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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}}
{{EDO intro|31}}
{{ED intro}}


31edo is also referred to as the ''tricesimoprimal meantone temperament''. The term ''tricesimoprimal'' was first used by [[Adriaan Fokker]].
31edo is also referred to as the ''tricesimoprimal meantone temperament''. The term ''tricesimoprimal'' was first used by [[Adriaan Fokker]].
{{Wikipedia| 31 equal temperament }}
{{Wikipedia| 31 equal temperament }}


== Basic theory ==
== Theory ==
{{Harmonics in equal|31|prec=2||columns=16}}
31edo's [[3/2|perfect fifth]] is flat of just by 5.2{{c}}, as befits a tuning of [[meantone]]. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31edo's approximation of [[7/4]], a cent flat, is also very close to just. Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).
 
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 [[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.
 
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 ===
{{Harmonics in equal|31|columns=11}}
{{Harmonics in equal|31|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 31edo (continued)}}
 
=== 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. 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]].


31edo's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning [[support|supporting]] [[meantone]], but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of 7/4, a cent flat, is also very close to just. It is 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. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of 13/9, 13/11, 18/13, and 22/13.
[[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]].


Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31edo is relatively quite accurate and is [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap EDO, a zeta peak EDO, and a zeta peak integer EDO]], meaning it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|strict zeta EDO]]. Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the 7-, 9-, and [[11-odd-limit]], which it is [[consistent]] through, and that it is the first [[Trivial temperament|non-trivial]] EDO to be consistent in the 11-[[Odd prime sum limit|odd-prime-sum-limit]].
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.


One step of 31edo, measuring about 38.7¢, 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.  
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 is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L 3s) [[mos]] accurate 13:17:19 chords.
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 is the 11th [[prime numbers|prime]] EDO, following [[29edo]] and coming before [[37edo]].
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 ===
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 ==
{{Main|31edo/Individual degrees}}
{{See also|Table of 31edo intervals|31edo/Individual degrees}}
{{See also|Table of 31edo intervals}}
 
{| 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}}
|-
| 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}}
{{See also|31edo solfege}}


