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<span style="display: block; text-align: right;">[[:de:31edo|Deutsch]]</span>
{{Interwiki
| en = 31edo
| de = 31-EDO
| es = 31 EDO
| ja = 31平均律
| zh = 31平均律
}}
{{Infobox ET}}
{{ED intro}}


__FORCETOC__
31edo is also referred to as the ''tricesimoprimal meantone temperament''. The term ''tricesimoprimal'' was first used by [[Adriaan Fokker]].
-----
{{Wikipedia| 31 equal temperament }}
''Thirty-one tone equal temperament'', also called ''31-tET'', ''31-EDO'', ''31-et'', or ''tricesimoprimal meantone temperament'', is the scale derived by dividing the octave into 31 [[Equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan_Fokker|Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents|cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit|7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit|11-limit]] fairly well, and is consistent through it, but is the [[Optimal_patent_val|optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. 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.


31edo is the 11th [[prime_numbers|prime]] edo, following [[29edo|29edo]] and coming before [[37edo|37edo]].
== Theory ==
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).  


For more encyclopedic info, see [http://en.wikipedia.org/wiki/31_equal_temperament Wikipedia's article].
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.


=Linear temperaments=
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.
[[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]]
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.


{| class="wikitable"
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]].
 
[[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 ===
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 ==
{{See also|Table of 31edo intervals|31edo/Individual degrees}}
 
{| class="wikitable center-1 right-2"
|-
|-
! | Generator
! #
! | Cents
! Cents
! | Temperaments
! Interval categories
! | [[pergen|Pergen]]
! 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]]
|-
|-
| | 1\31
| 0
| | 38.71
| 0.0
| | [[Slender|Slender]]
| Unison
| | (P8, P4/13)
| [[1/1]]
| {{UDnote|step=0}}
|-
|-
| | 2\31
| 1
| | 77.42
| 38.7
| | [[Valentine|Valentine]]/[[Lupercalia|Lupercalia]]
| Super-unison
| | (P8, P5/9)
| [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[64/63]], [[128/125]]
| {{UDnote|step=1}}
|-
|-
| | 3\31
| 2
| | 116.13
| 77.4
| | [[Miracle|Miracle]]
| Subminor second
| | (P8, P5/6)
| [[21/20]], [[22/21]], [[23/22]], [[25/24]], [[28/27]]
| {{UDnote|step=2}}
|-
|-
| | 4\31
| 3
| | 154.84
| 116.1
| | [[Nusecond|Nusecond]]
| Minor second
| | (P8, P11/11)
| [[14/13]], [[15/14]], [[16/15]]
| {{UDnote|step=3}}
|-
|-
| | 5\31
| 4
| | 193.55
| 154.8
| | [[Luna|Luna]]/[[Hemithirds|Hemithirds]]/[[Hemiwürschmidt|Hemiwürschmidt]]
| Neutral second
| | (P8, WWP4/15)
| [[11/10]], [[12/11]], [[13/12]], [[35/32]]
| {{UDnote|step=4}}
|-
|-
| | 6\31
| 5
| | 232.26
| 193.5
| | [[Mothra|Mothra]]/[[Mosura|Mosura]]
| Major second
| | (P8, P5/3)
| [[9/8]], [[10/9]], [[19/17]], [[28/25]]
| {{UDnote|step=5}}
|-
|-
| | 7\31
| 6
| | 270.97
| 232.3
| | [[Orson|Orson]]/[[Orwell|Orwell]]/[[Semicomma_family#Orwell-Winston|Winston]]
| Supermajor second
| | (P8, P12/7)
| [[8/7]]
| {{UDnote|step=6}}
|-
|-
| | 8\31
| 7
| | 309.68
| 271.0
| | [[Myna|Myna]]
| Subminor third
| | (P8, WWP5/10)
| [[7/6]]
| {{UDnote|step=7}}
|-
|-
| | 9\31
| 8
| | 348.39
| 309.7
| | [[Vicentino|Vicentino]]/[[Mohajira|Mohajira]]/[[Migration|Migration]]
| Minor third
| | (P8, P5/2)
| [[6/5]], [[25/21]], ''[[13/11]]''
| {{UDnote|step=8}}
|-
|-
| | 10\31
| 9
| | 387.10
| 348.4
| | [[Würschmidt|Würschmidt]]/[[Worschmidt|Worschmidt]]
| Neutral third
| | (P8, WWP5/8)
| [[11/9]], [[16/13]]
| {{UDnote|step=9}}
|-
|-
| | 11\31
| 10
| | 425.81
| 387.1
| | [[Squares|Squares]]/[[Sentinel|Sentinel]]
| Major third
| | (P8, P11/4)
| [[5/4]]
| {{UDnote|step=10}}
|-
|-
| | 12\31
| 11
| | 464.52
| 425.8
| | [[Semisept|Semisept]]
| Supermajor third
| | (P8, W<span style="vertical-align: super;">5</span>P4/14)
| [[9/7]], [[14/11]], [[23/18]], [[32/25]]
| {{UDnote|step=11}}
|-
|-
| | 13\31
| 12
| | 503.23
| 464.5
| | [[Meantone|Meantone]]/[[Meanpop|Meanpop]]
| Subfourth
| | (P8, P5)
| [[13/10]], [[17/13]], [[21/16]]
| {{UDnote|step=12}}
|-
|-
| | 14\31
| 13
| | 541.94
| 503.2
| | [[casablanca|Casablanca]]/[[Cypress|Cypress]]/[[Oracle|Oracle]]
| Perfect fourth
| | (P8, W<span style="vertical-align: super;">5</span>P4/12)
| [[4/3]]
| {{UDnote|step=13}}
|-
|-
| | 15\31
| 14
| | 580.65
| 541.9
| | [[Tritonic|Tritonic]]/[[Tritoni|Tritoni]]
| Superfourth
| | (P8, WWP4/5)
| [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'', [[48/35]]
|}
| {{UDnote|step=14}}
 
=Intervals=
 
==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 31edo (ordered by absolute error).
 
===Best direct mapping, even if inconsistent===
 
{| class="wikitable"
|-
|-
! | Interval, complement
| 15
! | Error (abs., in [[cent|cents]])
| 580.6
| Augmented fourth
| [[7/5]], [[25/18]], [[45/32]]
| {{UDnote|step=15}}
|-
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| 16
| style="text-align:center;" | 0.783
| 619.4
| Diminished fifth
| [[10/7]], [[36/25]], [[64/45]]
| {{UDnote|step=16}}
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| 17
| style="text-align:center;" | 0.979
| 658.1
| Subfifth
| [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'', [[35/24]]
| {{UDnote|step=17}}
|-
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| 18
| style="text-align:center;" | 1.084
| 696.8
| Perfect fifth
| [[3/2]]
| {{UDnote|step=18}}
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| 19
| style="text-align:center;" | 1.867
| 735.5
| Superfifth
| [[20/13]], [[26/17]], [[32/21]]
| {{UDnote|step=19}}
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| 20
| style="text-align:center;" | 3.314
| 774.2
| Subminor sixth
| [[11/7]], [[14/9]], [[25/16]]
| {{UDnote|step=20}}
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| 21
| style="text-align:center;" | 4.097
| 812.9
| Minor sixth
| [[8/5]]
| {{UDnote|step=21}}
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| 22
| style="text-align:center;" | 4.202
| 851.6
| Neutral sixth
| [[13/8]], [[18/11]]
| {{UDnote|step=22}}
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| 23
| style="text-align:center;" | 4.398
| 890.3
| Major sixth
| [[5/3]], [[42/25]], ''[[22/13]]''
| {{UDnote|step=23}}
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| 24
| style="text-align:center;" | 4.985
| 929.0
| Supermajor sixth
| [[12/7]]
| {{UDnote|step=24}}
|-
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| 25
| style="text-align:center;" | 5.181
| 967.7
| Subminor seventh
| [[7/4]]
| {{UDnote|step=25}}
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| 26
| style="text-align:center;" | 5.964
| 1006.5
| Minor seventh
| [[9/5]], [[16/9]], [[25/14]], [[34/19]]
| {{UDnote|step=26}}
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| 27
| style="text-align:center;" | 8.298
| 1045.2
| Neutral seventh
| [[11/6]], [[20/11]], [[24/13]], [[64/35]]
| {{UDnote|step=27}}
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| 28
| style="text-align:center;" | 9.278
| 1083.9
| Major seventh
| [[13/7]], [[15/8]], [[28/15]]
| {{UDnote|step=28}}
|-
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| 29
| style="text-align:center;" | 9.382
| 1122.6
| Supermajor seventh
| [[21/11]], [[27/14]], [[40/21]], [[44/23]], [[48/25]]
| {{UDnote|step=29}}
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| 30
| style="text-align:center;" | 10.166
| 1161.3
| Sub-octave
| [[35/18]], [[49/25]], [[63/32]], [[88/45]], [[96/49]], [[125/64]]
| {{UDnote|step=30}}
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| 31
| style="text-align:center;" | 10.302
| 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
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
! #
| style="text-align:center;" | 10.362
! Cents
! 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" | Extended pythagorean notation
! colspan="3" | [[SKULO interval names|SKULO notation]]<br>(S or {{nowrap|U {{=}} 1}})
|-
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| 0
| style="text-align:center;" | 11.085
| 0.0
| P1
| perfect unison
| D
| P1
| perfect unison
| D
| P1
| perfect unison
| D
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| 1
| style="text-align:center;" | 11.145
| 38.7
| ^1, d2
| up-unison, dim 2nd
| ^D, Ebb
| d2
| dim 2nd
| Ebb
| S1/U1
| super/uber unison
| SD/UD
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| 2
| style="text-align:center;" | 12.169
| 77.4
| A1, vm2
| aug 1sn, downminor 2nd
| D#, vEb
| A1
| aug 1sn
| D#
| sm2
| subminor 2nd
| sEb
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| 3
| style="text-align:center;" | 15.483
| 116.1
| m2
| minor 2nd
| Eb
| m2
| minor 2nd
| Eb
| m2
| minor 2nd
| Eb
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| 4
| style="text-align:center;" | 16.266
| 154.8
| ~2
| mid 2nd
| vE
| AA1, dd3
| double-aug 1sn, double-dim 3rd
| Dx, Fbb
| N2
| neutral 2nd
| UEb/uE
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| 5
| style="text-align:center;" | 17.263
| 193.5
| M2
| major 2nd
| E
| M2
| major 2nd
| E
| M2
| major 2nd
| E
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| 6
| style="text-align:center;" | 18.242
| 232.3
|}
| ^M2
 
