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{{interwiki
{{Interwiki
| en = 31edo
| de = 31-EDO
| de = 31-EDO
| en = 31edo
| es = 31 EDO
| es = 31 EDO
| ja = 31平均律
| ja = 31平均律
| zh = 31平均律
}}
}}
{{Infobox ET}}
{{Infobox ET}}
Line 18: Line 19:
Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[the Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.
Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[the Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.


One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.  
One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.
 
In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds each, which is much more than the two varieties available in 12edo.


=== Prime harmonics ===
=== Prime harmonics ===
Line 25: Line 28:


=== As a tuning of other temperaments ===
=== As a tuning of other temperaments ===
Besides meantone, 31edo can be used as a tuning for [[mohajira]], [[mothra]], or less optimally [[miracle]] and [[valentine]]. These temperaments split 31edo's fifth, at 18 steps, into two, three, six, and nine equal parts. Mohajira and its alternative, called [[migration]], merge in 31edo, and create a near-optimal 11-limit meantone structure in one unified system. In fact, 31edo can be defined as the unique temperament that [[tempering out|tempers out]] [[81/80]], [[99/98]], [[121/120]], and [[126/125]].  
Besides meantone, 31edo can be used as a tuning for [[mohajira]], [[mothra]] or less optimally [[miracle]] and [[valentine]]. These temperaments split 31edo's fifth, at 18 steps, into two, three, six, and nine equal parts. In fact, 31edo can be defined as the unique temperament that [[tempering out|tempers out]] [[81/80]], [[99/98]], [[121/120]], and [[126/125]].


31edo also [[support]]s [[orwell]], which splits the [[3/1|perfect twelfth]] into seven equal parts of ~7/6. Another notable temperament it supports is [[myna]], which divides the category of thirds into five different intervals: subminor, minor, neutral, major, and supermajor, representing [[7/6]], [[6/5]], 11/9~[[16/13]], 5/4, and 9/7, respectively.  
If we split the meantone [[generator]] of ~3/2 into two neutral thirds, each representing [[11/9]]~[[27/22]], then we get the [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament [[mohaha]], tempering out [[121/120]] and [[243/242]]. We can then map [[7/4]] to the semi-diminished seventh (-13 generators), tempering out [[385/384]], to get the full 11-limit mohajira temperament, which maps 7/6, 6/5, 11/9, 5/4, and 9/7 equidistant from each other. Alternatively, we can use the septimal meantone mapping of 7/4 (+20 generators) to get [[migration]]. Mohajira and [[migration]] merge in 31edo, and create a near-optimal 11-limit meantone structure in one unified system.
 
The supermajor second [[8/7]] is mapped to a third of the perfect fifth in 31edo, thus tempering out [[1029/1024]], supporting [[slendric]] in the [[2.3.7 subgroup|2.3.7-subgroup]]. Slendric is a [[cluster temperament]] with 5 clusters of notes in an octave, each with nearby intervals separated by the interval found at -5 generators, or 1 step of 31edo, representing [[49/48]]~[[64/63]]. For example, 9/8, 8/7, and 7/6 are one step apart from each other, as well as 9/7, 21/16, and 4/3. 31edo supports the full 7-limit extension mothra, which tempers out 81/80, thus equating the 49/48~64/63 spacer with [[36/35]], so that 9/8~10/9, 8/7, 7/6, and 6/5 are all mapped equidistantly, as well as 5/4, 9/7, 21/16, and 4/3. Mothra splits into two 11-limit extensions: [[Gamelismic clan#Undecimal mothra|undecimal mothra]] (26 & 31) tempering out [[99/98]], and [[mosura]] (31 & 36) tempering out [[176/175]].
 
[[Miracle]] temperament splits the slendric generator in two parts and the perfect fifth in six, each representing [[15/14]]~[[16/15]], thus tempering out [[225/224]], so that 5/4 is found at -7 generators. The 11-limit version of miracle sets 11/9 to the neutral third, with prime 11 mapped at +15 generators. While 31edo supports miracle, a more accurate tuning is [[72edo]]. [[Valentine]] temperament splits the slendric generator in three parts and the perfect fifth in nine, each representing [[21/20]], tempering out [[126/125]]. Valentine can also be seen as [[Carlos Alpha]] but with octaves added. The canonical 11-limit extension equates the step with [[22/21]], thus tempering out [[121/120]], [[176/175]], and [[441/440]].
 
