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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 16: Line 17:
Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents. However, intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.
Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents. However, intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.


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 [[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 ===
{{Harmonics in equal|31|columns=9}}
{{Harmonics in equal|31|columns=11}}
{{Harmonics in equal|31|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 31edo (continued)}}
{{Harmonics in equal|31|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 31edo (continued)}}


=== 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]].
 
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.


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.  
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}}
|-
| 7
| 271.0
| Subminor third
| [[7/6]]
| {{UDnote|step=7}}
|-
|-
! Degree
| 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.
{{Sharpness-sharp2a}}
{{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 710: Line 879:


[[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&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)


== Approximation to JI ==
== Approximation to JI ==
Line 926: 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 ===
Line 1,216: Line 1,233:
| 0.42
| 0.42
| Sathurugu
| Sathurugu
| Schismina
| Minisma
|}
|}
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 1,325: Line 1,343:
| (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 ==
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.
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.
Good options include:
* [[zpi|127zpi]]: Good [[13-limit]] option
* [[80ed6]]: Great 11-limit option but bad harmonic 13
* [[49edt]]: Good 13-limit option for the opposite mapping of 13


; 31edo
== Scales ==
* Step size: 38.710{{c}}, octave size: 1200.0{{c}}  
* [[Meantone5]]
Pure-octaves 31edo approximates all harmonics up to 16 within NNN{{c}}.
* [[Meantone7]]
{{Harmonics in equal|31|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo}}
* [[Meantone12]]
{{Harmonics in equal|31|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo (continued)}}
 
=== 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
|}


; [[WE|31et, 13-limit WE tuning]]
In mode 16, the most closely-matched harmonics are the composite ones, 21 and 25. Of the other harmonics:
* Step size: 38.725{{c}}, octave size: 1200.5{{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}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|38.725|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 13-limit WE tuning}}
{{Harmonics in cet|38.725|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)}}


; [[zpi|127zpi]]
* 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.
* Step size: 38.737{{c}}, octave size: 1200.8{{c}}
* 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]]).
Stretching the octave of 31edo by slightly less than 1{{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.
* 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.
{{Harmonics in cet|38.737|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}}
* 27 is quite flat, as it's 3<sup>3</sup> and the error from the meantone fifths accumulates.
{{Harmonics in cet|38.737|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}}
* 29 and 31 are both almost critically sharp, and intervals involving them are unlikely to play any major role.


; [[WE|31et, 11-limit WE tuning]]
{| class="wikitable"
* Step size: 38.748{{c}}, octave size: 1201.2{{c}}
|-
Stretching the octave of 31edo by slightly more than 1{{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]].
! Odd overtones in "Mode 16":
{{Harmonics in cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}}
| 17
{{Harmonics in cet|38.748|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning (continued)}}
| 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
|}


; [[80ed6]]  
=== Various subsets ===
* Step size: 38.774{{c}}, octave size: 1202.0{{c}}
; Lists of scales
Stretching the octave of 31edo by about 2{{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.
* [[31edo modes]]
{{Harmonics in equal|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}}
* [[Strictly proper]] [[Strictly proper 7-tone 31edo scales|7-tone 31edo scales]]
{{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued)}}
* Interesting (to somebody) [[9-tone 31edo scales]]
* the [[Erose–McClain double mode]]s, which are [[nonoctave]]


; [[49edt]]  
; Individual scales
* Step size: 38.815{{c}}, octave size: 1203.3{{c}}
* the [[Euler–Fokker genus]] (technically [[JI]] but representable in 31)
Stretching the octave of 31edo by about 3.5{{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.
* the [[altered pentad]]
{{Harmonics in equal|49|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt}}
* [[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)
{{Harmonics in equal|49|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49edt (continued)}}
* the [[moon dust]] scale{{idio}} (technically [[JI]] but representable in 31)


== Instruments ==
== Instruments ==
Line 1,378: Line 1,518:


=== 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 ===
Line 1,385: Line 1,525:
=== 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]].
Line 1,427: Line 1,565:
* [[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"