| upmajor 2nd
===Patent val mapping===
| ^E
 
| d3
{| class="wikitable"
| dim 3rd
| Fb
| SM2
| supermajor 2nd
| SE
|-
|-
! | Interval, complement
| 7
! | Error (abs., in [[cent|cents]])
| 271.0
| vm3
| downminor 3rd
| vF
| A2
| aug 2nd
| E#
| sm3
| subminor 3rd
| sF
|-
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| 8
| style="text-align:center;" | 0.783
| 309.7
| m3
| minor 3rd
| F
| m3
| minor 3rd
| F
| m3
| minor 3rd
| F
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| 9
| style="text-align:center;" | 0.979
| 348.4
| ~3
| mid 3rd
| ^F
| AA2, dd4
| double-aug 2nd, double-dim 4th
| Ex, Gbb
| N3
| neutral 3rd
| UF/uF#
|-
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| 10
| style="text-align:center;" | 1.084
| 387.1
| M3
| major 3rd
| F#
| M3
| major 3rd
| F#
| M3
| major 3rd
| F#
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| 11
| style="text-align:center;" | 1.867
| 425.8
| ^M3
| upmajor 3rd
| ^F#
| d4
| dim 4th
| Gb
| SM3
| supermajor 3rd
| SF#
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| 12
| style="text-align:center;" | 3.314
| 464.5
| v4
| down-4th
| vG
| A3
| aug 3rd
| Fx
| s4
| sub 4th
| sG
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| 13
| style="text-align:center;" | 4.097
| 503.2
| P4
| perfect 4th
| G
| P4
| perfect 4th
| G
| P4
| perfect 4th
| G
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| 14
| style="text-align:center;" | 4.202
| 541.9
| ^4, ~4
| up-4th, mid 4th
| ^G
| AA3, dd5
| double-aug 3rd, double-dim 5th
| Fx#, Abb
| U4/N4
| uber/neutral 4th
| UG
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| 15
| style="text-align:center;" | 4.398
| 580.6
| A4, vd5
| aug 4th, downdim 5th
| G#, vAb
| A4
| aug 4th
| G#
| A4
| aug 4th
| G#
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| 16
| style="text-align:center;" | 4.985
| 619.4
| ^A4, d5
| upaug 4th, dim 5th
| ^G#, Ab
| d5
| dim 5th
| Ab
| d5
| dim 5th
| Ab
|-
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| 17
| style="text-align:center;" | 5.181
| 658.1
| v5, ~5
| down-5th, mid 5th
| vA
| AA4, dd6
| double-aug 4th, double-dim 6th
| Gx, Bbbb
| u5/N5
| unter/neutral 5th
| uA
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| 18
| style="text-align:center;" | 5.964
| 696.8
| P5
| perfect 5th
| A
| P5
| perfect 5th
| A
| P5
| perfect 5th
| A
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| 19
| style="text-align:center;" | 8.298
| 735.5
| ^5
| up-5th
| ^A
| d6
| dim 6th
| Bbb
| S5
| super 5th
| SA
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| 20
| style="text-align:center;" | 9.278
| 774.2
| vm6
| downminor 6th
| vBb
| A5
| aug 5th
| A#
| sm6
| subminor 6th
| sBb
|-
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| 21
| style="text-align:center;" | 9.382
| 812.9
| m6
| minor 6th
| Bb
| m6
| minor 6th
| Bb
| m6
| minor 6th
| Bb
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| 22
| style="text-align:center;" | 10.166
| 851.6
| ~6
| mid 6th
| vB
| AA5, dd7
| double-aug 5th, double-dim 7th
| Ax, Cbb
| N6
| neutral 6th
| UBb/uB
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| 23
| style="text-align:center;" | 10.302
| 890.3
| M6
| major 6th
| B
| M6
| major 6th
| B
| M6
| major 6th
| B
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| 24
| style="text-align:center;" | 10.362
| 929.0
| ^M6
| upmajor 6th
| ^B
| d7
| dim 7th
| Cb
| SM6
| supermajor 6th
| SB
|-
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| 25
| style="text-align:center;" | 11.085
| 967.7
| vm7
| downminor 7th
| vC
| A6
| aug 6th
| B#
| sm7
| subminor 7th
| sC
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| 26
| style="text-align:center;" | 11.145
| 1006.5
| m7
| minor 7th
| C
| m7
| minor 7th
| C
| m7
| minor 7th
| C
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| 27
| style="text-align:center;" | 12.169
| 1045.2
| ~7
| mid 7th
| ^C
| AA6, dd8
| double-aug 6th, double-dim 8ve
| Bx, Dbb
| N7
| neutral 7th
| UC/uC#
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| 28
| style="text-align:center;" | 15.483
| 1083.9
| M7
| major 7th
| C#
| M7
| major 7th
| C#
| M7
| major 7th
| C#
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| 29
| style="text-align:center;" | 16.266
| 1122.6
| ^M7
| upmajor 7th
| ^C#
| d8
| dim 8ve
| Db
| SM7
| supermajor 7th
| SC#
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| 30
| style="text-align:center;" | 20.468
| 1161.3
| v8
| down-8ve
| vD
| A7
| aug 7th
| Cx
| s8/u8
| sub 8th, unter 8ve
| sD/uD
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| 31
| style="text-align:center;" | 21.447
| 1200.0
| P8
| perfect 8ve
| D
| P8
| perfect 8ve
| D
| P8
| perfect 8ve
| D
|}
|}


===1\31 octave - approx. 38.71¢ - Diesis or up-unison===
=== Interval quality and chord names in color notation ===
A single step of 31-edo is about 38.71¢. Intervals around this size are called ''dieses'' (singular '''diesis'''). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the [[11-limit|11-limit]], the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. 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. Demonstrated in [[SpiralProgressions|SpiralProgressions]].
Combining [[ups and downs notation]] with [[color notation]], qualities can be loosely associated with colors:


===2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second or downminor 2nd===
{| class="wikitable center-all"
The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two "microtones". In meantone, it is the ''chromatic semitone'', the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates [[Starling_temperaments#Valentine temperament|valentine temperament]] - aka [[Armodue_theory#Semi-equalized Armodue|semi-equalized Armodue]].
|-
 
! Quality
====MOS Scales generated by 2\31:====
! [[Color name]]
 
! Monzo format
{| class="wikitable"
! Examples
|-
|-
! | number of tones
| downminor
! | MOS class
| zo
! | 0
| {{monzo| a b 0 1 }}
! | 1
| 7/6, 7/4
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | 15-tone ([[Maximal_evenness|ME]] or quasi-equal)
| rowspan="2" | minor
| | [[1L_14s|1L 14s]]
| fourthward wa
| | 2
| {{monzo| a b }} where {{nowrap| b > −1 }}
| |
| 32/27, 16/9
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
|-
|-
| | 16-tone
| gu
| | [[15L_1s|15L 1s]]
| {{monzo| a b -1 }}
| | 2
| 6/5, 9/5
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
|}
 
===3\31 octave - approx. 116.13¢- Major Semitone or Diatonic Semitone or Large Minor Second or minor 2nd===
The larger and clunkier of the 31edo semitones. In meantone, it is the ''diatonic semitone'' which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a ''secor'' and generates [[Gamelismic_clan|miracle temperament]].
 
====MOS Scales generated by 3\31:====
 
{| class="wikitable"
|-
|-
! | number of tones
| rowspan="2" | mid
! | MOS class
| ilo
! | 0
| {{monzo| a b 0 0 1 }}
! | 1
| 11/9, 11/6
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | nonatonic
| lu
| | [[1L_8s|1L 8s]]
| {{monzo| a b 0 0 -1 }}
| | 3
| 12/11, 18/11
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
|-
| | decatonic (quasi-equal)
| rowspan="2" | major
| | [[9L_1s|9L 1s]]
| yo
| | 3
| {{monzo| a b 1 }}
| |
| 5/4, 5/3
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 4
| |
| |
| |
|-
|-
| | 11-tone
| fifthward wa
| | [[10L_1s|10L 1s]]
| {{monzo| a b }} where {{nowrap| b > 1 }}
| | 3
| 9/8, 27/16
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 1
|-
|-
| | 21-tone (Blackjack)
| upmajor
| | [[11L_10s|11L 10s]]
| ru
| | 2
| {{monzo| a b 0 -1 }}
| |
| 9/7, 12/7
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 1
|}
|}


===4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second or mid 2nd===
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:
Exactly one half of the minor third and twice the minor semitone. 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Although neutral seconds are typically associated with the 11-limit, 4\31 approximates the [[7-limit|7-limit]] interval 35/32 quite well, as the 5th harmonic of the 7th harmonic or vice versa, both of which are closely approximated in 31edo. And although 31 is not extremely accurate in the 11-limit, it is notable that since 11 and 3 are both flat, the interval that distinguishes them (12/11) is only about 4.5¢ off. Generates [[Starling_temperaments|nusecond temperament]].