31edo also [[support]]s [[orwell]], which splits the [[3/1|perfect twelfth]] into seven equal parts of ~7/6. Three of these reach [[8/5]], and two reach [[11/8]], with 1–7/6–11/8–8/5 being the [[orwell tetrad]]. Commas tempered out by orwell include [[99/98]], [[121/120]], [[176/175]], and [[385/384]], among others.
 
Another notable temperament it supports is [[myna]], which is generated by the minor third, and sets the intervals [[7/6]], [[6/5]], 11/9~[[16/13]], 5/4, and 9/7 being equidistant. Like mohajira, it creates five interval categories, but with 126/125 tempered out instead of 81/80.
 
31edo also supports [[squares]], which splits the [[8/3|perfect eleventh]] into four equal parts, each representing [[14/11]]~9/7, two of which make [[18/11]], and four of which make [[8/3]]. The [[2.3.7.11-subgroup|2.3.7.11 subgroup]] version of this temperament is sometimes known as ''skwares'', tempering out 99/98 and 243/242. Then, prime [[5/1|5]] is found by tempering out [[81/80]], completing the 11-limit.
 
Another temperament supported by 31edo is [[würschmidt]], which is generated by 5/4, such that 8 intervals of 5/4 reach [[6/1]]. Würschmidt extends to the 7- and 11-limit through the skwares mapping, also creating 5 interval categories, with the thirds being 7/6, 6/5, 11/9, 5/4, and 14/11~9/7, each equidistant from each other.


=== 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]] and [[93edo]], which double and triple it, respectively, provide alternative ways to extend the temperament to the 13-, 17-, and 19-limits, and in the case of 93edo, even to the 23-limit.
 
[[217edo]], which slices the edostep in seven, provides a very good correction of primes 3, 13, 17 and 31, and is consistent in the 21-odd-limit.


== Intervals ==
== Intervals ==
{{See also|Table of 31edo intervals|31edo/Individual degrees|31edo solfege}}
{{See also|Table of 31edo intervals|31edo/Individual degrees}}


{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Interval categories
! Approximate ratios<ref group="note">As a 13-limit temperament, with additional ratios of 17, 19, and 23. Inconsistent intervals are in ''italics''.</ref>
! [[Kite's ups and downs notation|Ups and downs notation]]
|-
| 0
| 0.0
| Unison
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 38.7
| Super-unison
| [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[64/63]], [[128/125]]
| {{UDnote|step=1}}
|-
| 2
| 77.4
| Subminor second
| [[21/20]], [[22/21]], [[23/22]], [[25/24]], [[28/27]]
| {{UDnote|step=2}}
|-
| 3
| 116.1
| Minor second
| [[14/13]], [[15/14]], [[16/15]]
| {{UDnote|step=3}}
|-
| 4
| 154.8
| Neutral second
| [[11/10]], [[12/11]], [[13/12]], [[35/32]]
| {{UDnote|step=4}}
|-
| 5
| 193.5
| Major second
| [[9/8]], [[10/9]], [[19/17]], [[28/25]]
| {{UDnote|step=5}}
|-
| 6
| 232.3
| Supermajor second
| [[8/7]]
| {{UDnote|step=6}}
|-
|-
! Degree
| 7
| 271.0
| Subminor third
| [[7/6]]
| {{UDnote|step=7}}
|-
| 8
| 309.7
| Minor third
| [[6/5]], [[25/21]], ''[[13/11]]''
| {{UDnote|step=8}}
|-
| 9
| 348.4
| Neutral third
| [[11/9]], [[16/13]]
| {{UDnote|step=9}}
|-
| 10
| 387.1
| Major third
| [[5/4]]
| {{UDnote|step=10}}
|-
| 11
| 425.8
| Supermajor third
| [[9/7]], [[14/11]], [[23/18]], [[32/25]]
| {{UDnote|step=11}}
|-
| 12
| 464.5
| Subfourth
| [[13/10]], [[17/13]], [[21/16]]
| {{UDnote|step=12}}
|-
| 13
| 503.2
| Perfect fourth
| [[4/3]]
| {{UDnote|step=13}}
|-
| 14
| 541.9
| Superfourth
| [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'', [[48/35]]
| {{UDnote|step=14}}
|-
| 15
| 580.6
| Augmented fourth
| [[7/5]], [[25/18]], [[45/32]]
| {{UDnote|step=15}}
|-
| 16
| 619.4
| Diminished fifth
| [[10/7]], [[36/25]], [[64/45]]
| {{UDnote|step=16}}
|-
| 17
| 658.1
| Subfifth
| [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'', [[35/24]]
| {{UDnote|step=17}}
|-
| 18
| 696.8
| Perfect fifth
| [[3/2]]
| {{UDnote|step=18}}
|-
| 19
| 735.5
| Superfifth
| [[20/13]], [[26/17]], [[32/21]]
| {{UDnote|step=19}}
|-
| 20
| 774.2
| Subminor sixth
| [[11/7]], [[14/9]], [[25/16]]
| {{UDnote|step=20}}
|-
| 21
| 812.9
| Minor sixth
| [[8/5]]
| {{UDnote|step=21}}
|-
| 22
| 851.6
| Neutral sixth
| [[13/8]], [[18/11]]
| {{UDnote|step=22}}
|-
| 23
| 890.3
| Major sixth
| [[5/3]], [[42/25]], ''[[22/13]]''
| {{UDnote|step=23}}
|-
| 24
| 929.0
| Supermajor sixth
| [[12/7]]
| {{UDnote|step=24}}
|-
| 25
| 967.7
| Subminor seventh
| [[7/4]]
| {{UDnote|step=25}}
|-
| 26
| 1006.5
| Minor seventh
| [[9/5]], [[16/9]], [[25/14]], [[34/19]]
| {{UDnote|step=26}}
|-
| 27
| 1045.2
| Neutral seventh
| [[11/6]], [[20/11]], [[24/13]], [[64/35]]
| {{UDnote|step=27}}
|-
| 28
| 1083.9
| Major seventh
| [[13/7]], [[15/8]], [[28/15]]
| {{UDnote|step=28}}
|-
| 29
| 1122.6
| Supermajor seventh
| [[21/11]], [[27/14]], [[40/21]], [[44/23]], [[48/25]]
| {{UDnote|step=29}}
|-
| 30
| 1161.3
| Sub-octave
| [[35/18]], [[49/25]], [[63/32]], [[88/45]], [[96/49]], [[125/64]]
| {{UDnote|step=30}}
|-
| 31
| 1200.0
| Octave
| [[2/1]]
| {{UDnote|step=31}}
|}
<references group="note" />
 
