[3L 1s]], [[4L 3s]], [[4L 7s]]
| [[3L 1s]], [[4L 3s]], [[4L 7s]]
|-
|-
|1
| 1
|7\15
| 7\15
|7/5
| 7/5
|[[Progress]]<br>[[Parakangaroo]]
| [[Progress]]<br>[[Parakangaroo]]
|[[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]] [[2L 9s]], [[2L 11s]]
| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]] [[2L 9s]], [[2L 11s]]
|-
|-
| 3
| 3
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| 16/15
| 16/15
| [[Augmented (temperament)|Augmented]] / [[augene]]
| [[Augmented (temperament)|Augmented]] / [[augene]]
|[[3L 3s]], [[3L 6s]], [[3L 9s]]
| [[3L 3s]], [[3L 6s]], [[3L 9s]]
|-
|-
|3
| 3
|2\15
| 2\15
|7/6
| 7/6
|[[Triforce]]
| [[Triforce]]
|[[3L 3s]], [[6L 3s]]
| [[3L 3s]], [[6L 3s]]
|-
|-
|5
| 5
| 1\15
| 1\15
|16/15
| 16/15
|[[Blackwood]] / [[blacksmith]]
| [[Blackwood]]
| [[5L 5s]]
| [[5L 5s]]
|}
|}
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