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! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | Extended pythagorean notation
! colspan="3" | Extended pythagorean notation
! colspan="3" |[[SKULO interval names|SKULO]] notation (S or U = 1)
! colspan="3" | [[SKULO interval names|SKULO]] notation (S or U = 1)
|-
|-
| 0
| 0
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* F𝄫 = E{{demiflat2}}
* F𝄫 = E{{demiflat2}}


===Sagittal notation===
=== Sagittal notation ===
The Revo flavor of Sagittal notation from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]]:
The Revo flavor of Sagittal notation from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]]:


[[File:31edo Sagittal.png|800px]]
[[File:31edo Sagittal.png|800px]]


==Approximation to JI==
== Approximation to JI ==
[[File:31-edo.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 31edo]]
[[File:31-edo.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 31edo]]
===15-odd-limit interval mappings===
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 31edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  
The following table shows how [[15-odd-limit intervals]] are represented in 31edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  


{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space:nowrap" |15-odd-limit intervals by direct mapping (even if inconsistent)
|+style="white-space:nowrap"| 15-odd-limit intervals by direct mapping (even if inconsistent)
|-
|-
!Interval, complement
! Interval, complement
!Error (abs, [[Cent|¢]])
! Error (abs, [[Cent|¢]])
!Error (rel, [[Relative cent|%]])
! Error (rel, [[Relative cent|%]])
|-
|-
|'''[[5/4]], [[8/5]]'''
| '''[[5/4]], [[8/5]]'''
|'''0.783'''
| '''0.783'''
|'''2.0'''
| '''2.0'''
|-
|-
|[[11/9]], [[18/11]]
| [[11/9]], [[18/11]]
|0.979
| 0.979
|2.5
| 2.5
|-
|-
|'''[[7/4]], [[8/7]]'''
| '''[[7/4]], [[8/7]]'''
|'''1.084'''
| '''1.084'''
|'''2.8'''
| '''2.8'''
|-
|-
|[[7/5]], [[10/7]]
| [[7/5]], [[10/7]]
|1.867
| 1.867
|4.8
| 4.8
|-
|-
|[[15/14]], [[28/15]]
| [[15/14]], [[28/15]]
|3.314
| 3.314
|8.6
| 8.6
|-
|-
|[[7/6]], [[12/7]]
| [[7/6]], [[12/7]]
|4.097
| 4.097
|10.6
| 10.6
|-
|-
|[[11/6]], [[12/11]]
| [[11/6]], [[12/11]]
|4.202
| 4.202
|10.9
| 10.9
|-
|-
|[[15/8]], [[16/15]]
| [[15/8]], [[16/15]]
|4.398
| 4.398
|11.4
| 11.4
|-
|-
|[[15/11]], [[22/15]]
| [[15/11]], [[22/15]]
|4.985
| 4.985
|12.9
| 12.9
|-
|-
|'''[[3/2]], [[4/3]]'''
| '''[[3/2]], [[4/3]]'''
|'''5.181'''
| '''5.181'''
|'''13.4'''
| '''13.4'''
|-
|-
|[[5/3]], [[6/5]]
| [[5/3]], [[6/5]]
|5.964
| 5.964
|15.4
| 15.4
|-
|-
|[[11/7]], [[14/11]]
| [[11/7]], [[14/11]]
|8.298
| 8.298
|21.4
| 21.4
|-
|-
|[[9/7]], [[14/9]]
| [[9/7]], [[14/9]]
|9.278
| 9.278
|24.0
| 24.0
|-
|-
|'''[[11/8]], [[16/11]]'''
| '''[[11/8]], [[16/11]]'''
|'''9.382'''
| '''9.382'''
|'''24.2'''
| '''24.2'''
|-
|-
|[[11/10]], [[20/11]]
| [[11/10]], [[20/11]]
|10.166
| 10.166
|26.3
| 26.3
|-
|-
|[[13/10]], [[20/13]]
| [[13/10]], [[20/13]]
|10.302
| 10.302
|26.6
| 26.6
|-
|-
|[[9/8]], [[16/9]]
| [[9/8]], [[16/9]]
|10.362
| 10.362
|26.8
| 26.8
|-
|-
|'''[[13/8]], [[16/13]]'''
| '''[[13/8]], [[16/13]]'''
|'''11.085'''
| '''11.085'''
|'''28.6'''
| '''28.6'''
|-
|-
|[[9/5]], [[10/9]]
| [[9/5]], [[10/9]]
|11.145
| 11.145
|28.8
| 28.8
|-
|-
|[[13/7]], [[14/13]]
| [[13/7]], [[14/13]]
|12.169
| 12.169
|31.4
| 31.4
|-
|-
|[[15/13]], [[26/15]]
| [[15/13]], [[26/15]]
|15.483
| 15.483
|40.0
| 40.0
|-
|-
|[[13/12]], [[24/13]]
| [[13/12]], [[24/13]]
|16.266
| 16.266
|42.0
| 42.0
|-
|-
|''[[13/9]], [[18/13]]''
| ''[[13/9]], [[18/13]]''
|''17.263''
| ''17.263''
|''44.6''
| ''44.6''
|-
|-
|''[[13/11]], [[22/13]]''
| ''[[13/11]], [[22/13]]''
|''18.242''
| ''18.242''
|''47.1''
| ''47.1''
|}
|}
{{15-odd-limit|31}}
{{15-odd-limit|31}}