====MOS Scales generated by 4\31:====
{| class="wikitable center-all"
 
|-
{| class="wikitable"
! [[Color notation|Color of the 3rd]]
! JI chord
! Edosteps
! Notes of C chord
! Written name
! Spoken name
|-
|-
! | number of tones
| zo (7-over)
! | MOS class
| 6:7:9
! | 0
| {{dash|0, 7, 18|s=hair|d=med}}
! | 1
|{{dash|C, vE{{flat}}, G|s=hair|d=med}} or {{dash|C, E{{sesquiflat}}, G|s=hair|d=med}}
! | 2
| Cvm
! | 3
| C downminor
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | heptatonic
| gu (5-under)
| | [[1L_6s|1L 6s]]
| 10:12:15
| | 4
| {{dash|0, 8, 18|s=hair|d=med}}
| |
| {{dash|C, E{{flat}}, G|s=hair|d=med}}
| |
| Cm
| |
| C minor
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
|-
| | octatonic (quasi-equal)
| ilo (11-over)
| | [[7L_1s|7L 1s]]
| 18:22:27
| | 4
| {{dash|0, 9, 18|s=hair|d=med}}
| |
|{{dash|C, vE, G|s=hair|d=med}} or {{dash|C, E{{demiflat}}, G|s=hair|d=med}}
| |
| C~
| |
| C mid
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 3
| |
| |
|-
|-
| | 15-tone
| yo (5-over)
| | [[8L_7s|8L 7s]]
| 4:5:6
| | 1
| {{dash|0, 10, 18|s=hair|d=med}}
| | 3
| {{dash|C, E, G|s=hair|d=med}}
| |
| C, Cmaj
| |
| C, C major
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 3
| |
| |
|-
|-
| | 23-tone
| ru (7-under)
| | [[8L_15s|8L 15s]]
| 14:18:21
| | 1
| {{dash|0, 11, 18|s=hair|d=med}}
| | 1
|{{dash|C, ^E, G|s=hair|d=med}} or {{dash|C, E{{demisharp}}, G|s=hair|d=med}}
| | 2
| C^
| |
| C up, C upmajor
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 2
| |
|}
|}


===5\31 octave - approx. 193.55¢ - Whole Tone or Major Second or major 2nd===
For a more complete list of chords, see [[31edo Chord Names]] and [[Ups and downs notation #Chords and chord progressions]].
A rather smallish whole tone. Sometimes called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a "meantone". Two meantones make a near-just major third. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat "active" (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps 0-18-36-45). Generates [[Gamelismic_clan|hemithirds temperament]] and [[Wuerschmidt_family|hermiwuerschmidt temperament]].


====MOS Scales generated by 5\31:====
== 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.
{{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"
|-
|-
! | number of tones
! Degree
! | MOS class
! Letter
! | 0
! Solfège
! | 1
! English full name
! | 2
|-
! | 3
| 0
! | 4
| C
! | 5
| do
! | 6
| C
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | hexatonic (quasi-equal)
| 1
| | [[1L_5s|1L 5s]]
| C{{demisharp2}}
| | 5
| do {{demisharp2}}
| |
| C half-sharp
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
|-
|-
| | heptatonic
| 2
| | [[6L_1s|6L 1s]]
| C♯
| | 5
| do ♯
| |
| C sharp
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 1
|-
|-
| | 13-tone
| 3
| | [[6L_7s|6L 7s]]
| D♭
| | 4
| re ♭
| |
| D flat
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
|-
|-
| | 19-tone
| 4
| | [[6L_13s|6L 13s]]
| D{{demiflat2}}
| | 3
| re {{demiflat2}}
| |
| D half-flat
| |
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
|-
|-
| | 25-tone
| 5
| | [[6L_19s|6L 19s]]
| D
| | 2
| re
| |
| D
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
|}
|}


===6\31 octave - approx. 232.26¢ - Supermajor Second or upmajor 2nd===
==== Stein–Zimmermann accidentals ====
Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 closely represents 8:7 (231.17¢). In meantone, it is a diminished third, eg. C to Ebb. Generates [[Meantone_family|mothra temperament]].
{{Sharpness-sharp2}}
 
====MOS Scales generated by 6\31:====


=== Chain-of-fifths notation ===
[[Chain-of-fifths notation]] uses double sharps and double flats only:
{| class="wikitable"
{| class="wikitable"
|-
|-
! | number of tones
! Degree
! | MOS class
! Letter
! | 0
! Solfège
! | 1
! English full name
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | pentatonic (quasi-equal)
| 0
| | [[1L_4s|1L 4s]]
| C
| | 6
| do
| |
| C
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
|-
| | hexatonic
| 1
| | [[5L_1s|5L 1s]]
| D𝄫
| | 6
| re 𝄫
| |
| D double flat
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 1
|-
|-
| | 11-tone
| 2
| | [[5L_6s|5L 6s]]
| C♯
| | 5
| do ♯
| |
| C sharp
| |
| |
| |
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
|-
|-
| | 16-tone
| 3
| | [[5L_11s|5L 11s]]
| D♭
| | 4
| re ♭
| |
| D flat
| |
| |
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 1
|-
|-
| | 21-tone
| 4
| | [[5L_16s|5L 16s]]
| C𝄪
| | 3
| do 𝄪
| |
| C double sharp
| |
| | 1
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 26-tone
| 5
| | [[5L_21s|5L 21s]]
| D
| | 2
| re
| |
| D
| | 1
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
| | 1
|}
|}


===7\31 octave - approx. 270.97¢ - Subminor Third or downminor 3rd===
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:
Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). In meantone temperament, it is an augmented 2nd, eg. C to D#. Generates [[Semicomma_family|orwell temperament]].
* C / D♯ / G / A♯
* C♯ / D𝄪 / G♯ / A𝄪
* D♭ / E / A♭ / B
* D / E♯ / A / B♯


====MOS Scales generated by 7\31:====
In 12edo, the enharmonic equivalences include {{nowrap|C♯ {{=}} D♭|E♯ {{=}} F}}, and {{nowrap|E {{=}} F♭}}. But in 31edo we have:
* C𝄪 = D{{demiflat2}}
* D𝄫 = C{{demisharp2}}
* E♯ = F{{demiflat2}}
* F♭ = E{{demisharp2}}
* E𝄪 = F{{demisharp2}}
* F𝄫 = E{{demiflat2}}


{| class="wikitable"
=== Sagittal notation ===
|-
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]].
! | number of tones
 
! | MOS class
==== Evo flavor ====
! | 0
{{Sagittal chart|Evo}}
! | 1
 
! | 2
==== Evo-SZ flavor ====
! | 3
{{Sagittal chart|Evo-SZ}}
! | 4
 
! | 5
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.
! | 6
 
! | 7
==== Revo flavor ====
! | 8
{{Sagittal chart}}
! | 9
 
! | 10
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:
! | 11
 
! | 12
[[File:31edo Sagittal.png|800px]]
! | 13
 
! | 14
== Relationship to 12edo ==
! | 15
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]]).
! | 16
 
! | 17
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.
! | 18
 
! | 19
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.
! | 20
 
! | 21
[[File:31-edo spiral.png|582x582px]]
! | 22
 
! | 23
== Approximation to JI ==
! | 24
[[File:31ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 31edo]]
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
| | pentatonic
| | [[4L_1s|4L 1s]]
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
|-
| | nonatonic (quasi-equal; Orwell[9])
| | [[4L_5s|4L 5s]]
| | 4
| |
| |
| |
| | 3
| |
| |
| | 4
| |
| |
| |
| | 3
| |
| |
| | 4
| |
| |
| |
| | 3
| |
| |
| | 4
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
| | 13-tone (Orwell[13])
| | [[9L_4s|9L 4s]]
| | 1
| | 3
| |
| |
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
| | 22-tone (Orwell[22])
| | [[9L_13s|9L 13s]]
| | 1
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
|}


===8\31 octave - approx. 309.68¢ - Minor Third===
=== Interval mappings ===
A minor third, closer to the just 6:5 (315.64¢) than 12-edo, but still on the flat side. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates [[Starling_temperaments|myna temperament]].
{{Q-odd-limit intervals}}


====MOS Scales generated by 8\31:====
=== Consistent circles ===
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.


