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Added: "Coda" --> 31-EDO music by NullPointerException Music
It's funny how "MOS scales" wasn't a subsection of "Scales"
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[[File:31edo CoF semi and sesqui.png|none|thumb|500x500px]]
[[File:31edo CoF semi and sesqui.png|none|thumb|500x500px]]


== MOS scales ==
== Scales ==
* [[Meantone5]]
* [[Meantone7]]
* [[Meantone12]]
 
=== MOS scale ===
{{main| 31edo MOS scales }}
 
The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains are also useful:
The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains are also useful:
* 31edo's 9\31 neutral third generator generates [[Step ratio|ultrasoft]] [[3L 4s|mosh]] and [[Step ratio|superhard]] [[7L 3s|dicoid]] MOSes.
* 31edo's 9\31 neutral third generator generates [[Step ratio|ultrasoft]] [[3L 4s|mosh]] and [[Step ratio|superhard]] [[7L 3s|dicoid]] MOSes.
Line 740: Line 747:
See [[31edo#MOSes and rank-2 temperaments]] for a table of MOSes and some temperament interpretations.
See [[31edo#MOSes and rank-2 temperaments]] for a table of MOSes and some temperament interpretations.


== As a regular temperament ==
=== Harmonic Scale ===
31edo approximates Mode 8 of the [[OverToneSeries|harmonic series]] O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the [[13-limit]].
 
{| class="wikitable"
|-
| Overtones in "Mode 8":
| 8
| 9
| 10
| 11
| 12
| 13
| 14
| 15
| 16
|-
| …as JI Ratio from 1/1:
| 1/1
| 9/8
| 5/4
| 11/8
| 3/2
| 13/8
| 7/4
| 15/8
| 2/1
|-
| …in cents:
| 0
| 203.9
| 386.3
| 551.3
| 702.0
| 840.5
| 968.8
| 1088.3
| 1200.0
|-
| Nearest degree of 31edo:
| 0
| 5
| 10
| 14
| 18
| 22
| 25
| 28
| 31
|-
| …in cents:
| 0
| 193.5
| 387.1
| 541.9
| 696.8
| 851.6
| 967.7
| 1083.9
| 1200.0
|}
 
In mode 16, the most closely-matched harmonics are the composite ones, 21 and 25. Of the other harmonics:
 
* 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.
* 19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent.
* 23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent.
* 27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.
* 29 and 31 are both ''very'' sharp, and intervals involving them are unlikely to play any major role.
 
{| class="wikitable"
|-
| Odd overtones in "Mode 16":
| 17
| 19
| 21
| 23
| 25
| 27
| 29
| 31
|-
| …as JI Ratio from 1/1:
| 17/16
| 19/16
| 21/16
| 23/16
| 25/16
| 27/16
| 29/16
| 31/16
|-
| …in cents:
| 105.0
| 297.5
| 470.8
| 628.3
| 772.6
| 905.9
| 1029.6
| 1145.0
|-
| Nearest degree of 31edo:
| 3
| 8
| 12
| 16
| 20
| 23
| 27
| 30
|-
| …in cents:
| 116.1
| 309.7
| 464.5
| 619.4
| 774.2
| 890.3
| 1045.1
| 1161.3
|}
 
=== Various subsets ===
A large open list of subsets from 31edo that people have named:
* [[31edo modes]].
* [[Strictly proper]] [[Strictly proper 7-note 31edo scales|7-note 31edo scales]].
* Interesting (to somebody) [[9-note 31edo scales]].
 