==Relationship to 12-edo==
== Relationship to 12-edo ==
Whereas 12-edo has a circle of twelve 5ths, 31-edo has a spiral of twelve 5ths (since 18\31 is on the 7\12 kite in the scale tree). This spiral of 5th shows 31-edo in a 12-edo-friendly format. Excellent for introducing 31-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.
Whereas 12-edo has a circle of twelve 5ths, 31-edo has a spiral of twelve 5ths (since 18\31 is on the 7\12 kite in the scale tree). This spiral of 5th shows 31-edo in a 12-edo-friendly format. Excellent for introducing 31-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.


Line 768: Line 768:
[[File:31edo CoF semi and sesqui.png|none|thumb|500x500px]]
[[File:31edo CoF semi and sesqui.png|none|thumb|500x500px]]


==Scales==
== Scales ==
*[[Meantone5]]
* [[Meantone5]]
*[[Meantone7]]
* [[Meantone7]]
*[[Meantone12]]
* [[Meantone12]]


===MOS scales===
=== MOS scales ===
{{main| 31edo MOS scales }}
{{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{{clarify}} are also useful:
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 neutral third generator generates [[ultrasoft]] [[mosh]] and [[superhard]] [[dicotonic]] MOSes.
* 9\31 neutral third generator generates [[ultrasoft]] [[mosh]] and [[superhard]] [[dicotonic]] MOSes.
*11\31 generator 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.
* 11\31 generator 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]] 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]].
* 12\31 generator generates a [[semihard]] [[oneirotonic]] 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.
* 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¢) subminor third generator. The [[ultrasoft]] 9-tone [[4L 5s|orwelloid (4L 5s)]] MOS could be treated as a 9-tone well temperament.
* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[ultrasoft]] 9-tone [[4L 5s|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.
* 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.
See [[#Rank-2 temperaments]] for a table of MOSes and their temperament interpretations.


===Harmonic scales===
=== 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 even better. 31's version of 13/8 is quite wide and only vaguely suggests the [[13-limit]].  
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 even better. 31's version of 13/8 is quite wide and only vaguely suggests the [[13-limit]].  


Line 793: Line 793:
{| class="wikitable"
{| class="wikitable"
|-
|-
|Overtones in "Mode 8":
| Overtones in "Mode 8":
|8
| 8
|9
| 9
|10
| 10
|11
| 11
|12
| 12
|13
| 13
|14
| 14
|15
| 15
|16
| 16
|-
|-
|…as JI Ratio from 1/1:
| …as JI Ratio from 1/1:
|1/1
| 1/1
|9/8
| 9/8
|5/4
| 5/4
|11/8
| 11/8
|3/2
| 3/2
|13/8
| 13/8
|7/4
| 7/4
|15/8
| 15/8
|2/1
| 2/1
|-
|-
|…in cents:
| …in cents:
|0
| 0
|203.9
| 203.9
|386.3
| 386.3
|551.3
| 551.3
|702.0
| 702.0
|840.5
| 840.5
|968.8
| 968.8
|1088.3
| 1088.3
|1200.0
| 1200.0
|-
|-
|Nearest degree of 31edo:
| Nearest degree of 31edo:
|0
| 0
|5
| 5
|10
| 10
|14
| 14
|18
| 18
|22
| 22
|25
| 25
|28
| 28
|31
| 31
|-
|-
|…in cents:
| …in cents:
|0
| 0
|193.5
| 193.5
|387.1
| 387.1
|541.9
| 541.9
|696.8
| 696.8
|851.6
| 851.6
|967.7
| 967.7
|1083.9
| 1083.9
|1200.0
| 1200.0
|}
|}