{| class="wikitable"
== Regular temperament properties ==
|-
{| class="wikitable center-4 center-5 center-6"
! | number of tones
! | MOS class
! | 0
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | tetratonic (quasi-equal)
! rowspan="2" | [[Subgroup]]
| | [[3L_1s|3L 1s]]
! rowspan="2" | [[Comma list]]
| | 8
! rowspan="2" | [[Mapping]]
| |
! rowspan="2" | Optimal<br>8ve stretch (¢)
| |
! colspan="2" | Tuning error
| |
| |
| |
| |
| |
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
|-
| | heptatonic
! [[TE error|Absolute]] (¢)
| | [[4L_3s|4L 3s]]
! [[TE simple badness|Relative]] (%)
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
|-
| | 11-tone
| 2.3
| | [[4L_7s|4L 7s]]
| {{monzo| -49 31 }}
| | 1
| {{mapping| 31 49 }}
| | 1
| +1.637
| | 6
| 1.637
| |
| 4.228
| |
| |
| |
| |
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 6
| |
| |
| |
| |
| |
|-
|-
| | 15-tone
| 2.3.5
| | [[4L_11s|4L 11s]]
| 81/80, 393216/390625
| | 1
| {{mapping| 31 49 72 }}
| | 1
| +0.976
| | 1
| 1.628
| | 5
| 4.204
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
| | 5
| |
| |
| |
| |
|-
|-
| | 19-tone
| 2.3.5.7
| | [[4L_15s|4L 15s]]
| 81/80, 126/125, 1029/1024
| | 1
| {{mapping| 31 49 72 87 }}
| | 1
| +0.828
| | 1
| 1.432
| | 1
| 3.700
| | 4
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 1
| | 4
| |
| |
| |
|-
|-
| | 23-tone
| 2.3.5.7.11
| | [[4L_19s|4L 19s]]
| 81/80, 99/98, 121/120, 126/125
| | 1
| {{mapping| 31 49 72 87 107 }}
| | 1
| +1.205
| | 1
| 1.487
| | 1
| 3.841
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 3
| |
| |
|-
|-
| | 27-tone
| 2.3.5.7.11.13
| | [[4L_23s|4L 23s]]
| 66/65, 81/80, 99/98, 105/104, 121/120
| | 1
| {{mapping| 31 49 72 87 107 115 }}
| | 1
| +0.502
| | 1
| 2.072
| | 1
| 5.353
| | 1
|- style="border-top: double;"
| | 1
| 2.3.5.7.11.23
| | 2
| 81/80, 99/98, 126/125, 161/160, 231/230
| |
| {{mapping| 31 49 72 87 107 140 }}
| | 1
| +1.333
| | 1
| 1.387
| | 1
| 3.584
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 2
| |
|}
|}
* 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.


===9\31 octave - approx. 348.39¢ - Neutral Third or mid 3rd===
=== Uniform maps ===
A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for [[Mohajira|mohajira temperament]]), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a [[13-limit|13-limit]] interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more "major-like" than "minor-like"). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, which leaves the 143rd harmonic only about 2¢ wide after cancelling with the narrow 11th harmonic, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.
{{Uniform map|edo=31}}


====MOS Scales generated by 9\31:====
=== Commas ===
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="wikitable"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! | number of tones
! | MOS class
! | 0
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
| | tetratonic
| | [[3L_1s|3L 1s]]
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 4
| |
| |
| |
|-
| | heptatonic (quasi-equal)
| | [[3L_4s|3L 4s]]
| | 5
| |
| |
| |
| |
| | 4
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 4
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
|-
|-
| | 10-tone
! [[Harmonic limit|Prime<br>limit]]
| | [[7L_3s|7L 3s]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
| | 1
! [[Monzo]]
| | 4
! [[Cents]]
| |
! [[Color name]]
| |
! Name
| |
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
|-
|-
| | 17-tone
| 3
| | [[7L_10s|7L 10s]]
| <abbr title="617673396283947/562949953421312">(30 digits)</abbr>
| | 1
| {{monzo| -49 31}}
| | 1
| 160.605
| | 3
| Quadlawa
| |
| 31-comma
| |
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
|-
|-
| | 24-tone
| 5
| | [[7L_17s|7L 17s]]
| [[34171875/33554432|(16 digits)]]
| | 1
| {{monzo| -25 7 6 }}
| | 1
| 31.567
| | 1
| Lala-tribiyo
| | 2
| [[Ampersand comma]]
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
|}
 
===10\31 octave - approx. 387.10¢ - Major Third===
A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as "smooth". Generates [[Wuerschmidt_family|wurshmidt/worshmidt temperaments]].
 
====MOS Scales generated by 10\31:====
 
{| class="wikitable"
|-
|-
! | number of tones
| 5
! | MOS class
| [[81/80]]
! | 0
| {{monzo| -4 4 -1 }}
! | 1
| 21.506
! | 2
| Gu
! | 3
| [[Syntonic comma]]
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | tritonic (quasi-equal)
| 5
| | [[1L_2s|1L 2s]]
| <abbr title="393216/390625">(12 digits)</abbr>
| | 10
| {{monzo| 17 1 -8 }}
| |
| 11.445
| |
| Saquadbigu
| |
| [[Würschmidt comma]]
| |
| |
| |
| |
| |
| |
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
|-
| | tetratonic
| 5
| | [[3L_1s|3L 1s]]
| <abbr title="2109375/2097152">(14 digits)</abbr>
| | 10
| {{monzo| -21 3 7 }}
| |
| 10.061
| |
| Lasepyo
| |
| [[Semicomma]]
| |
| |
| |
| |
| |
| |
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
|-
|-
| | heptatonic
| 5
| | [[3L_4s|3L 4s]]
| <abbr title="274877906944/274658203125">(24 digits)</abbr>
| | 9
| {{monzo| 38 -2 -15 }}
| |
| 1.3843
| |
| Sasa-quintrigu
| |
| [[Hemithirds comma]]
| |
| |
| |
| |
| |
| | 1
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
|-
|-
| | 10-tone
| 7
| | [[3L_7s|3L 7s]]
| [[59049/57344]]
| | 8
| {{monzo| -13 10 0 -1 }}
| |
| 50.72
| |
| Laru
| |
| Harrison's comma
| |
| |
| |
| |
| | 1
| | 1
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
|-
|-
| | 13-tone
| 7
| | [[3L_10s|3L 10s]]
| [[3645/3584]]
| | 7
| {{monzo| -9 6 1 -1 }}
| |
| 29.22
| |
| Laruyo
| |
| Schismean comma
| |
| |
| |
| | 1
| | 1
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 16-tone
| 7
| | [[3L_13s|3L 13s]]
| <abbr title="854296875/843308032">(18 digits)</abbr>
| | 6
| {{monzo| -10 7 8 -7 }}
| |
| 22.413
| |
| Lasepru-aquadbiyo
| |
| [[Blackjackisma]]
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 19-tone
| 7
| | [[3L_16s|3L 16s]]
| [[64827/64000]]
| | 5
| {{monzo| -9 3 -3 4 }}
| |
| 22.227
| |
| Laquadzo-atrigu
| |
| Squalentine comma
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 22-tone
| 7
| | [[3L_19s|3L 19s]]
| [[2430/2401]]
| | 4
| {{monzo| 1 5 1 -4 }}
| |
| 20.785
| |
| Quadru-ayo
| |
| Nuwell comma
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 25-tone
| 7
| | [[3L_22s|3L 22s]]
| [[50421/50000]]
| | 3
| {{monzo| -4 1 -5 5 }}
| |
| 14.516
| |
| Quinzogu
| | 1
| Trimyna comma
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 28-tone
| 7
| | [[3L_25s|3L 25s]]
| [[126/125]]
| | 2
| {{monzo| 1 2 -3 1 }}
| |
| 13.795
| | 1
| Zotrigu
| | 1
| Starling comma, septimal semicomma
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|}
 
===11\31 octave - approx. 425.806¢ - Supermajor Third or upmajor 3rd===
11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating an interval of the [[23-limit|23-limit]], suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular [[17edo|17edo]]. Generates [[Meantone_family|squares temperament]].
 
====MOS Scales generated by 11\31:====
 
{| class="wikitable"
|-
|-
! | number of tones
| 7
! | MOS class
| [[1728/1715]]
! | 0
| {{monzo| 6 3 -1 -3 }}
! | 1
| 13.074
! | 2
| Trizo-agu
! | 3
| Orwellisma
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | tritonic
| 7
| | [[2L_1s|2L 1s]]
| [[1029/1024]]
| | 11
| {{monzo| -10 1 0 3 }}
| |
| 8.4327
| |
| Latrizo
| |
| Gamelisma
| |
| |
| |
| |
| |
| |
| |
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
|-
|-
| | pentatonic
| 7
| | [[3L_2s|3L 2s]]
| [[225/224]]
| | 2
| {{monzo| -5 2 2 -1 }}
| |
| 7.7115
| | 9
| Ruyoyo
| |
| Marvel comma, septimal kleisma
| |
| |
| |
| |
| |
| |
| |
| | 2
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
|-
|-
| | octatonic
| 7
| | [[3L_5s|3L 5s]]
| [[16875/16807]]
| | 2
| {{monzo| 0 3 4 -5 }}
| |
| 6.9903
| | 2
| Quinru-aquadyo
| |
| Mirkwai comma
| | 7
| |
| |
| |
| |
| |
| |
| | 2
| |
| | 2
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 2
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
|-
| | 11-tone
| 7
| | [[3L_8s|3L 8s]]
| [[3136/3125]]
| | 2
| {{monzo| 6 0 -5 2 }}
| |
| 6.0832
| | 2
| Zozoquingu
| |
| Hemimean comma
| | 2
| |
| | 5
| |
| |
| |
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 5
| |
| |
| |
| |
| | 2
| |
| | 2
| |
| | 5
| |
| |
| |
| |
|-
|-
| | 14-tone (quasi-equal)
| 7
| | [[3L_11s|3L 11s]]
| [[6144/6125]]
| | 2
| {{monzo| 11 1 -3 -2 }}
| |
| 5.3621
| | 2
| Sarurutrigu
| |
| Porwell comma
| | 2
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
|-
|-
| | 17-tone
| 7
| | [[3L_14s|3L 14s]]
| <abbr title="201768035/201326592">(18 digits)</abbr>
| | 2
| {{monzo| -26 -1 1 9 }}
| |
| 3.7919
| | 2
| Latritrizo-ayo
| |
| [[Wadisma]]
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| | 1
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
|}
 
===12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth or down 4th===
Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). It is also quite close to the [[17-limit|17-limit]] interval 17/13 (464.43¢), although 31edo does not offer up reasonable approximations of the 17th or 13th harmonics to help make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-limit consonances, and the only intervals in the 31edo octave that are not 15-limit consonances. Generates [[Semisept|semisept]] temperament.
 
