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=== Subsets and supersets ===
=== Subsets and supersets ===
31edo is the 11th [[prime edo]], following [[29edo]] and coming before [[37edo]]. It does not contain any nontrivial subset edos, though it contains [[31ed4]]. [[62edo]], which doubles it, provides an alternative way to extend the temperament to the 13- and 17- and 19-limit.
31edo is the 11th [[prime edo]], following [[29edo]] and coming before [[37edo]]. It does not contain any nontrivial subset edos, though it contains [[31ed4]]. [[62edo]], which doubles it, provides an alternative way to extend the temperament to the 13- and 17- and 19-limit.
=== Stretched and compressed tunings ===
31edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13. Tunings such as [[80ed6]] and [[111ed12]] are great demonstrations of this.
229ed169 has an octave stretched by 2.23893{{c}}. Since the 13th harmonic is exactly halfway between 114 and 115 steps, this difference is the absolute maximum amount of octave stretch 31edo can tolerate before a discrepancy for the 13th harmonic occurs.


== Intervals ==
== Intervals ==
Line 715: Line 710:


[[File:31-edo spiral.png|582x582px]]
[[File:31-edo spiral.png|582x582px]]
== Scales ==
* [[Meantone5]]
* [[Meantone7]]
* [[Meantone12]]
=== MOS scales ===
{{main| List of MOS scales in 31edo }}
The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains{{clarify}} are also useful:
* 9\31, the neutral third, generates [[ultrasoft]] [[mosh]] and [[superhard]] [[dicotonic]] MOSes.
* 11\31, the supermajor third or diminished fourth, generates a [[TAMNAMS|parahard]] [[sensoid]] scale with resolution from neutral thirds, sixths, and sevenths to perfect fourths, fifths, and octaves, and a [[semihard]] [[3L 8s]] scale with a jagged-but-chromatic feel.
* 12\31 generator generates a [[semihard]] oneirotonic ([[5L 3s]]) scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4, and 7/6 better in tune and with the flat fifth close to [[19/13]].
* A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo.
* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0{{c}}) subminor third generator. The [[ultrasoft]] 9-tone orwelloid [[4L 5s]] MOS could be treated as a 9-tone well temperament.
* It has close approximations to [[6edf]] (→ [[miracle]]) and [[9edf]] (→ [[Carlos Alpha]]), fifth-equivalent equal divisions that hit many good JI approximations.
See [[#Rank-2 temperaments]] for a table of MOSes and their temperament interpretations.
=== Harmonic scales ===
31edo approximates Mode 8 of the [[harmonic series]] okay, 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 okay, but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated much better. 31edo's closest approximation of 13/8, the neutral sixth, is significantly sharper than just and only vaguely suggests the [[13-limit]].
The steps are: 5 5 4 4 4 3 3 3.
{| 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 (see [[No-threes subgroup temperaments#Mercy|mercy temperament]]).
* 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<sup>3</sup> and the error from the meantone fifths accumulates.
* 29 and 31 are both almost critically 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-tone 31edo scales|7-tone 31edo scales]]
* Interesting (to somebody) [[9-tone 31edo scales]]
* the [[Erose–McClain double mode]]s, which are [[nonoctave]]
* the [[Euler–Fokker genus]] (technically [[JI]] but representable in 31)
* the [[altered pentad]]
* [[diasem]] (2.3.7 subgroup scale; 5 2 5 1 5 2 5 1 5 or 5 1 5 2 5 1 5 2 5 in 31edo)