In mode 16, the most closely-matched harmonics are the composite ones, 21 and 25. Of the other harmonics:
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.
* 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]]).
* 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.
* 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.
* 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.
* 29 and 31 are both ''very'' sharp, and intervals involving them are unlikely to play any major role.


{| class="wikitable"
{| class="wikitable"
|-
|-
|Odd overtones in "Mode 16":
| Odd overtones in "Mode 16":
|17
| 17
|19
| 19
|21
| 21
|23
| 23
|25
| 25
|27
| 27
|29
| 29
|31
| 31
|-
|-
|…as JI Ratio from 1/1:
| …as JI Ratio from 1/1:
|17/16
| 17/16
|19/16
| 19/16
|21/16
| 21/16
|23/16
| 23/16
|25/16
| 25/16
|27/16
| 27/16
|29/16
| 29/16
|31/16
| 31/16
|-
|-
|…in cents:
| …in cents:
|105.0
| 105.0
|297.5
| 297.5
|470.8
| 470.8
|628.3
| 628.3
|772.6
| 772.6
|905.9
| 905.9
|1029.6
| 1029.6
|1145.0
| 1145.0
|-
|-
|Nearest degree of 31edo:
| Nearest degree of 31edo:
|3
| 3
|8
| 8
|12
| 12
|16
| 16
|20
| 20
|23
| 23
|27
| 27
|30
| 30
|-
|-
|…in cents:
| …in cents:
|116.1
| 116.1
|309.7
| 309.7
|464.5
| 464.5
|619.4
| 619.4
|774.2
| 774.2
|890.3
| 890.3
|1045.1
| 1045.1
|1161.3
| 1161.3
|}
|}


===Various subsets===
=== Various subsets ===
A large open list of subsets from 31edo that people have named:  
A large open list of subsets from 31edo that people have named:  
*[[31edo modes]]
* [[31edo modes]]
*[[Strictly proper]] [[Strictly proper 7-tone 31edo scales|7-tone 31edo scales]]
* [[Strictly proper]] [[Strictly proper 7-tone 31edo scales|7-tone 31edo scales]]
*Interesting (to somebody) [[9-tone 31edo scales]]
* Interesting (to somebody) [[9-tone 31edo scales]]
*the [[Euler-Fokker genus]] (technically [[JI]] but representable in 31)
* the [[Euler-Fokker genus]] (technically [[JI]] but representable in 31)
*the [[altered pentad]]
* 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)
* [[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)


==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|Comma List]]
! rowspan="2" | [[Comma list|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
| 2.3
|{{monzo| -49 31 }}
| {{monzo| -49 31 }}
|[{{val| 31 49 }}]
| [{{val| 31 49 }}]
| +1.63
| +1.63
|1.64
| 1.64
|4.22
| 4.22
|-
|-
|2.3.5
| 2.3.5
|81/80, 393216/390625
| 81/80, 393216/390625
|[{{val| 31 49 72 }}]
| [{{val| 31 49 72 }}]
| +0.98
| +0.98
|1.63
| 1.63
|4.20
| 4.20
|-
|-
|2.3.5.7
| 2.3.5.7
|81/80, 126/125, 1029/1024
| 81/80, 126/125, 1029/1024
|[{{val| 31 49 72 87 }}]
| [{{val| 31 49 72 87 }}]
| +0.83
| +0.83
|1.43
| 1.43
|3.70
| 3.70
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|81/80, 99/98, 121/120, 126/125
| 81/80, 99/98, 121/120, 126/125
|[{{val| 31 49 72 87 107 }}]
| [{{val| 31 49 72 87 107 }}]
| +1.21
| +1.21
|1.49
| 1.49
|3.84
| 3.84
|}
|}


Line 963: Line 963:
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 the quince comma [[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. In the 17-limit it tempers out [[120/119]], equating the otonal tetrad of 4:5:6:7 and the inversion of the 10:12:15:17 minor tetrad.
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 the quince comma [[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. In the 17-limit it tempers out [[120/119]], equating the otonal tetrad of 4:5:6:7 and the inversion of the 10:12:15:17 minor tetrad.