====MOS Scales generated by 12\31:====
 
{| class="wikitable"
|-
|-
! | number of tones
| 7
! | MOS class
| [[65625/65536]]
! | 0
| {{monzo| -16 1 5 1 }}
! | 1
| 2.3495
! | 2
| Lazoquinyo
! | 3
| Horwell comma
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | tritonic
| 7
| | [[2L_1s|2L 1s]]
| [[703125/702464|(12 digits)]]
| | 12
| {{monzo| -11 2 7 -3 }}
| |
| 1.6283
| |
| Latriru-asepyo
| |
| [[Metric comma]]
| |
| |
| |
| |
| |
| |
| |
| |
| | 12
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
|-
| | pentatonic
| 7
| | [[3L_2s|3L 2s]]
| [[2401/2400]]
| | 5
| {{monzo| -5 -1 -2 4 }}
| |
| 0.72120
| |
| Bizozogu
| |
| Breedsma
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
|-
| | octatonic
| 11
| | [[5L_3s|5L 3s]]
| [[99/98]]
| | 5
| {{monzo| -1 2 0 -2 1 }}
| |
| 17.576
| |
| Loruru
| |
| Mothwellsma
| |
| | 5
| |
| |
| |
| |
| | 2
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 2
| |
| | 5
| |
| |
| |
| |
| | 2
| |
|-
|-
| | 13-tone (quasi-equal)
| 11
| | [[5L_8s|5L 8s]]
| [[121/120]]
| | 3
| {{monzo| -3 -1 -1 0 2 }}
| |
| 14.367
| |
| Lologu
| | 2
| Biyatisma
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
|-
|-
| | 18-tone
| 11
| | [[13L_5s|13L 5s]]
| [[176/175]]
| | 1
| {{monzo| 4 0 -2 -1 1 }}
| | 2
| 9.8646
| |
| Lorugugu
| | 2
| Valinorsma
| |
| | 1
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 2
| |
|}
 
===13\31 octave - approx. 503.23¢ - Perfect Fourth===
A slightly wide perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for [[Meantone|meantone]] temperament.
 
====MOS Scales generated by 13\31:====
 
{| class="wikitable"
|-
|-
! | number of tones
| 11
! | MOS class
| [[243/242]]
! | 0
| {{monzo| -1 5 0 0 -2 }}
! | 1
| 7.1391
! | 2
| Lulu
! | 3
| Rastma
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | tritonic
| 11
| | [[2L_1s|2L 1s]]
| [[385/384]]
| | 13
| {{monzo| -7 -1 1 1 1 }}
| |
| 4.5026
| |
| Lozoyo
| |
| Keenanisma
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 13
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
|-
|-
| | pentatonic
| 11
| | [[2L_3s|2L 3s]]
| [[441/440]]
| | 8
| {{monzo| -3 2 -1 2 -1 }}
| |
| 3.9302
| |
| Luzozogu
| |
| Werckisma
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
|-
|-
| | heptatonic
| 11
| | [[5L_2s|5L 2s]]
| [[540/539]]
| | 3
| {{monzo| 2 3 1 -2 -1 }}
| |
| 3.2090
| |
| Lururuyo
| | 5
| Swetisma
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 3
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
|-
|-
| | 12-tone (quasi-equal)
| 11
| | [[7L_5s|7L 5s]]
| [[3025/3024]]
| | 3
| {{monzo| -4 -3 2 -1 2 }}
| |
| 0.57240
| |
| Loloruyoyo
| | 3
| Lehmerisma
| |
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
|-
|-
| | 19-tone
| 13
| | [[12L_7s|12L 7s]]
| [[105/104]]
| | 1
| {{monzo| -3 1 1 1 0 -1 }}
| | 2
| 16.567
| |
| Thuzoyo
| | 1
| Animist comma
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
|}
 
===14\31 octave - approx. 541.94¢ - Superfourth or up 4th===
Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates [[Starling_temperaments|casablanca temperament]].
 
====MOS Scales generated by 14\31:====
 
{| class="wikitable"
|-
|-
! | number of tones
| 13
! | MOS class
| [[144/143]]
! | 0
| {{monzo| 4 2 0 0 -1 -1 }}
! | 1
| 12.064
! | 2
| Thulu
! | 3
| Grossma
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | tritonic
| 13
| | [[2L_1s|2L 1s]]
| [[196/195]]
| | 14
| {{monzo| 2 -1 -1 2 0 -1 }}
| |
| 8.8554
| |
| Thuzozogu
| |
| Mynucuma
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 14
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
|-
|-
| | pentatonic
| 13
| | [[2L_3s|2L 3s]]
| [[351/350]]
| | 11
| {{Monzo| -1 3 -2 -1 0 1 }}
| |
| 4.94
| |
| Thorugugu
| |
| Ratwolfsma
| |
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
|-
| | heptatonic
| 13
| | [[2L_5s|2L 5s]]
| [[352/351]]
| | 8
| {{monzo| 5 -3 0 0 1 -1 }}
| |
| 4.93
| |
| Thulo
| |
| Minor minthma
| |
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
|-
| | nonatonic
| 13
| | [[2L_7s|2L 7s]]
| [[625/624]]
| | 5
| {{monzo| -4 -1 4 0 0 -1 }}
| |
| 2.77
| |
| Thuquadyo
| |
| Tunbarsma
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 5
| |
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
|-
| | 11-tone (quasi-equal)
| 13
| | [[9L_2s|9L 2s]]
| [[1001/1000]]
| | 2
| {{monzo| -3 0 -3 1 1 1 }}
| |
| 1.73
| | 3
| Tholozotrigu
| |
| Fairytale comma, sinbadma
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
|-
| | 20-tone
| 13
| | [[11L_9s|11L 9s]]
| [[4096/4095]]
| | 2
| {{monzo| 12 -2 -1 -1 0 -1 }}
| |
| 0.42
| | 2
| Sathurugu
| |
| Minisma
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
|}
|}
<references group="note" />


===15\31 octave - approx. 580.65¢ - Small Tritone or Augmented 4th or Subdiminished Fifth or downdim 5th===
=== Rank-2 temperaments ===
In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates [[Tritonic|tritonic]] temperament.
* [[List of 31et rank two temperaments by badness]]
* [[List of edo-distinct 31et rank two temperaments]]
* [[Syntonic–31 equivalence continuum]]


====MOS Scales generated by 15\31:====
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"
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Rank-2 temperaments by generators
|-
|-
! | number of tones
! Generator*
! | MOS class
! Cents*
! | 0
! Mos scales
! | 1
! Temperaments
! | 2
! [[Pergen]]
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
|-
| | tritonic
| 1\31
| | [[2L_1s|2L 1s]]
| 38.71
| | 15
|  
| |
| [[Slender]]
| |
| (P8, P4/13)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 15
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
|-
|-
| | pentatonic
| 2\31
| | [[2L_3s|2L 3s]]
| 77.42
| | 14
| [[1L&nbsp;14s]], [[15L&nbsp;1s]]
| |
| [[Valentine]] / [[lupercalia]]
| |
| (P8, P5/9)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 14
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
|-
|-
| | heptatonic
| 3\31
| | [[2L_5s|2L 5s]]
| 116.13
| | 13
| [[1L&nbsp;9s]], [[10L&nbsp;1s]], [[10L&nbsp;11s]]
| |
| [[Mercy]] / [[miracle]]
| |
| (P8, P5/6)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 13
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
|-
|-
| | nonatonic
| 4\31
| | [[2L_7s|2L 7s]]
| 154.84
| | 12
| [[1L&nbsp;6s]], [[7L&nbsp;1s]], <br>[[8L&nbsp;7s]], [[8L&nbsp;15s]]
| |
| [[Greeley]] / [[nusecond]]
| |
| (P8, P11/11)
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 12
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 11-tone
| 5\31
| | [[2L_9s|2L 9s]]
| 193.55
| | 11
| [[1L&nbsp;5s]], [[6L&nbsp;1s]], [[6L&nbsp;7s]], <br>[[6L&nbsp;13s]], [[6L&nbsp;19s]]
| |
| [[Luna]] / [[didacus]] / [[hemithirds]] /<br>[[hemiwürschmidt]] / [[tutone]]
| |
| (P8, ccP4/15)
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 13-tone
| 6\31
| | [[2L_11s|2L 11s]]
| 232.26
| | 10
| [[1L&nbsp;4s]], [[5L&nbsp;1s]], [[5L&nbsp;6s]], <br>[[5L&nbsp;11s]], [[5L&nbsp;16s]], [[5L&nbsp;21s]]
| |
| [[Mothra]] / [[mosura]]<br>[[Quadrawell]]
| |
| (P8, P5/3)
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 15-tone
| 7\31
| | [[2L_13s|2L 13s]]
| 270.97
| | 9
| [[1L&nbsp;3s]], [[4L&nbsp;1s]], [[4L&nbsp;5s]], <br>[[9L&nbsp;4s]], [[9L&nbsp;13s]]
| |
| [[Orson]] / [[orwell]] / [[winston]]
| |
| (P8, P12/7)
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 17-tone
| 8\31
| | [[2L_15s|2L 15s]]
| 309.68
| | 8
| [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[4L&nbsp;7s]], <br>[[4L&nbsp;11s]], [[4L&nbsp;15s]], [[4L&nbsp;19s]], <br>[[4L&nbsp;23s]]
| |
| [[Myna]]<br>[[Triwell]]
| |
| (P8, ccP5/10)
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 19-tone
| 9\31
| | [[2L_17s|2L 17s]]
| 348.39
| | 7
| [[3L&nbsp;1s]], [[3L&nbsp;4s]], [[7L&nbsp;3s]], <br>[[7L&nbsp;10s]], [[7L&nbsp;17s]]
| |
| [[Mohaha]] / [[vicentino]] /<br>[[mohajira]] / [[migration]]
| |
| (P8, P5/2)
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 21-tone
| 10\31
| | [[2L_19s|2L 19s]]
| 387.10
| | 6
| [[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]]
| |
| (P8, ccP5/8)
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 23-tone
| 11\31
| | [[2L_21s|2L 21s]]
| 425.81
| | 5
| [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[3L&nbsp;8s]], <br>[[3L&nbsp;11s]], [[14L&nbsp;3s]]
| |
| [[Squares]] / [[sentinel]]
| |
| (P8, P11/4)
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 25-tone
| 12\31
| | [[2L_23s|2L 23s]]
| 464.52
| | 4
| [[3L&nbsp;2s]], [[5L&nbsp;3s]], <br>[[5L&nbsp;8s]], [[13L&nbsp;5s]]
| |
| [[A-Team]]<br>[[Semisept]]
| |
| (P8, c<sup>5</sup>P4/14)
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 27-tone
| 13\31
| | [[2L_25s|2L 25s]]
| 503.23
| | 3
| [[2L&nbsp;3s]], [[5L&nbsp;2s]], <br>[[7L&nbsp;5s]], [[12L&nbsp;7s]]
| |
| [[Meantone]] / [[meanpop]]
| |
| (P8, P5)
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
|-
| | 29-tone
| 14\31
| | [[2L_27s|2L 27s]]
| 541.94
| | 2
| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[9L&nbsp;2s]], [[11L&nbsp;9s]]
| |
| [[Casablanca]]<br>[[Cypress]]<br>[[Oracle]]
| | 1
| (P8, c<sup>5</sup>P4/12)
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 2
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|}
 