=== Proposed interval names and solfeges ===
{{See also|31edo solfege}}
 
{| class="wikitable center-all right-2 left-4 left-7 left-10 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
|-
! #
! Cents
! Cents
! Approximate ratios<ref group="note">{{sg|limit=23-limit}} Inconsistent intervals are in ''italics''.</ref>
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vd2)
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vd2)
! colspan="3" | Extended pythagorean notation
! colspan="3" | Extended pythagorean notation
! colspan="3" | [[SKULO interval names|SKULO notation]] (S or {{nowrap|U {{=}} 1}})
! colspan="3" | [[SKULO interval names|SKULO notation]]<br>(S or {{nowrap|U {{=}} 1}})
|-
|-
| 0
| 0
| 0.0
| 0.0
| 1/1
| P1
| P1
| perfect unison
| perfect unison
Line 59: Line 280:
| 1
| 1
| 38.7
| 38.7
| 45/44, 49/48, 46/45, 128/125, 36/35
| ^1, d2
| ^1, d2
| up-unison, dim 2nd
| up-unison, dim 2nd
Line 72: Line 292:
| 2
| 2
| 77.4
| 77.4
| 25/24, 21/20, 22/21, 23/22
| A1, vm2
| A1, vm2
| aug 1sn, downminor 2nd
| aug 1sn, downminor 2nd
Line 85: Line 304:
| 3
| 3
| 116.1
| 116.1
| 15/14, 16/15
| m2
| m2
| minor 2nd
| minor 2nd
Line 98: Line 316:
| 4
| 4
| 154.8
| 154.8
| 12/11, 11/10, 35/32
| ~2
| ~2
| mid 2nd
| mid 2nd
Line 111: Line 328:
| 5
| 5
| 193.5
| 193.5
| 9/8, 10/9, 19/17, 28/25
| M2
| M2
| major 2nd
| major 2nd
Line 124: Line 340:
| 6
| 6
| 232.3
| 232.3
| 8/7, 144/125
| ^M2
| ^M2
| upmajor 2nd
| upmajor 2nd
Line 137: Line 352:
| 7
| 7
| 271.0
| 271.0
| 7/6, 75/64
| vm3
| vm3
| downminor 3rd
| downminor 3rd
Line 150: Line 364:
| 8
| 8
| 309.7
| 309.7
| 6/5, ''13/11'', 25/21
| m3
| m3
| minor 3rd
| minor 3rd
Line 163: Line 376:
| 9
| 9
| 348.4
| 348.4
| 11/9, 27/22, 16/13, 60/49, 49/40
| ~3
| ~3
| mid 3rd
| mid 3rd
Line 176: Line 388:
| 10
| 10
| 387.1
| 387.1
| 5/4
| M3
| M3
| major 3rd
| major 3rd
Line 189: Line 400:
| 11
| 11
| 425.8
| 425.8
| 9/7, 14/11, 23/18, 32/25
| ^M3
| ^M3
| upmajor 3rd
| upmajor 3rd
Line 202: Line 412:
| 12
| 12
| 464.5
| 464.5
| 21/16, 64/49, 13/10, 17/13, 125/96
| v4
| v4
| down-4th
| down-4th
Line 215: Line 424:
| 13
| 13
| 503.2
| 503.2
| 4/3
| P4
| P4
| perfect 4th
| perfect 4th
Line 228: Line 436:
| 14
| 14
| 541.9
| 541.9
| 175/128, 11/8, 15/11, ''18/13'', 26/19
| ^4, ~4