===Commas===
=== Commas ===
31edo [[tempers out]] the following [[commas]]. This assumes the [[val]] {{val| 31 49 72 87 107 115 }}, comma values rounded to 5 significant digits.
31edo [[tempers out]] the following [[commas]]. This assumes the [[val]] {{val| 31 49 72 87 107 115 }}, comma values rounded to 5 significant digits.


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic limit| Prime<br>Limit]]
! [[Harmonic limit|Prime<br>Limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
! [[Color name| Color Name]]
! [[Color name|Color Name]]
! Name
! Name
|-
|-
Line 987: Line 987:
| 31.567
| 31.567
| Lala-tribiyo
| Lala-tribiyo
| [[Ampersand comma]]
| [[Ampersand]]
|-
|-
| 5
| 5
Line 1,199: Line 1,199:
| Mynucuma
| Mynucuma
|}
|}
<references />
<references/>


===Rank-2 temperaments===
=== Rank-2 temperaments ===
*[[List of 31et rank two temperaments by badness]]
* [[List of 31et rank two temperaments by badness]]
*[[List of edo-distinct 31et rank two temperaments]]
* [[List of edo-distinct 31et rank two temperaments]]
*[[Syntonic-31 equivalence continuum]]
* [[Syntonic-31 equivalence continuum]]