{| class="wikitable center-all right-2 left-3"
{| 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
|-
|-
! Degree
! #
! Cents
! Cents
! Approximate Ratios
! 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]]
! 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]]<br>(S or {{nowrap|U {{=}} 1}})
|-
|-
| 0
| 0
| 0.00
| 0.0
| 1/1
| P1
| P1
| perfect unison
| perfect unison
Line 52: Line 279:
|-
|-
| 1
| 1
| 38.71
| 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 65: Line 291:
|-
|-
| 2
| 2
| 77.42
| 77.4
| 25/24, 21/20, 22/21, 23/22
| A1, vm2
| A1, vm2
| aug 1sn, downminor 2nd
| aug 1sn, downminor 2nd
Line 78: Line 303:
|-
|-
| 3
| 3
| 116.13
| 116.1
| 15/14, 16/15
| m2
| m2
| minor 2nd
| minor 2nd
Line 91: Line 315:
|-
|-
| 4
| 4
| 154.84
| 154.8
| 12/11, 11/10, 35/32
| ~2
| ~2
| mid 2nd
| mid 2nd
Line 104: Line 327:
|-
|-
| 5
| 5
| 193.55
| 193.5
| 9/8, 10/9, 19/17, 28/25
| M2
| M2
| major 2nd
| major 2nd
Line 117: Line 339:
|-
|-
| 6
| 6
| 232.26
| 232.3
| 8/7, 144/125
| ^M2
| ^M2
| upmajor 2nd
| upmajor 2nd
Line 130: Line 351:
|-
|-
| 7
| 7
| 270.97
| 271.0
| 7/6, 75/64
| vm3
| vm3
| downminor 3rd
| downminor 3rd
Line 143: Line 363:
|-
|-
| 8
| 8
| 309.68
| 309.7
| 6/5, 25/21
| m3
| m3
| minor 3rd
| minor 3rd
Line 156: Line 375:
|-
|-
| 9
| 9
| 348.39
| 348.4
| 11/9, 27/22, 16/13, 60/49, 49/40
| ~3
| ~3
| mid 3rd
| mid 3rd
Line 169: Line 387:
|-
|-
| 10
| 10
| 387.10
| 387.1
| 5/4
| M3
| M3
| major 3rd
| major 3rd
Line 182: Line 399:
|-
|-
| 11
| 11
| 425.81
| 425.8
| 9/7, 14/11, 23/18, 32/25
| ^M3
| ^M3
| upmajor 3rd
| upmajor 3rd
Line 195: Line 411:
|-
|-
| 12
| 12
| 464.52
| 464.5
| 21/16, 64/49, 13/10, 17/13, 125/96
| v4
| v4
| down-4th
| down-4th
Line 208: Line 423:
|-
|-
| 13
| 13
| 503.23
| 503.2
| 4/3
| P4
| P4
| perfect 4th
| perfect 4th
Line 221: Line 435:
|-
|-
| 14
| 14
| 541.94
| 541.9
| 175/128, 11/8, 15/11, 26/19
| ^4, ~4
| ^4, ~4
| up-4th, mid 4th
| up-4th, mid 4th
Line 234: Line 447:
|-
|-
| 15
| 15
| 580.65
| 580.6
| 7/5, 45/32, 25/18
| A4, vd5
| A4, vd5
| aug 4th, downdim 5th
| aug 4th, downdim 5th
Line 247: Line 459:
|-
|-
| 16
| 16
| 619.35
| 619.4
| 10/7, 64/45, 36/25
| ^A4, d5
| ^A4, d5
| upaug 4th, dim 5th
| upaug 4th, dim 5th
Line 260: Line 471:
|-
|-
| 17
| 17
| 658.06
| 658.1
| 256/175, 16/11, 22/15, 19/13
| v5, ~5
| v5, ~5
| down-5th, mid 5th
| down-5th, mid 5th
Line 273: Line 483:
|-
|-
| 18
| 18
| 696.77
| 696.8
| 3/2
| P5
| P5
| perfect 5th
| perfect 5th
Line 286: Line 495:
|-
|-
| 19
| 19
| 735.48
| 735.5
| 32/21, 49/32, 20/13, 26/17, 192/125
| ^5
| ^5
| up-5th
| up-5th
Line 299: Line 507:
|-
|-
| 20
| 20
| 774.19
| 774.2
| 14/9, 11/7, 25/16
| vm6
| vm6
| downminor 6th
| downminor 6th
Line 312: Line 519:
|-
|-
| 21
| 21
| 812.90
| 812.9
| 8/5
| m6
| m6
| minor 6th
| minor 6th
Line 325: Line 531:
|-
|-
| 22
| 22
| 851.61
| 851.6
| 18/11, 44/27, 13/8, 49/30, 80/49
| ~6
| ~6
| mid 6th
| mid 6th
Line 338: Line 543:
|-
|-
| 23
| 23
| 890.32
| 890.3
| 5/3, 42/25
| M6
| M6
| major 6th
| major 6th
Line 351: Line 555:
|-
|-
| 24
| 24
| 929.03
| 929.0
| 12/7, 128/75
| ^M6
| ^M6
| upmajor 6th
| upmajor 6th
Line 364: Line 567:
|-
|-
| 25
| 25
| 967.74
| 967.7
| 7/4, 125/72
| vm7
| vm7
| downminor 7th
| downminor 7th
Line 377: Line 579:
|-
|-
| 26
| 26
| 1006.45
| 1006.5
| 16/9, 9/5, 34/19, 25/14
| m7
| m7
| minor 7th
| minor 7th
Line 390: Line 591:
|-
|-
| 27
| 27
| 1045.16
| 1045.2
| 11/6, 20/11, 64/35
| ~7
| ~7
| mid 7th
| mid 7th
Line 403: Line 603:
|-
|-
| 28
| 28
| 1083.87
| 1083.9
| 28/15, 15/8
| M7
| M7
| major 7th
| major 7th
Line 416: Line 615:
|-
|-
| 29
| 29
| 1122.58
| 1122.6
| 48/25, 40/21, 21/11, 44/23
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
Line 429: Line 627:
|-
|-
| 30
| 30
| 1161.29
| 1161.3
| 88/45, 96/49, 45/23, 125/64, 35/18
| v8
| v8
| down-8ve
| down-8ve
Line 442: Line 639:
|-
|-
| 31
| 31
| 1200.00
| 1200.0
| 2/1
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 456: Line 652:


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
Combining [[ups and downs notation]] with [[color notation]], qualities can be loosely associated with colors:


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 462: Line 658:
! Quality
! Quality
! [[Color name]]
! [[Color name]]
! Monzo Format
! Monzo format
! Examples
! Examples
|-
|-
| downminor
| downminor
| zo
| zo
| {{monzo|a b 0 1}}
| {{monzo| a b 0 1 }}
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| rowspan="2" | minor
| rowspan="2" | minor
| fourthward wa
| fourthward wa
| {{monzo|a b}} where b &lt; -1
| {{monzo| a b }} where {{nowrap| b > −1 }}
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| gu
| gu
| {{monzo|a b -1}}
| {{monzo| a b -1 }}
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| rowspan="2" | mid
| rowspan="2" | mid
| ilo
| ilo
| {{monzo|a b 0 0 1}}
| {{monzo| a b 0 0 1 }}
| 11/9, 11/6
| 11/9, 11/6
|-
|-
| lu
| lu
| {{monzo|a b 0 0 -1}}
| {{monzo| a b 0 0 -1 }}
| 12/11, 18/11
| 12/11, 18/11
|-
|-
| rowspan="2" | major
| rowspan="2" | major
| yo
| yo
| {{monzo|a b 1}}
| {{monzo| a b 1 }}
| 5/4, 5/3
| 5/4, 5/3
|-
|-
| fifthward wa
| fifthward wa
| {{monzo|a b}} where b &gt; 1
| {{monzo| a b }} where {{nowrap| b > 1 }}
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| upmajor
| upmajor
| ru
| ru
| {{monzo|a b 0 -1}}
| {{monzo| a b 0 -1 }}
| 9/7, 12/7
| 9/7, 12/7
|}
|}
All 31edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:
All 31edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:


Line 507: Line 704:
|-
|-
! [[Color notation|Color of the 3rd]]
! [[Color notation|Color of the 3rd]]
! JI Chord
! JI chord
! Edosteps
! Edosteps
! Notes of C Chord
! Notes of C chord
! Written name
! Written name
! Spoken name
! Spoken name
|-
|-
| zo
| zo (7-over)
| 6:7:9
| 6:7:9
| {{dash|0, 7, 18|s=hair|d=med}}
| {{dash|0, 7, 18|s=hair|d=med}}
| {{dash|C, E{{sesquiflat2}}, G|s=hair|d=med}}
|{{dash|C, vE{{flat}}, G|s=hair|d=med}} or {{dash|C, E{{sesquiflat}}, G|s=hair|d=med}}
| Cvm
| Cvm
| C subminor
| C downminor
|-
|-
| gu
| gu (5-under)
| 10:12:15
| 10:12:15
| {{dash|0, 8, 18|s=hair|d=med}}
| {{dash|0, 8, 18|s=hair|d=med}}
| {{dash|C, E♭, G|s=hair|d=med}}
| {{dash|C, E{{flat}}, G|s=hair|d=med}}
| Cm
| Cm
| C minor
| C minor
|-
|-
| ilo
| ilo (11-over)
| 18:22:27
| 18:22:27
| {{dash|0, 9, 18|s=hair|d=med}}
| {{dash|0, 9, 18|s=hair|d=med}}
| {{dash|C, E{{demiflat2}}, G|s=hair|d=med}}
|{{dash|C, vE, G|s=hair|d=med}} or {{dash|C, E{{demiflat}}, G|s=hair|d=med}}
| C~
| C~
| C neutral
| C mid
|-
|-
| yo
| yo (5-over)
| 4:5:6
| 4:5:6
| {{dash|0, 10, 18|s=hair|d=med}}
| {{dash|0, 10, 18|s=hair|d=med}}
Line 541: Line 738:
| C, C major
| C, C major
|-
|-
| ru
| ru (7-under)
| 14:18:21
| 14:18:21
| {{dash|0, 11, 18|s=hair|d=med}}
| {{dash|0, 11, 18|s=hair|d=med}}
| {{dash|C, E{{demisharp2}}, G|s=hair|d=med}}
|{{dash|C, ^E, G|s=hair|d=med}} or {{dash|C, E{{demisharp}}, G|s=hair|d=med}}
| C^
| C^
| C supermajor
| C up, C upmajor
|}
|}