===16\31 Large Tritone or dim 5th===
 
Etc.
 
=Notation=
 
31edo can be notated with [[Ups_and_Downs_Notation|ups and downs notation]] like so:
 
{| class="wikitable"
|-
|-
| style="text-align:center;" | 0
| 15\31
| style="text-align:center;" | 1
| 580.65
| style="text-align:center;" | 2
| [[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]]
| style="text-align:center;" | 3
| [[Tritonic]] / [[tritoni]]
| style="text-align:center;" | 4
| (P8, ccP4/5)
| style="text-align:center;" | 5
| style="text-align:center;" | 6
| style="text-align:center;" | 7
| style="text-align:center;" | 8
| style="text-align:center;" | 9
| style="text-align:center;" | 10
| style="text-align:center;" | 11
| style="text-align:center;" | 12
| style="text-align:center;" | 13
| style="text-align:center;" | 14
| style="text-align:center;" | 15
| style="text-align:center;" | 16
| style="text-align:center;" | 17
| style="text-align:center;" | 18
| style="text-align:center;" | 19
| style="text-align:center;" | 20
| style="text-align:center;" | 21
| style="text-align:center;" | 22
| style="text-align:center;" | 23
| style="text-align:center;" | 24
| style="text-align:center;" | 25
| style="text-align:center;" | 26
| style="text-align:center;" | 27
| style="text-align:center;" | 28
| style="text-align:center;" | 29
| style="text-align:center;" | 30
| style="text-align:center;" | 31
|-
| style="text-align:center;" | P1
| style="text-align:center;" | ^P1
 
d2
| style="text-align:center;" | A1
 
vm2
| style="text-align:center;" | m2
| style="text-align:center;" | ~2
| style="text-align:center;" | M2
| style="text-align:center;" | ^M2
 
d3
| style="text-align:center;" | A2
 
vm3
| style="text-align:center;" | m3
| style="text-align:center;" | ~3
| style="text-align:center;" | M3
| style="text-align:center;" | ^M3
 
d4
| style="text-align:center;" | A3
 
v4
| style="text-align:center;" | P4
| style="text-align:center;" | ^4
| style="text-align:center;" | A4
| style="text-align:center;" | d5
| style="text-align:center;" | v5
| style="text-align:center;" | P5
| style="text-align:center;" | ^5
 
d6
| style="text-align:center;" | A5
 
vm6
| style="text-align:center;" | m6
| style="text-align:center;" | ~6
| style="text-align:center;" | M6
| style="text-align:center;" | ^M6
 
d7
| style="text-align:center;" | A6
 
vm7
| style="text-align:center;" | m7
| style="text-align:center;" | ~7
| style="text-align:center;" | M7
| style="text-align:center;" | ^M7
 
d8
| style="text-align:center;" | A7
 
v8
| style="text-align:center;" | P8
|}
|}
^ = up, v = down, ~ = mid
<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
 
vm2 = downminor 2nd, ~2 = mid 2nd, ^M2 = upmajor 2nd, etc.
 
{| class="wikitable"
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 1
| style="text-align:center;" | 2
| style="text-align:center;" | 3
| style="text-align:center;" | 4
| style="text-align:center;" | 5
| style="text-align:center;" | 6
| style="text-align:center;" | 7
| style="text-align:center;" | 8
| style="text-align:center;" | 9
| style="text-align:center;" | 10
| style="text-align:center;" | 11
| style="text-align:center;" | 12
| style="text-align:center;" | 13
| style="text-align:center;" | 14
| style="text-align:center;" | 15
| style="text-align:center;" | 16
| style="text-align:center;" | 17
| style="text-align:center;" | 18
| style="text-align:center;" | 19
| style="text-align:center;" | 20
| style="text-align:center;" | 21
| style="text-align:center;" | 22
| style="text-align:center;" | 23
| style="text-align:center;" | 24
| style="text-align:center;" | 25
| style="text-align:center;" | 26
| style="text-align:center;" | 27
| style="text-align:center;" | 28
| style="text-align:center;" | 29
| style="text-align:center;" | 30
| style="text-align:center;" | 31
|-
| style="text-align:center;" | C
| style="text-align:center;" | C^
| style="text-align:center;" | C#


Dbv
== Octave stretch or compression ==
| style="text-align:center;" | C#^
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.


Db
Good options include:
| style="text-align:center;" | Dv
* [[zpi|127zpi]]: Good [[13-limit]] option
| style="text-align:center;" | D
* [[80ed6]]: Great 11-limit option but bad harmonic 13
| style="text-align:center;" | D^
* [[49edt]]: Good 13-limit option for the opposite mapping of 13
| style="text-align:center;" | D#


Ebv
== Scales ==
| style="text-align:center;" | D#^
* [[Meantone5]]
* [[Meantone7]]
* [[Meantone12]]


Eb
=== MOS scales ===
| style="text-align:center;" | Ev
{{main| List of MOS scales in 31edo }}
| style="text-align:center;" | E
| style="text-align:center;" | E^


Fb
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:
| style="text-align:center;" | E#
* 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.


Fv
See [[#Rank-2 temperaments]] for a table of MOSes and their temperament interpretations.
| style="text-align:center;" | F
| style="text-align:center;" | F^
| style="text-align:center;" | F#


Gbv
=== Harmonic scales ===
| style="text-align:center;" | F#^
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]].


Gb
The steps are: 5 5 4 4 4 3 3 3.
| style="text-align:center;" | Gv
| style="text-align:center;" | G
| style="text-align:center;" | G^
| style="text-align:center;" | G#
 
Abv
| style="text-align:center;" | G#^
 
Ab
| style="text-align:center;" | Av
| style="text-align:center;" | A
| style="text-align:center;" | A^
| style="text-align:center;" | A#
 
Bbv
| style="text-align:center;" | A#^
 
Bb
| style="text-align:center;" | Bv
| style="text-align:center;" | B
| style="text-align:center;" | B^
 
Cb
| style="text-align:center;" | B#
 
Cv
| style="text-align:center;" | C
|}
 
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:


{| class="wikitable"
{| class="wikitable"
|-
|-
! | quality
! Overtones in "Mode 8":
! | [[Kite's color notation|color]]
| 8
! | monzo format
| 9
! | examples
| 10
|-
| 11
| style="text-align:center;" | downminor
| 12
| style="text-align:center;" | zo
| 13
| style="text-align:center;" | {a, b, 0, 1}
| 14
| style="text-align:center;" | 7/6, 7/4
| 15
|-
| 16
| style="text-align:center;" | minor
| style="text-align:center;" | fourthward wa
| style="text-align:center;" | {a, b}, b &lt; -1
| style="text-align:center;" | 32/27, 16/9
|-
| style="text-align:center;" | "
| style="text-align:center;" | gu
| style="text-align:center;" | {a, b, -1}
| style="text-align:center;" | 6/5, 9/5
|-
| style="text-align:center;" | mid
| style="text-align:center;" | lova
| style="text-align:center;" | {a, b, 0, 0, 1}
| style="text-align:center;" | 11/9, 11/6
|-
|-
| style="text-align:center;" | "
! …as JI Ratio from 1/1:
| style="text-align:center;" | lu
| 1/1
| style="text-align:center;" | {a, b, 0, 0, -1}
| 9/8
| style="text-align:center;" | 12/11, 18/11
| 5/4
| 11/8
| 3/2
| 13/8
| 7/4
| 15/8
| 2/1
|-
|-
| style="text-align:center;" | major
! …in cents:
| style="text-align:center;" | yo
| 0
| style="text-align:center;" | {a, b, 1}
| 203.9
| style="text-align:center;" | 5/4, 5/3
| 386.3
| 551.3
| 702.0
| 840.5
| 968.8
| 1088.3
| 1200.0
|-
|-
| style="text-align:center;" | "
! Nearest degree of 31edo:
| style="text-align:center;" | fifthward wa
| 0
| style="text-align:center;" | {a, b}, b &gt; 1
| 5
| style="text-align:center;" | 9/8, 27/16
| 10
| 14
| 18
| 22
| 25
| 28
| 31
|-
|-
| style="text-align:center;" | upmajor
! …in cents:
| style="text-align:center;" | ru
| 0
| style="text-align:center;" | {a, b, 0, -1}
| 193.5
| style="text-align:center;" | 9/7, 12/7
| 387.1
| 541.9
| 696.8
| 851.6
| 967.7
| 1083.9
| 1200.0
|}
|}
All 31edo chords can be named using ups and downs. Here are the zo, gu, lova, yo and ru triads:
 