{| class="wikitable center-1"
{| class="wikitable center-1"
|+Rank-2 temperaments by generators
|+ Rank-2 temperaments by generators
|-
|-
!Generator
! Generator
!Cents
! Cents
!MOSes
! MOSes
!Temperaments
! Temperaments
![[Pergen]]
! [[Pergen]]
|-
|-
|1\31
| 1\31
|38.71
| 38.71
|
|  
|[[Slender]]
| [[Slender]]
|(P8, P4/13)
| (P8, P4/13)
|-
|-
|2\31
| 2\31
|77.42
| 77.42
|[[1L 14s]], [[15L 1s]]
| [[1L 14s]], [[15L 1s]]
|[[Valentine]] / [[lupercalia]]
| [[Valentine]] / [[lupercalia]]
|(P8, P5/9)
| (P8, P5/9)
|-
|-
|3\31
| 3\31
|116.13
| 116.13
|[[1L 9s]], [[10L 1s]], [[10L 11s]]
| [[1L 9s]], [[10L 1s]], [[10L 11s]]
|[[Mercy]] / [[miracle]]
| [[Mercy]] / [[miracle]]
|(P8, P5/6)
| (P8, P5/6)
|-
|-
|4\31
| 4\31
|154.84
| 154.84
|[[1L 6s]], [[7L 1s]], <br>[[8L 7s]], [[8L 15s]]
| [[1L 6s]], [[7L 1s]], <br>[[8L 7s]], [[8L 15s]]
|[[Nusecond]] / [[greeley]]
| [[Nusecond]] / [[greeley]]
|(P8, P11/11)
| (P8, P11/11)
|-
|-
|5\31
| 5\31
|193.55
| 193.55
|[[1L 5s]], [[6L 1s]], [[6L 7s]], <br>[[6L 13s]], [[6L 19s]]
| [[1L 5s]], [[6L 1s]], [[6L 7s]], <br>[[6L 13s]], [[6L 19s]]
|[[Luna]] / [[didacus]] / [[hemithirds]] /  
| [[Luna]] / [[didacus]] / [[hemithirds]] /  
[[hemiwürschmidt]] / [[tutone]]
[[hemiwürschmidt]] / [[tutone]]
|(P8, ccP4/15)
| (P8, ccP4/15)
|-
|-
|6\31
| 6\31
|232.26
| 232.26
|[[1L 4s]], [[5L 1s]], [[5L 6s]], <br>[[5L 11s]], [[5L 16s]], [[5L 21s]]
| [[1L 4s]], [[5L 1s]], [[5L 6s]], <br>[[5L 11s]], [[5L 16s]], [[5L 21s]]
|[[Mothra]] / [[mosura]]<br>[[Quadrawell]]
| [[Mothra]] / [[mosura]]<br>[[Quadrawell]]
|(P8, P5/3)
| (P8, P5/3)
|-
|-
|7\31
| 7\31
|270.97
| 270.97
|[[1L 3s]], [[4L 1s]], [[4L 5s]], <br>[[9L 4s]], [[9L 13s]]
| [[1L 3s]], [[4L 1s]], [[4L 5s]], <br>[[9L 4s]], [[9L 13s]]
|[[Orson]] / [[orwell]] / [[winston]]
| [[Orson]] / [[orwell]] / [[winston]]
|(P8, P12/7)
| (P8, P12/7)
|-
|-
|8\31
| 8\31
|309.68
| 309.68
|[[3L 1s]], [[4L 3s]], [[4L 7s]], <br>[[4L 11s]], [[4L 15s]], [[4L 19s]], <br>[[4L 23s]]
| [[3L 1s]], [[4L 3s]], [[4L 7s]], <br>[[4L 11s]], [[4L 15s]], [[4L 19s]], <br>[[4L 23s]]
|[[Myna]] / [[triwell]]
| [[Myna]] / [[triwell]]
|(P8, ccP5/10)
| (P8, ccP5/10)
|-
|-
|9\31
| 9\31
|348.39
| 348.39
|[[3L 1s]], [[3L 4s]], [[7L 3s]], <br>[[7L 10s]], [[7L 17s]]
| [[3L 1s]], [[3L 4s]], [[7L 3s]], <br>[[7L 10s]], [[7L 17s]]
|[[Mohaha]] / [[vicentino]] /<br>[[mohajira]] / [[migration]]
| [[Mohaha]] / [[vicentino]] /<br>[[mohajira]] / [[migration]]
|(P8, P5/2)
| (P8, P5/2)
|-
|-
|10\31
| 10\31
|387.10
| 387.10
|[[3L 1s]], [[3L 4s]], [[3L 7s]], <br>[[3L 10s]], [[3L 13s]], [[3L 16s]], <br>[[3L 19s]], [[3L 22s]], [[3L 25s]]
| [[3L 1s]], [[3L 4s]], [[3L 7s]], <br>[[3L 10s]], [[3L 13s]], [[3L 16s]], <br>[[3L 19s]], [[3L 22s]], [[3L 25s]]
|[[Würschmidt]] / [[worschmidt]]
| [[Würschmidt]] / [[worschmidt]]
|(P8, ccP5/8)
| (P8, ccP5/8)
|-
|-
|11\31
| 11\31
|425.81
| 425.81
|[[3L 2s]], [[3L 5s]], [[3L 8s]], <br>[[3L 11s]], [[14L 3s]]
| [[3L 2s]], [[3L 5s]], [[3L 8s]], <br>[[3L 11s]], [[14L 3s]]
|[[Squares]] / [[Sentinel]]
| [[Squares]] / [[Sentinel]]
|(P8, P11/4)
| (P8, P11/4)
|-
|-
|12\31
| 12\31
|464.52
| 464.52
|[[3L 2s]], [[5L 3s]], <br>[[5L 8s]], [[13L 5s]]
| [[3L 2s]], [[5L 3s]], <br>[[5L 8s]], [[13L 5s]]
|[[5L 3s/Temperaments#A-Team|A-Team]]<br>[[Semisept]]
| [[5L 3s/Temperaments#A-Team|A-Team]]<br>[[Semisept]]
|(P8, c<sup>5</sup>P4/14)
| (P8, c<sup>5</sup>P4/14)
|-
|-
|13\31
| 13\31
|503.23
| 503.23
|[[2L 3s]], [[5L 2s]], <br>[[7L 5s]], [[12L 7s]]
| [[2L 3s]], [[5L 2s]], <br>[[7L 5s]], [[12L 7s]]
|[[Meantone]] / [[meanpop]]
| [[Meantone]] / [[meanpop]]
|(P8, P5)
| (P8, P5)
|-
|-
|14\31
| 14\31
|541.94
| 541.94
|[[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[9L 2s]], [[11L 9s]]
| [[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[9L 2s]], [[11L 9s]]
|[[Casablanca]]<br>[[Cypress]]<br>[[Oracle]]
| [[Casablanca]]<br>[[Cypress]]<br>[[Oracle]]
|(P8, c<sup>5</sup>P4/12)
| (P8, c<sup>5</sup>P4/12)
|-
|-
|15\31
| 15\31
|580.65
| 580.65
|[[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[2L 9s]], [[2L 11s]], [[2L 13s]], <br>[[2L 15s]], [[2L 17s]], [[2L 19s]], <br>[[2L 21s]], [[2L 23s]], [[2L 25s]], <br>[[2L 27s]]
| [[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[2L 9s]], [[2L 11s]], [[2L 13s]], <br>[[2L 15s]], [[2L 17s]], [[2L 19s]], <br>[[2L 21s]], [[2L 23s]], [[2L 25s]], <br>[[2L 27s]]
|[[Tritonic]] / [[tritoni]]
| [[Tritonic]] / [[tritoni]]
|(P8, ccP4/5)
| (P8, ccP4/5)
|}
|}