For a more complete list of chords, see [[31edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].
For a more complete list of chords, see [[31edo Chord Names]] and [[Ups and downs notation #Chords and chord progressions]].


== Notation ==
== Notation ==
{{Sharpness-sharp2}}
=== Ups and downs notation ===
=== Neutral circle-of-fifths 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.
[[Neutral circle-of-fifths notation]] is much like ups and downs notation. Between C and D (do and re) for example, we have the following notes:
{{Ups and downs sharpness}}
 
=== Neutral chain-of-fifths notation ===
[[File:31edo CoF semi and sesqui.png|thumb|500x500px|Circle of fifths in 31edo showing equivalences and quartertone accidentals]]
 
Since a sharp raises by 2 steps, 31edo can be notated using quarter-tone accidentals. Between C and D (do and re) for example, we have the following notes:
 
{| class="wikitable"
{| class="wikitable"
|-
|-
! Degree
! Degree
! Letter
! Letter
! Name
! Solfège
! English full name
! English full name
|-
|-
Line 593: Line 796:
|}
|}


=== Circle-of-fifths notation ===
==== Stein–Zimmermann accidentals ====
[[Circle-of-fifths notation]] uses double sharps and double flats:
{{Sharpness-sharp2}}
 
=== Chain-of-fifths notation ===
[[Chain-of-fifths notation]] uses double sharps and double flats only:
{| class="wikitable"
{| class="wikitable"
|-
|-
! Degree
! Degree
! Letter
! Letter
! Name
! Solfège
! English full name
! English full name
|-
|-
Line 633: Line 839:
|}
|}


While using double sharp and double flat may seem confusing because it alternates between C and D, it provides a way of writing chords that is consistent with traditional notation. For example, the subminor7 chord 12:14:18:21 is written like so:
While using double sharps and double flats may seem confusing because it alternates between C and D, it provides a way of writing chords that is consistent with traditional notation. For example, the subminor7 chord 12:14:18:21 is written like so:
* C / D♯ / G / A♯
* C / D♯ / G / A♯
* C♯ / D𝄪 / G♯ / A𝄪
* C♯ / D𝄪 / G♯ / A𝄪
Line 639: Line 845:
* D / E♯ / A / B♯
* D / E♯ / A / B♯


In 12edo, the enharmonic equivalences include C♯ = D♭, E♯ = F, and E = F♭. But in 31edo we have:
In 12edo, the enharmonic equivalences include {{nowrap|C♯ {{=}} D♭|E♯ {{=}} F}}, and {{nowrap|E {{=}} F♭}}. But in 31edo we have:
* C𝄪 = D{{demiflat2}}
* C𝄪 = D{{demiflat2}}
* D𝄫 = C{{demisharp2}}
* D𝄫 = C{{demisharp2}}
Line 648: Line 854:


=== Sagittal notation ===
=== Sagittal notation ===
The Revo flavor of Sagittal notation from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]]:
This notation uses the same sagittal sequence as edos [[17edo #Sagittal notation|17]], [[24edo #Sagittal notation|24]], and [[38edo #Sagittal notation|38]], and is a subset of the notation for [[62edo #Sagittal notation|62edo]].