| ^4, ~4
| up-4th, mid 4th
| up-4th, mid 4th
Line 241: Line 448:
| 15
| 15
| 580.6
| 580.6
| 7/5, 45/32, 25/18
| A4, vd5
| A4, vd5
| aug 4th, downdim 5th
| aug 4th, downdim 5th
Line 254: Line 460:
| 16
| 16
| 619.4
| 619.4
| 10/7, 64/45, 36/25
| ^A4, d5
| ^A4, d5
| upaug 4th, dim 5th
| upaug 4th, dim 5th
Line 267: Line 472:
| 17
| 17
| 658.1
| 658.1
| 256/175, ''13/9'', 16/11, 22/15, 19/13
| v5, ~5
| v5, ~5
| down-5th, mid 5th
| down-5th, mid 5th
Line 280: Line 484:
| 18
| 18
| 696.8
| 696.8
| 3/2
| P5
| P5
| perfect 5th
| perfect 5th
Line 293: Line 496:
| 19
| 19
| 735.5
| 735.5
| 32/21, 49/32, 20/13, 26/17, 192/125
| ^5
| ^5
| up-5th
| up-5th
Line 306: Line 508:
| 20
| 20
| 774.2
| 774.2
| 14/9, 11/7, 25/16
| vm6
| vm6
| downminor 6th
| downminor 6th
Line 319: Line 520:
| 21
| 21
| 812.9
| 812.9
| 8/5
| m6
| m6
| minor 6th
| minor 6th
Line 332: Line 532:
| 22
| 22
| 851.6
| 851.6
| 18/11, 44/27, 13/8, 49/30, 80/49
| ~6
| ~6
| mid 6th
| mid 6th
Line 345: Line 544:
| 23
| 23
| 890.3
| 890.3
| 5/3, 42/25
| M6
| M6
| major 6th
| major 6th
Line 358: Line 556:
| 24
| 24
| 929.0
| 929.0
| 12/7, 128/75
| ^M6
| ^M6
| upmajor 6th
| upmajor 6th
Line 371: Line 568:
| 25
| 25
| 967.7
| 967.7
| 7/4, 125/72
| vm7
| vm7
| downminor 7th
| downminor 7th
Line 384: Line 580:
| 26
| 26
| 1006.5
| 1006.5
| 16/9, 9/5, 34/19, 25/14
| m7
| m7
| minor 7th
| minor 7th
Line 397: Line 592:
| 27
| 27
| 1045.2
| 1045.2
| 11/6, 20/11, 64/35
| ~7
| ~7
| mid 7th
| mid 7th
Line 410: Line 604:
| 28
| 28
| 1083.9
| 1083.9
| 28/15, 15/8
| M7
| M7
| major 7th
| major 7th
Line 423: Line 616:
| 29
| 29
| 1122.6
| 1122.6
| 48/25, 40/21, 21/11, 44/23
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
Line 436: Line 628:
| 30
| 30
| 1161.3
| 1161.3
| 88/45, 96/49, 45/23, 125/64, 35/18
| v8
| v8
| down-8ve
| down-8ve
Line 449: Line 640:
| 31
| 31
| 1200.0
| 1200.0
| 2/1
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 561: Line 751:
=== Ups and downs notation ===
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp. The gamut runs D, ^D/vD#, D#, Eb, ^Eb/vE, E, ^E, vF, F etc.
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp. The gamut runs D, ^D/vD#, D#, Eb, ^Eb/vE, E, ^E, vF, F etc.
{{Ups and downs sharpness|31}}
{{Ups and downs sharpness}}