==Music==
== Music ==
{{Main| 31edo/Music }}
{{Main| 31edo/Music }}
{{Catrel|31edo tracks}}
{{Catrel|31edo tracks}}


==See also==
== See also ==
*[[Lumatone mapping for 31edo]]
* [[Lumatone mapping for 31edo]]
*[[List of 31edo Chords]]
* [[List of 31edo Chords]]
*[[Skip fretting system 31 2 9]]
* [[Skip fretting system 31 2 9]]
*[[Pentachords of 31edo]]
* [[Pentachords of 31edo]]
*[[Tricesimoprimal Tetrachordal Tesseract]]
* [[Tricesimoprimal Tetrachordal Tesseract]]
*[[MicroPedagogyCollective]] - is at work (as of 2012) producing demonstrative material which will encourage and enable more people to learn this system. There have been two [[ThirtyOneToneSinginCamp]]s as well.
* [[MicroPedagogyCollective]] - is at work (as of 2012) producing demonstrative material which will encourage and enable more people to learn this system. There have been two [[ThirtyOneToneSinginCamp]]s as well.


==Further reading==
== Further reading ==
===Books===
=== Books ===
*Coates, Bill. ''[https://scribd.com/document/32296502/31-tone-equal-temperament Diesis: An Introduction to the Temperament of 31 Notes to Each Octave]''. Self-published, 1992.
* Coates, Bill. ''[https://scribd.com/document/32296502/31-tone-equal-temperament Diesis: An Introduction to the Temperament of 31 Notes to Each Octave]''. Self-published, 1992.
*[[Sword, Ron]]. ''[https://ronsword.bigcartel.com/product/tricesimoprimal-scales-for-guitar Tricesimoprimal Scales for Guitar: Scales for 31-EDO]''. 2009. ([http://www.metatonalmusic.com/books.html Metatonal Music link]) (A comprehensive approach to 31edo and all the families associated for guitar. Features over 300 scale charts/scale examples.)
* [[Sword, Ron]]. ''[https://ronsword.bigcartel.com/product/tricesimoprimal-scales-for-guitar Tricesimoprimal Scales for Guitar: Scales for 31-EDO]''. 2009. ([http://www.metatonalmusic.com/books.html Metatonal Music link]) (A comprehensive approach to 31edo and all the families associated for guitar. Features over 300 scale charts/scale examples.)