[[File:31edo Sagittal.png|800px]]
==== Evo flavor ====
{{Sagittal chart|Evo}}


== Relationship to 12-edo ==
==== Evo-SZ flavor ====
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.
{{Sagittal chart|Evo-SZ}}


[[File:31-edo spiral.png|582x582px]]
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.


31edo can be notated with a seperate semi/sesqui sharp/flat chain (like 17edo), with its own enharmonic circle of fifths.
==== Revo flavor ====
[[File:31edo CoF semi and sesqui.png|none|thumb|500x500px]]
{{Sagittal chart}}


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


=== MOS scales ===
[[File:31edo Sagittal.png|800px]]
{{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:
== Relationship to 12edo ==
* 9\31 neutral third generator generates [[ultrasoft]] [[mosh]] and [[superhard]] [[dicotonic]] MOSes.
31edo’s [[circle of fifths|circle of 31 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. In Kite Giedraitis' theory, this is possible because going up 12 fifths in 31edo yields a difference (the absolute value of the [[Sharpness|dodeca-sharpness]]) of 1 edostep (which also implies that 18\31 is on the 7\12 kite in the [[scale tree]]).  
* 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]].
* 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.
* 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.
This "spiral of fifths" can be a useful construct for introducing 31edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.


=== Harmonic scales ===
The two innermost and two outermost intervals on the spiral are duplicates, reflecting the fact that it is a repeating circle at heart and the spiral shape is only a helpful illusion.
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]].  


The steps are: 5 5 4 4 4 3 3 3.
[[File:31-edo spiral.png|582x582px]]


{| class="wikitable"
== Approximation to JI ==
|-
[[File:31ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 31edo]]
| 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:
=== Interval mappings ===
 
{{Q-odd-limit intervals}}
* 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 3^3 and the error from the meantone fifths accumulates.
* 29 and 31 are both ''very'' sharp, and intervals involving them are unlikely to play any major role.
 
{| class="wikitable"
|-
| Odd overtones in "Mode 16":
| 17
| 19
| 21
| 23
| 25
| 27
| 29
| 31
|-
| …as JI Ratio from 1/1:
| 17/16
| 19/16
| 21/16
| 23/16
| 25/16
| 27/16
| 29/16
| 31/16
|-
| …in cents:
| 105.0
| 297.5
| 470.8
| 628.3
| 772.6
| 905.9
| 1029.6
| 1145.0
|-
| Nearest degree of 31edo:
| 3
| 8
| 12
| 16
| 20
| 23
| 27
| 30
|-
| …in cents:
| 116.1
| 309.7
| 464.5
| 619.4
| 774.2
| 890.3
| 1045.1
| 1161.3
|}
 
=== Various subsets ===
A large open list of subsets from 31edo that people have named:
* [[31edo modes]]
* [[Strictly proper]] [[Strictly proper 7-tone 31edo scales|7-tone 31edo scales]]
* Interesting (to somebody) [[9-tone 31edo scales]]
* 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)


== JI approximation ==
=== Consistent circles ===
[[File:31ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 31edo]]
31edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L&nbsp;3s) [[mos]] accurate 13:17:19 chords.
=== Interval mappings ===
{{Q-odd-limit intervals|31}}