== Approximation to JI ==
== Approximation to JI ==
Line 1,331: Line 1,174:
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Octave stretch or compression ==
31edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13.
What follows is a comparison of stretched-octave 31edo tunings.
; 31edo
* Step size: 38.710{{c}}, octave size: 1200.000{{c}}
Pure-octaves 31edo approximates all harmonics up to 16 within 11.1{{c}}.
{{Harmonics in equal|31|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 31edo}}
{{Harmonics in equal|31|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31edo (continued)}}
; [[WE|31et, 13-limit WE tuning]]
* Step size: 38.725{{c}}, octave size: 1200.481{{c}}
Stretching the octave of 31edo by around 0.5{{c}} results in slightly improved primes 3, 7 and 11, but slightly worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 12.8{{c}}. Both the 13-limit TE and WE tunings do this.
{{Harmonics in cet|38.725188|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 31et, 13-limit WE tuning}}
{{Harmonics in cet|38.725188|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)}}
; [[ZPI|127zpi]]
* Step size: 38.737{{c}}, octave size: 1200.837{{c}}
Stretching the octave of 31edo by around 0.8{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 127zpi does this.
{{Harmonics in cet|38.736691|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 127zpi}}
{{Harmonics in cet|38.736691|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 127zpi (continued)}}
; [[WE|31et, 11-limit WE tuning]]
* Step size: 38.748{{c}}, octave size: 1201.196{{c}}
Stretching the octave of 31edo by around 1.2{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This approximates all harmonics up to 16 within 15.5{{c}} Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this, so does the tuning [[111ed12]].
{{Harmonics in cet|38.748261|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 31et, 11-limit WE tuning}}
{{Harmonics in cet|38.748261|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued)}}
; [[80ed6]]
* Step size: 38.774{{c}}, octave size: 1202.008{{c}}
Stretching the octave of 31edo by around 2.0{{c}} results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This is approaching 2.239{{c}} – the most octave stretch 31edo can tolerate before the mapping of the 13th harmonic changes. This approximates all harmonics up to 16 within 18.5{{c}}. The tuning 80ed6 does this.
{{Harmonics in equal|80|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 80ed6}}
{{Harmonics in equal|80|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 80ed6 (continued)}}
; [[49edt]]
* Step size: 38.815{{c}}, octave size: 1203.278{{c}}
Stretching the octave of 31edo by around 3.3{{c}} results in improved primes 3 and 11, especially 11, but slightly worse primes 2, 5, 7 and 13. The 13 is now differently mapped than – and much better than – 80ed6's (but not as good as the pure octaves 13). This approximates all harmonics up to 16 within 15.6{{c}}. The tuning 49edt does this.
{{Harmonics in equal|49|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 49edt}}
{{Harmonics in equal|49|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 49edt (continued)}}
== Scales ==
* [[Meantone5]]
* [[Meantone7]]
* [[Meantone12]]
=== MOS scales ===
{{main| List of MOS scales in 31edo }}
The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains{{clarify}} are also useful:
* 9\31, the neutral third, generates [[ultrasoft]] [[mosh]] and [[superhard]] [[dicotonic]] MOSes.
* 11\31, the supermajor third or diminished fourth, generates a [[TAMNAMS|parahard]] [[sensoid]] scale with resolution from neutral thirds, sixths, and sevenths to perfect fourths, fifths, and octaves, and a [[semihard]] [[3L&nbsp;8s]] scale with a jagged-but-chromatic feel.
* 12\31 generator generates a [[semihard]] oneirotonic ([[5L&nbsp;3s]]) scale, similar to the 5L&nbsp;3s scale in [[13edo]] but with the 9/8, 5/4, and 7/6 better in tune and with the flat fifth close to [[19/13]].
* A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo.
* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0{{c}}) subminor third generator. The [[ultrasoft]] 9-tone orwelloid [[4L&nbsp;5s]] MOS could be treated as a 9-tone well temperament.
* It has close approximations to [[6edf]] (→&nbsp;[[miracle]]) and [[9edf]] (→&nbsp;[[Carlos Alpha]]), fifth-equivalent equal divisions that hit many good JI approximations.
See [[#Rank-2 temperaments]] for a table of MOSes and their temperament interpretations.
=== Harmonic scales ===
31edo approximates Mode 8 of the [[harmonic series]] okay, 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 okay, but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated much better. 31edo's closest approximation of 13/8, the neutral sixth, is significantly sharper than just and only vaguely suggests the [[13-limit]].
The steps are: 5 5 4 4 4 3 3 3.
{| 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 (see [[No-threes subgroup temperaments#Mercy|mercy temperament]]).
* 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<sup>3</sup> and the error from the meantone fifths accumulates.
* 29 and 31 are both almost critically 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-tone 31edo scales|7-tone 31edo scales]]
* Interesting (to somebody) [[9-tone 31edo scales]]
* the [[Erose–McClain double mode]]s, which are [[nonoctave]]
* the [[Euler–Fokker genus]] (technically [[JI]] but representable in 31)
* the [[altered pentad]]
* [[diasem]] (2.3.7 subgroup scale; 5 2 5 1 5 2 5 1 5 or 5 1 5 2 5 1 5 2 5 in 31edo)


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