===Articles===
=== Articles ===
*[http://www.huygens-fokker.org/docs/beerart.html ''The Development of 31-tone Music''] [https://www.webcitation.org/5xeFzBM9b Permalink] by [[Anton de Beer]]
* [http://www.huygens-fokker.org/docs/beerart.html ''The Development of 31-tone Music''] [https://www.webcitation.org/5xeFzBM9b Permalink] by [[Anton de Beer]]
*[http://www.huygens-fokker.org/docs/fokkerorg.html ''Equal Temperament and the Thirty-one-keyed organ''] [https://www.webcitation.org/5xeG6Tmli Permalink] by [[Adriaan Daniël Fokker]]
* [http://www.huygens-fokker.org/docs/fokkerorg.html ''Equal Temperament and the Thirty-one-keyed organ''] [https://www.webcitation.org/5xeG6Tmli Permalink] by [[Adriaan Daniël Fokker]]
*''New Music with 31 Notes'' by Adriaan Daniël Fokker, translated by Leigh Gerdine
* ''New Music with 31 Notes'' by Adriaan Daniël Fokker, translated by Leigh Gerdine
*[http://www.huygens-fokker.org/docs/rap31.html ''About 31-tone Equal Temperament''] [https://www.webcitation.org/5xeGH4uBH Permalink] by [[Paul Rapoport]]
* [http://www.huygens-fokker.org/docs/rap31.html ''About 31-tone Equal Temperament''] [https://www.webcitation.org/5xeGH4uBH Permalink] by [[Paul Rapoport]]
*[http://www.huygens-fokker.org/docs/terp31.html ''Toward a Theory of Meantone (and 31-et) Harmony''] [https://www.webcitation.org/5xeGMeCMd Permalink] by [[Siemen Terpstra]]
* [http://www.huygens-fokker.org/docs/terp31.html ''Toward a Theory of Meantone (and 31-et) Harmony''] [https://www.webcitation.org/5xeGMeCMd Permalink] by [[Siemen Terpstra]]
*[http://tonalsoft.com/enc/number/31edo.aspx Tonalsoft Encyclopedia | ''31edo''] [https://www.webcitation.org/5xeGYj7QU Permalink]
* [http://tonalsoft.com/enc/number/31edo.aspx Tonalsoft Encyclopedia | ''31edo''] [https://www.webcitation.org/5xeGYj7QU Permalink]
*[http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Harmonic-Resources-31Et-EMT-31EBMT.pdf ''Harmonic Resources of 31Et EMT and 31EBMT''] by [[Juhan Puhm]] (2016)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Harmonic-Resources-31Et-EMT-31EBMT.pdf ''Harmonic Resources of 31Et EMT and 31EBMT''] by [[Juhan Puhm]] (2016)


==External links==
== External links ==
===Videos===
=== Videos ===
*[https://youtu.be/E_VD3tqwCAM ''Quarter sharps and flats in the same diatonic key signature'' – Youtube] by [[Stephen Weigel]] – a list of diatonic key signatures and major scales in 31edo (including semi- and sesqui-sharps); and docs in its description.
* [https://youtu.be/E_VD3tqwCAM ''Quarter sharps and flats in the same diatonic key signature'' – Youtube] by [[Stephen Weigel]] – a list of diatonic key signatures and major scales in 31edo (including semi- and sesqui-sharps); and docs in its description.


===Software===
=== Software ===
*[http://31et.com/keyboard.php Virtual Piano Keyboard in 31-Tone Equal Temperament]
* [http://31et.com/keyboard.php Virtual Piano Keyboard in 31-Tone Equal Temperament]
*[http://www.warmplace.ru/forum/viewtopic.php?f=9&t=4750 31EDO Piano — Mini synthesizer in Pixilang]
* [http://www.warmplace.ru/forum/viewtopic.php?f=9&t=4750 31EDO Piano — Mini synthesizer in Pixilang]


===Diagrams===
=== Diagrams ===
*[http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keys-and-Modes-of-31Et.pdf ''Keys and Modes of 31Et''] by Juhan Puhm (2016)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keys-and-Modes-of-31Et.pdf ''Keys and Modes of 31Et''] by Juhan Puhm (2016)
*[http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keyboard-Mapping-for-31Et.pdf ''Keyboard Mapping for 31Et''] by Juhan Puhm (2017)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keyboard-Mapping-for-31Et.pdf ''Keyboard Mapping for 31Et''] by Juhan Puhm (2017)
*[http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Mapping-Range-for-31Et.pdf ''Mapping Range for 31Et''] by Juhan Puhm (2017)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Mapping-Range-for-31Et.pdf ''Mapping Range for 31Et''] by Juhan Puhm (2017)


[[Category:Golden meantone]]
[[Category:Golden meantone]]
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