== 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]]
! 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]] (¢)
Line 829: Line 903:
| 2.3
| 2.3
| {{monzo| -49 31 }}
| {{monzo| -49 31 }}
| [{{val| 31 49 }}]
| {{mapping| 31 49 }}
| +1.63
| +1.637
| 1.64
| 1.637
| 4.22
| 4.228
|-
|-
| 2.3.5
| 2.3.5
| 81/80, 393216/390625
| 81/80, 393216/390625
| [{{val| 31 49 72 }}]
| {{mapping| 31 49 72 }}
| +0.98
| +0.976
| 1.63
| 1.628
| 4.20
| 4.204
|-
|-
| 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 }}]
| {{mapping| 31 49 72 87 }}
| +0.83
| +0.828
| 1.43
| 1.432
| 3.70
| 3.700
|-
|-
| 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 }}]
| {{mapping| 31 49 72 87 107 }}
| +1.21
| +1.205
| 1.49
| 1.487
| 3.84
| 3.841
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 66/65, 81/80, 99/98, 105/104, 121/120
| 66/65, 81/80, 99/98, 105/104, 121/120
| [{{val| 31 49 72 87 107 115 }}]
| {{mapping| 31 49 72 87 107 115 }}
| +0.50
| +0.502
| 2.07
| 2.072
| 5.35
| 5.353
|- style="border-top: double;"
| 2.3.5.7.11.23
| 81/80, 99/98, 126/125, 161/160, 231/230
| {{mapping| 31 49 72 87 107 140 }}
| +1.333
| 1.387
| 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 [[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]].
* 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.


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.
=== Uniform maps ===
 
{{Uniform map|edo=31}}
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.
31et [[tempering out|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 group="note">{{rd}}</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
! [[Color name|Color Name]]
! [[Color name]]
! Name
! Name
|-
|-
Line 891: Line 974:
| 31.567
| 31.567
| Lala-tribiyo
| Lala-tribiyo
| [[Ampersand]]
| [[Ampersand comma]]
|-
|-
| 5
| 5
Line 947: Line 1,030:
| 22.227
| 22.227
| Laquadzo-atrigu
| Laquadzo-atrigu
| Squalentine
| Squalentine comma
|-
|-
| 7
| 7
Line 954: Line 1,037:
| 20.785
| 20.785
| Quadru-ayo
| Quadru-ayo
| Nuwell
| Nuwell comma
|-
|-
| 7
| 7
Line 961: Line 1,044:
| 14.516
| 14.516
| Quinzogu
| Quinzogu
| Trimyna
| Trimyna comma
|-
|-
| 7
| 7
Line 996: Line 1,079:
| 6.9903
| 6.9903
| Quinru-aquadyo
| Quinru-aquadyo
| Mirkwai
| Mirkwai comma
|-
|-
| 7
| 7
Line 1,031: Line 1,114:
| 1.6283
| 1.6283
| Latriru-asepyo
| Latriru-asepyo
| [[Meter comma]]
| [[Metric comma]]
|-
|-
| 7
| 7
Line 1,116: Line 1,199:
| Thuzozogu
| Thuzozogu
| Mynucuma
| Mynucuma
|-
| 13
| [[351/350]]
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.94
| Thorugugu
| Ratwolfsma
|-
| 13
| [[352/351]]
| {{monzo| 5 -3 0 0 1 -1 }}
| 4.93
| Thulo
| Minor minthma
|-
|-
| 13
| 13
Line 1,136: Line 1,233:
| 0.42
| 0.42
| Sathurugu
| Sathurugu
| Schismina
| Minisma
|}
|}
<references/>
<references group="note" />


=== 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]]
 
31edo provides the [[optimal patent val]] for the rank-5 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|huygens/meantone]], squares, [[winston]], [[lupercalia]], and [[nightengale]] in the 13-limit.