=== Neutral chain-of-fifths notation ===
=== Neutral chain-of-fifths notation ===
Line 667: Line 857:


==== Evo flavor ====
==== Evo flavor ====
<imagemap>
{{Sagittal chart|Evo}}
File:31-EDO_Evo_Sagittal.svg
 
desc none
==== Evo-SZ flavor ====
rect 80 0 300 50 [[Sagittal_notation]]
{{Sagittal chart|Evo-SZ}}
rect 567 0 727 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
 
rect 20 80 130 106 [[33/32]]
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.
default [[File:31-EDO_Evo_Sagittal.svg]]
</imagemap>


==== Revo flavor ====
==== Revo flavor ====
<imagemap>
{{Sagittal chart}}
File:31-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 599 0 759 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:31-EDO_Revo_Sagittal.svg]]
</imagemap>


We also have a diagram from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], which gives multiple spellings for each pitch, and up to the double-apotome:
We also have a diagram from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], which gives multiple spellings for each pitch, and up to the double-apotome:


[[File:31edo Sagittal.png|800px]]
[[File:31edo Sagittal.png|800px]]
==== Evo-SZ flavor ====
<imagemap>
File:31-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 519 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:31-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.


== Relationship to 12edo ==
== Relationship to 12edo ==
Line 774: Line 943:
| 3.584
| 3.584
|}
|}
* 31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next equal temperaments doing better in those subgroups are 72, 72, 41, and 46, respectively.
* 31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next equal temperaments doing better in those subgroups are [[72edo|72]], 72, [[41edo|41]], and [[46edo|46]], respectively.
* 31et 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]].  
* 31et 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 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.
* 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.


=== Uniform maps ===
=== Uniform maps ===
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| 0.42
| 0.42
| Sathurugu
| Sathurugu
| Schismina
| Minisma
|}
|}
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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| (P8, ccP4/5)
| (P8, ccP4/5)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Octave stretch or compression ==
== Octave stretch or compression ==
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* 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 [[Quince clan#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<sup>3</sup> and the error from the meantone fifths accumulates.
* 27 is quite flat, as it's 3<sup>3</sup> and the error from the meantone fifths accumulates.
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=== Various subsets ===
=== Various subsets ===
A large open list of subsets from 31edo that people have named:
; Lists of scales
* [[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 [[Erose–McClain double mode]]s, which are [[nonoctave]]
* the [[Erose–McClain double mode]]s, which are [[nonoctave]]
; Individual 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)
* the [[moon dust]] scale{{idio}} (technically [[JI]] but representable in 31)


== Instruments ==
== Instruments ==
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=== Other Instruments ===
=== Other Instruments ===
[[File:31edo array kalimba.jpg|none|thumb|640x640px|31edo array kalimba built by Tristan Bay; 3 octaves, 94 keys, and laid out in circle-of-fourths meantone tuning]]
[[File:31edo array kalimba.jpg|none|thumb|640x640px|31edo array kalimba built by [[Tristan Bay]]; 3 octaves, 94 keys, and laid out in circle-of-fourths meantone tuning]]


=== Lumatone ===
=== Lumatone ===
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=== Skip fretting ===
=== Skip fretting ===
'''[[Skip fretting system 31 2 9]]''' is a [[skip fretting]] system for 31edo.
'''[[Skip fretting system 31 2 9]]''' is a [[skip fretting]] system for 31edo.


'''[[Skip fretting system 31 3 7]]''' is another skip fretting system for 31edo.
'''[[Skip fretting system 31 3 7]]''' is another skip fretting system for 31edo.


'''Skip fretting system 31 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]].
'''Skip fretting system 31 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]].
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* [[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.
* [[CG-31]]
* [[CG-31]]
== Notes ==
<references group="note" />


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