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"
{| class="wikitable"
|-
|-
! | [[Kite's color notation|color of the 3rd]]
! Odd overtones in "Mode 16":
! | JI chord
| 17
! | notes as edosteps
| 19
! | notes of C chord
| 21
! | written name
| 23
! | spoken name
| 25
|-
| 27
| style="text-align:center;" | zo
| 29
| style="text-align:center;" | 6:7:9
| 31
| style="text-align:center;" | 0-7-18
| style="text-align:center;" | C Ebv G
| style="text-align:center;" | C.vm
| style="text-align:center;" | C downminor
|-
|-
| style="text-align:center;" | gu
! …as JI Ratio from 1/1:
| style="text-align:center;" | 10:12:15
| 17/16
| style="text-align:center;" | 0-8-18
| 19/16
| style="text-align:center;" | C Eb G
| 21/16
| style="text-align:center;" | Cm
| 23/16
| style="text-align:center;" | C minor
| 25/16
| 27/16
| 29/16
| 31/16
|-
|-
| style="text-align:center;" | lova
! …in cents:
| style="text-align:center;" | 18:22:27
| 105.0
| style="text-align:center;" | 0-9-18
| 297.5
| style="text-align:center;" | C Ev G
| 470.8
| style="text-align:center;" | C~
| 628.3
| style="text-align:center;" | C mid
| 772.6
| 905.9
| 1029.6
| 1145.0
|-
|-
| style="text-align:center;" | yo
! Nearest degree of 31edo:
| style="text-align:center;" | 4:5:6
| 3
| style="text-align:center;" | 0-10-18
| 8
| style="text-align:center;" | C E G
| 12
| style="text-align:center;" | C
| 16
| style="text-align:center;" | C major or C
| 20
| 23
| 27
| 30
|-
|-
| style="text-align:center;" | ru
! …in cents:
| style="text-align:center;" | 14:18:27
| 116.1
| style="text-align:center;" | 0-11-18
| 309.7
| style="text-align:center;" | C E^ G
| 464.5
| style="text-align:center;" | C.^
| 619.4
| style="text-align:center;" | C upmajor or C dot up
| 774.2
| 890.3
| 1045.1
| 1161.3
|}
|}
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].


=Harmonic Scale=
=== Various subsets ===
31edo approximates Mode 8 of the [[OverToneSeries|harmonic series]] O.K., 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 O.K., 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|13-limit]].
; 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]]


{| class="wikitable"
; Individual scales
|-
* the [[Euler–Fokker genus]] (technically [[JI]] but representable in 31)
| | Overtones in "Mode 8":
* the [[altered pentad]]
| | 8
* [[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)
| | 9
* the [[moon dust]] scale{{idio}} (technically [[JI]] but representable in 31)
| | 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:
== Instruments ==


<ul><li>17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.</li><li>19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp.</li><li>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 subset, on which it is consistent.</li><li>27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.</li><li>29 and 31 are both ''very'' sharp, and intervals involving them are unlikely to play any major role.</li></ul>
=== Keyboard Instruments ===
* [https://www.huygens-fokker.org/instruments/fokkerorgan.html Fokker Organ]
* [https://www.huygens-fokker.org/instruments/instrumentshuygensfokker/archiphone.html Archiphone]


{| class="wikitable"
=== String Instruments ===
|-
* [https://www.huygens-fokker.org/instruments/instrumentshuygensfokker/31-toneguitar.html Guitar]
| | 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
|}


=Commas=
=== Other Instruments ===
31 EDO tempers out the following commas. (Note: This assumes the val &lt; 31 49 72 87 107 115 |, comma values rounded to 5 significant digits.)
[[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]]


{| class="wikitable"
=== Lumatone ===
|-
* [[Lumatone mapping for 31edo]]
! | Ratio
! | Monzo
! | Value (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 34171875/33554432
| |<nowiki> | -25 7 6 </nowiki>&gt;
| style="text-align:right;" | 31.567
| style="text-align:center;" | [[ampersand|Ampersand's Comma]]
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 81/80
| |<nowiki> | -4 4 -1 </nowiki>&gt;
| style="text-align:right;" | 21.506
| style="text-align:center;" | [[Syntonic_Comma|Syntonic Comma]]
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 393216/390625
| |<nowiki> | 17 1 -8 </nowiki>&gt;
| style="text-align:right;" | 11.445
| style="text-align:center;" | [[Würschmidt_comma|Würschmidt Comma]]
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 2109375/2097152
| |<nowiki> | -21 3 7 </nowiki>&gt;
| style="text-align:right;" | 10.061
| style="text-align:center;" | [[semicomma|Semicomma]]
| style="text-align:center;" | Fokker Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 6719816/6714445
| |<nowiki> | 38 -2 -15 </nowiki>&gt;
| style="text-align:right;" | 1.3843
| style="text-align:center;" | Hemithirds Comma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 9859966/9733137
| |<nowiki> | -10 7 8 -7 </nowiki>&gt;
| style="text-align:right;" | 22.413
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 64827/64000
| |<nowiki> | -9 3 -3 4 </nowiki>&gt;
| style="text-align:right;" | 22.227
| style="text-align:center;" | Squalentine
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 2430/2401
| |<nowiki> | 1 5 1 -4 </nowiki>&gt;
| style="text-align:right;" | 20.785
| style="text-align:center;" | Nuwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 50421/50000
| |<nowiki> | -4 1 -5 5 </nowiki>&gt;
| style="text-align:right;" | 14.516
| style="text-align:center;" | Trimyna
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 126/125
| |<nowiki> | 1 2 -3 1 </nowiki>&gt;
| style="text-align:right;" | 13.795
| style="text-align:center;" | [[126/125|Septimal Semicomma]]
| style="text-align:center;" | Starling Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 1728/1715
| |<nowiki> | 6 3 -1 -3 </nowiki>&gt;
| style="text-align:right;" | 13.074
| style="text-align:center;" | Orwellisma
| style="text-align:center;" | Orwell Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 1029/1024
| |<nowiki> | -10 1 0 3 </nowiki>&gt;
| style="text-align:right;" | 8.4327
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 225/224
| |<nowiki> | -5 2 2 -1 </nowiki>&gt;
| style="text-align:right;" | 7.7115
| style="text-align:center;" | [[225/224|Septimal Kleisma]]
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 16875/16807
| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
| style="text-align:right;" | 6.9903
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 3136/3125
| |<nowiki> | 6 0 -5 2 </nowiki>&gt;
| style="text-align:right;" | 6.0832
| style="text-align:center;" | Hemimean
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 6144/6125
| |<nowiki> | 11 1 -3 -2 </nowiki>&gt;
| style="text-align:right;" | 5.3621
| style="text-align:center;" | Porwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 1065875/1063543
| |<nowiki> | -26 -1 1 9 </nowiki>&gt;
| style="text-align:right;" | 3.7919
| style="text-align:center;" | Wadisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 65625/65536
| |<nowiki> | -16 1 5 1 </nowiki>&gt;
| style="text-align:right;" | 2.3495
| style="text-align:center;" | Horwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 703125/702464
| |<nowiki> | -11 2 7 -3 </nowiki>&gt;
| style="text-align:right;" | 1.6283
| style="text-align:center;" | Meter
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 2401/2400
| |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
| style="text-align:right;" | 0.72120
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 99/98
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| style="text-align:right;" | 17.576
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 121/120
| |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
| style="text-align:right;" | 14.367
| style="text-align:center;" | Biyatisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 176/175
| |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
| style="text-align:right;" | 9.8646
| style="text-align:center;" | Valinorsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 243/242
| |<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
| style="text-align:right;" | 7.1391
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 385/384
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| style="text-align:right;" | 4.5026
| style="text-align:center;" | [[385/384|Keenanisma]]
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 441/440
| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
| style="text-align:right;" | 3.9302
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 540/539
| |<nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
| style="text-align:right;" | 3.2090
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 3025/3024
| |<nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
| style="text-align:right;" | 0.57240
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| style="text-align:center;" |
|}


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


A large open list of modes (subsets) from 31edo that people have named: [[31edo_modes|31edo modes]]. [http://en.wikipedia.org/wiki/Rothenberg_propriety Strictly proper] [[Strictly_proper_7-note_31edo_scales|7-note 31edo scales]] in the sense of [[David_Rothenberg|David Rothenberg]]. Interesting (to somebody) [[9-note_31edo_scales|9-note 31edo scales]]. See also [[31edo_MOS_scales|31edo MOS scales]]. Some of the popular ones:
'''[[Skip fretting system 31 3 7]]''' is another skip fretting system for 31edo.


<ul><li>31-tone major: 5 5 3 5 5 5 3</li><li>Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3</li><li>Harmonic scale 8: 5 5 4 4 4 3 3 3</li><li>the [[Euler-Fokker_genera|Euler-Fokker genera]] (technically [[JI|JI]] but representable in 31)</li></ul>
'''Skip fretting system 31 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]].