{| class="wikitable center-1"
{| class="wikitable center-1"
|+ Rank-2 temperaments by generators
|+ style="font-size: 105%;" | Rank-2 temperaments by generators
|-
|-
! Generator
! Generator*
! Cents
! Cents*
! MOSes
! Mos scales
! Temperaments
! Temperaments
! [[Pergen]]
! [[Pergen]]
Line 1,162: Line 1,261:
| 2\31
| 2\31
| 77.42
| 77.42
| [[1L 14s]], [[15L 1s]]
| [[1L&nbsp;14s]], [[15L&nbsp;1s]]
| [[Valentine]] / [[lupercalia]]
| [[Valentine]] / [[lupercalia]]
| (P8, P5/9)
| (P8, P5/9)
Line 1,168: Line 1,267:
| 3\31
| 3\31
| 116.13
| 116.13
| [[1L 9s]], [[10L 1s]], [[10L 11s]]
| [[1L&nbsp;9s]], [[10L&nbsp;1s]], [[10L&nbsp;11s]]
| [[Mercy]] / [[miracle]]
| [[Mercy]] / [[miracle]]
| (P8, P5/6)
| (P8, P5/6)
Line 1,174: Line 1,273:
| 4\31
| 4\31
| 154.84
| 154.84
| [[1L 6s]], [[7L 1s]], <br>[[8L 7s]], [[8L 15s]]
| [[1L&nbsp;6s]], [[7L&nbsp;1s]], <br>[[8L&nbsp;7s]], [[8L&nbsp;15s]]
| [[Nusecond]] / [[greeley]]
| [[Greeley]] / [[nusecond]]
| (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&nbsp;5s]], [[6L&nbsp;1s]], [[6L&nbsp;7s]], <br>[[6L&nbsp;13s]], [[6L&nbsp;19s]]
| [[Luna]] / [[didacus]] / [[hemithirds]] /  
| [[Luna]] / [[didacus]] / [[hemithirds]] /<br>[[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&nbsp;4s]], [[5L&nbsp;1s]], [[5L&nbsp;6s]], <br>[[5L&nbsp;11s]], [[5L&nbsp;16s]], [[5L&nbsp;21s]]
| [[Mothra]] / [[mosura]]<br>[[Quadrawell]]
| [[Mothra]] / [[mosura]]<br>[[Quadrawell]]
| (P8, P5/3)
| (P8, P5/3)
Line 1,193: Line 1,291:
| 7\31
| 7\31
| 270.97
| 270.97
| [[1L 3s]], [[4L 1s]], [[4L 5s]], <br>[[9L 4s]], [[9L 13s]]
| [[1L&nbsp;3s]], [[4L&nbsp;1s]], [[4L&nbsp;5s]], <br>[[9L&nbsp;4s]], [[9L&nbsp;13s]]
| [[Orson]] / [[orwell]] / [[winston]]
| [[Orson]] / [[orwell]] / [[winston]]
| (P8, P12/7)
| (P8, P12/7)
Line 1,199: Line 1,297:
| 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&nbsp;1s]], [[4L&nbsp;3s]], [[4L&nbsp;7s]], <br>[[4L&nbsp;11s]], [[4L&nbsp;15s]], [[4L&nbsp;19s]], <br>[[4L&nbsp;23s]]
| [[Myna]] / [[triwell]]
| [[Myna]]<br>[[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&nbsp;1s]], [[3L&nbsp;4s]], [[7L&nbsp;3s]], <br>[[7L&nbsp;10s]], [[7L&nbsp;17s]]
| [[Mohaha]] / [[vicentino]] /<br>[[mohajira]] / [[migration]]
| [[Mohaha]] / [[vicentino]] /<br>[[mohajira]] / [[migration]]
| (P8, P5/2)
| (P8, P5/2)
Line 1,211: Line 1,309:
| 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&nbsp;1s]], [[3L&nbsp;4s]], [[3L&nbsp;7s]], <br>[[3L&nbsp;10s]], [[3L&nbsp;13s]], [[3L&nbsp;16s]], <br>[[3L&nbsp;19s]], [[3L&nbsp;22s]], [[3L&nbsp;25s]]
| [[Würschmidt]] / [[worschmidt]]
| [[Würschmidt]] / [[worschmidt]]
| (P8, ccP5/8)
| (P8, ccP5/8)
Line 1,217: Line 1,315:
| 11\31
| 11\31
| 425.81
| 425.81
| [[3L 2s]], [[3L 5s]], [[3L 8s]], <br>[[3L 11s]], [[14L 3s]]
| [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[3L&nbsp;8s]], <br>[[3L&nbsp;11s]], [[14L&nbsp;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&nbsp;2s]], [[5L&nbsp;3s]], <br>[[5L&nbsp;8s]], [[13L&nbsp;5s]]
| [[5L 3s/Temperaments#A-Team|A-Team]]<br>[[Semisept]]
| [[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&nbsp;3s]], [[5L&nbsp;2s]], <br>[[7L&nbsp;5s]], [[12L&nbsp;7s]]
| [[Meantone]] / [[meanpop]]
| [[Meantone]] / [[meanpop]]
| (P8, P5)
| (P8, P5)
Line 1,235: Line 1,333:
| 14\31
| 14\31
| 541.94
| 541.94
| [[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[9L 2s]], [[11L 9s]]
| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[9L&nbsp;2s]], [[11L&nbsp;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)
Line 1,241: Line 1,339:
| 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&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]], <br>[[2L&nbsp;15s]], [[2L&nbsp;17s]], [[2L&nbsp;19s]], <br>[[2L&nbsp;21s]], [[2L&nbsp;23s]], [[2L&nbsp;25s]], <br>[[2L&nbsp;27s]]
| [[Tritonic]] / [[tritoni]]
| [[Tritonic]] / [[tritoni]]
| (P8, ccP4/5)
| (P8, ccP4/5)
|}
|}
<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 ==
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.
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
== 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 [[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.
* 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 ===
; Lists of scales
* [[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]]
; Individual scales
* 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)
* the [[moon dust]] scale{{idio}} (technically [[JI]] but representable in 31)
== Instruments ==