Some of the popular ones:
* 31-tone major: 5 5 3 5 5 5 3
* Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3
* Harmonic scale 8: 5 5 4 4 4 3 3 3
* the [[Euler-Fokker genera]] (technically [[JI]] but representable in 31)
 
== Regular temperament properties ==
31edo excels in the 2.5.7 subgroup (the JI chord 4:5:7 is represented highly [[consistent]]ly: to [[consistency#Consistency to distance d|distance]] 10.36). In 2.5.7 it tempers out the didacus comma [[3136/3125]] and {{monzo|-15 0 -2 7}} ([[823543/819200]]), thus also tempering out the very small [[rainy comma]], the simplest 2.5.7 comma tempered out by the 7-limit microtemperament [[171edo]]. In the 11-limit, 31edo can be defined as the unique temperament that tempers out [[81/80]], [[99/98]], [[121/120]] and [[126/125]], and it supports [[orwell]], [[mohajira]], and the relatively high-accuracy temperament [[miracle]]. In the [[13-limit]] 31edo doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
31edo excels in the 2.5.7 subgroup (the JI chord 4:5:7 is represented highly [[consistent]]ly: to [[consistency#Consistency to distance d|distance]] 10.36). In 2.5.7 it tempers out the didacus comma [[3136/3125]] and {{monzo|-15 0 -2 7}} ([[823543/819200]]), thus also tempering out the very small [[rainy comma]], the simplest 2.5.7 comma tempered out by the 7-limit microtemperament [[171edo]]. In the 11-limit, 31edo can be defined as the unique temperament that tempers out [[81/80]], [[99/98]], [[121/120]] and [[126/125]], and it supports [[orwell]], [[mohajira]], and the relatively high-accuracy temperament [[miracle]]. In the [[13-limit]] 31edo doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.


Line 1,057: Line 1,198:
| (P8, WWP4/5)
| (P8, WWP4/5)
|}
|}
== Scales  ==
=== Harmonic Scale ===
31edo approximates Mode 8 of the [[OverToneSeries|harmonic series]] O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the [[13-limit]].
{| class="wikitable"
|-
| Overtones in "Mode 8":
| 8
| 9
| 10
| 11
| 12
| 13
| 14
| 15
| 16
|-
| …as JI Ratio from 1/1:
| 1/1
| 9/8
| 5/4
| 11/8
| 3/2
| 13/8
| 7/4
| 15/8
| 2/1
|-
| …in cents:
| 0
| 203.9
| 386.3
| 551.3
| 702.0
| 840.5
| 968.8
| 1088.3
| 1200.0
|-
| Nearest degree of 31edo:
| 0
| 5
| 10
| 14
| 18
| 22
| 25
| 28
| 31
|-
| …in cents:
| 0
| 193.5
| 387.1
| 541.9
| 696.8
| 851.6
| 967.7
| 1083.9
| 1200.0
|}
In mode 16, the most closely-matched harmonics are the composite ones, 21 and 25. Of the other harmonics:
* 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.
* 19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent.
* 23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent.
* 27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.
* 29 and 31 are both ''very'' sharp, and intervals involving them are unlikely to play any major role.
{| class="wikitable"
|-
| Odd overtones in "Mode 16":
| 17
| 19
| 21
| 23
| 25
| 27
| 29
| 31
|-
| …as JI Ratio from 1/1:
| 17/16
| 19/16
| 21/16
| 23/16
| 25/16
| 27/16
| 29/16
| 31/16
|-
| …in cents:
| 105.0
| 297.5
| 470.8
| 628.3
| 772.6
| 905.9
| 1029.6
| 1145.0
|-
| Nearest degree of 31edo:
| 3
| 8
| 12
| 16
| 20
| 23
| 27
| 30
|-
| …in cents:
| 116.1
| 309.7
| 464.5
| 619.4
| 774.2
| 890.3
| 1045.1
| 1161.3
|}
=== Various subsets ===
A large open list of subsets from 31edo that people have named:
* [[31edo modes]].
* [[Strictly proper]] [[Strictly proper 7-note 31edo scales|7-note 31edo scales]].
* Interesting (to somebody) [[9-note 31edo scales]].
* See also [[31edo MOS scales]].
Some of the popular ones:
* 31-tone major: 5 5 3 5 5 5 3
* Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3
* Harmonic scale 8: 5 5 4 4 4 3 3 3
* the [[Euler-Fokker genera]] (technically [[JI]] but representable in 31)


== Trivia ==
== Trivia ==
Retrieved from "https://en.xen.wiki/w/31edo"