==Instruments==
=== Keyboard Instruments ===
=== Keyboard Instruments ===
* [https://www.huygens-fokker.org/instruments/fokkerorgan.html Fokker Organ]
* [https://www.huygens-fokker.org/instruments/fokkerorgan.html Fokker Organ]
Line 1,255: 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 mapping for 31edo]]
 
=== Skip fretting ===
'''[[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 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]].
 
; Prime harmonics
1/1: string 2 open
 
2/1: string 7 fret 3
 
3/2: string 4 fret 4
 
5/4: string 4 open
 
7/4: string 7 open
 
11/8: string 4 fret 2
 
13/8: string 6 fret 1
 
17/16: string 1 fret 4
 
19/16: string 2 fret 4
 
23/16: string 4 fret 3
 
29/16: string 7 fret 1
 
31/16: string 1 fret 2


== Music ==
== Music ==
Line 1,262: Line 1,560:


== See also ==
== See also ==
* [[Lumatone mapping for 31edo]]
* [[List of 31edo chords]]
* [[List of 31edo Chords]]
* [[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.
 
* [[CG-31]]
=== Notes ===
<references group="note" />


== Further reading ==
== Further reading ==
Line 1,296: Line 1,590:
=== 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 ===
Line 1,304: Line 1,598:


[[Category:Golden meantone]]
[[Category:Golden meantone]]
[[Category:Historical]]
[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Oneirotonic]]
[[Category:Orwell]]
[[Category:Semicomma]]
[[Category:Semicomma]]
[[Category:Oneirotonic]]
[[Category:Valentine]]
[[Category:Historical]]
[[Category:Würschmidt]]
[[Category:Würschmidt]]
Retrieved from "https://en.xen.wiki/w/31edo"