{| class="wikitable"
; Prime harmonics
|-
1/1: string 2 open
| colspan="2" |
====Some 31 tone equal modes:====
|-
| | <tt>'''2 3 3 2 3 2 3 2 3 3 2 3'''</tt>
| | Meantone Chromatic (53/220-comma)
|-
| | <tt>'''5 5 3 5 5 5 3'''</tt>
| | Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian
|-
| | <tt>'''5 3 5 5 3 5 5'''</tt>
| | Thirty-one tone Natural Minor, Intense Diatonic Hypodorian, Aeolian
|-
| | <tt>'''5 3 5 5 5 5 3'''</tt>
| | Thirty-one tone Melodic Minor
|-
| | <tt>'''5 3 5 5 3 7 3'''</tt>
| | Thirty-one tone Harmonic Minor
|-
| | <tt>'''5 5 3 5 3 7 3'''</tt>
| | Thirty-one tone Harmonic Major
|-
| | <tt>'''5 5 3 5 3 5 5'''</tt>
| | Thirty-one tone Major-Minor
|-
| | <tt>'''5 8 5 13'''</tt>
| | Genus primum
|-
| | <tt>'''10 3 5 5 5 3'''</tt>
| | Genus secundum
|-
| | <tt>'''8 2 8 3 7 3'''</tt>
| | Genus tertium
|-
| | <tt>'''10 10 10 1'''</tt>
| | Genus quartum
|-
| | <tt>'''5 7 6 7 5 1'''</tt>
| | Genus quintum
|-
| | <tt>'''4 6 2 6 4 3 3 3'''</tt>
| | Genus sextum
|-
| | <tt>'''4 6 5 6 4 6'''</tt>
| | Genus septimum
|-
| | <tt>'''6 6 6 1 6 6'''</tt>
| | Genus octavum
|-
| | <tt>'''4 6 9 6 4 2'''</tt>
| | Genus nonum
|-
| | <tt>'''13 6 6 6'''</tt>
| | Genus decimum
|-
| | <tt>'''5 5 3 5 5 3 2 3'''</tt>
| | Genus diatonicum
|-
| | <tt>'''3 5 2 3 5 3 2 5 3'''</tt>
| | Genus chromaticum
|-
| | <tt>'''5 5 2 1 5 5 2 3 3'''</tt>
| | Genus diatonicum cum septimis
|-
| | <tt>'''3 4 3 3 2 1 4 1 4 1 2 3'''</tt>
| | Genus enharmonicum vocale
|-
| | <tt>'''2 2 4 2 2 3 3 3 1 3 3 3'''</tt>
| | Genus enharmonicum instrumentale
|-
| | <tt>'''3 2 3 2 3 2 3 3 2 3 2 3'''</tt>
| | Genus diatonico-chromaticum
|-
| | <tt>'''5 2 1 2 5 3 2 1 4 1 2 3'''</tt>
| | Genus bichromaticum
|-
| | <tt>'''4 4 5 4 4 5 5'''</tt>
| | Neutral Diatonic Mixolydian
|-
| | <tt>'''4 5 4 4 5 5 4'''</tt>
| | Neutral Diatonic Lydian
|-
| | <tt>'''5 4 4 5 5 4 4'''</tt>
| | Neutral Diatonic Phrygian
|-
| | <tt>'''4 4 5 5 4 4 5'''</tt>
| | Neutral Diatonic Dorian
|-
| | <tt>'''4 5 5 4 4 5 4'''</tt>
| | Neutral Diatonic Hypolydian
|-
| | <tt>'''5 5 4 4 5 4 4'''</tt>
| | Neutral Diatonic Hypophrygian
|-
| | <tt>'''5 4 4 5 4 4 5'''</tt>
| | Neutral Diatonic Hypodorian
|-
| | <tt>'''4 5 4 4 5 4 5'''</tt>
| | Neutral Mixolydian
|-
| | <tt>'''5 4 4 5 4 5 4'''</tt>
| | Neutral Lydian
|-
| | <tt>'''4 4 5 4 5 4 5'''</tt>
| | Neutral Phrygian
|-
| | <tt>'''4 5 4 5 4 5 4'''</tt>
| | Neutral Dorian
|-
| | <tt>'''5 4 5 4 5 4 4'''</tt>
| | Neutral Hypolydian
|-
| | <tt>'''4 5 4 5 4 4 5'''</tt>
| | Neutral Hypophrygian
|-
| | <tt>'''5 4 5 4 4 5 4'''</tt>
| | Neutral Hypodorian
|-
| | <tt>'''2 2 9 2 2 9 5'''</tt>
| | Hemiolic Chromatic Mixolydian
|-
| | <tt>'''2 9 2 2 9 5 2'''</tt>
| | Hemiolic Chromatic Lydian
|-
| | <tt>'''9 2 2 9 5 2 2'''</tt>
| | Hemiolic Chromatic Phrygian
|-
| | <tt>'''2 2 9 5 2 2 9'''</tt>
| | Hemiolic Chromatic Dorian
|-
| | <tt>'''2 9 5 2 2 9 2'''</tt>
| | Hemiolic Chromatic Hypolydian
|-
| | <tt>'''9 5 2 2 9 2 2'''</tt>
| | Hemiolic Chromatic Hypophrygian
|-
| | <tt>'''5 2 2 9 2 2 9'''</tt>
| | Hemiolic Chromatic Hypodorian
|-
| | <tt>'''2 3 8 2 3 8 5'''</tt>
| | Ratio 2:3 Chromatic Mixolydian
|-
| | <tt>'''3 8 2 3 8 5 2'''</tt>
| | Ratio 2:3 Chromatic Lydian
|-
| | <tt>'''8 2 3 8 5 2 3'''</tt>
| | Ratio 2:3 Chromatic Phrygian
|-
| | <tt>'''2 3 8 5 2 3 8'''</tt>
| | Ratio 2:3 Chromatic Dorian
|-
| | <tt>'''3 8 5 2 3 8 2'''</tt>
| | Ratio 2:3 Chromatic Hypolydian
|-
| | <tt>'''8 5 2 3 8 2 3'''</tt>
| | Ratio 2:3 Chromatic Hypophrygian
|-
| | <tt>'''5 2 3 8 2 3 8'''</tt>
| | Ratio 2:3 Chromatic Hypodorian
|-
| | <tt>'''3 5 5 3 5 5 5'''</tt>
| | Intense Diatonic Mixolydian, M.Locrian
|-
| | <tt>'''5 3 5 5 5 3 5'''</tt>
| | Intense Diatonic Phrygian, M.Dorian
|-
| | <tt>'''3 5 5 5 3 5 5'''</tt>
| | Intense Diatonic Dorian, M.Phrygian
|-
| | <tt>'''5 5 5 3 5 5 3'''</tt>
| | Intense Diatonic Hypolydian, M.Lydian
|-
| | <tt>'''5 5 3 5 5 3 5'''</tt>
| | Intense Diatonic Hypophrygian, M.Mixolydian
|-
| | <tt>'''2 5 6 2 5 6 5'''</tt>
| | Soft Diatonic Mixolydian
|-
| | <tt>'''5 6 2 5 6 5 2'''</tt>
| | Soft Diatonic Lydian
|-
| | <tt>'''6 2 5 6 5 2 5'''</tt>
| | Soft Diatonic Phrygian
|-
| | <tt>'''2 5 6 5 2 5 6'''</tt>
| | Soft Diatonic Dorian
|-
| | <tt>'''5 6 5 2 5 6 2'''</tt>
| | Soft Diatonic Hypolydian
|-
| | <tt>'''6 5 2 5 6 2 5'''</tt>
| | Soft Diatonic Hypophrygian
|-
| | <tt>'''5 2 5 6 2 5 6'''</tt>
| | Soft Diatonic Hypodorian
|-
| | <tt>'''1 2 10 1 2 10 5'''</tt>
| | Enharmonic Mixolydian
|-
| | <tt>'''2 10 1 2 10 5 1'''</tt>
| | Enharmonic Lydian
|-
| | <tt>'''10 1 2 10 5 1 2'''</tt>
| | Enharmonic Phrygian
|-
| | <tt>'''1 2 10 5 1 2 10'''</tt>
| | Enharmonic Dorian
|-
| | <tt>'''2 10 5 1 2 10 1'''</tt>
| | Enharmonic Hypolydian
|-
| | <tt>'''10 5 1 2 10 1 2'''</tt>
| | Enharmonic Hypophrygian
|-
| | <tt>'''5 1 2 10 1 2 10'''</tt>
| | Enharmonic Hypodorian
|-
| | <tt>'''6 6 7 6 6'''</tt>
| | Quasi-equal Pentatonic
|-
| | <tt>'''3 2 2 3 3 2 3 3 2 2 3 3'''</tt>
| | Fokker 12-tone
|-
| | <tt>'''5 3 5 3 5 2 5 3'''</tt>
| | Modus conjunctus
|-
| | <tt>'''3 5 2 5 3 5 3 5'''</tt>
| | Octatonic
|-
| | <tt>'''3 3 4 3 5 3 4 3 3'''</tt>
| | Hahn symmetric pentachordal
|-
| | <tt>'''3 4 3 3 5 3 4 3 3'''</tt>
| | Hahn pentachordal
|-
| | <tt>'''4 4 2 5 3 3 4 3 3'''</tt>
| | Hahn Nonatonic
|-
| | <tt>'''5 1 5 1 5 1 5 1 5 1 1'''</tt>
| | de Vries 11-tone
|-
| | <tt>'''4 1 4 4 4 1 4 4 1 4'''</tt>
| | Breed 10-tone
|-
| | <tt>'''4 2 4 2 4 2 4 3 3 3'''</tt>
| | Lumma Decatonic
|-
| | <tt>'''5 3 3 3 3 5 3 3 3'''</tt>
| | Rothenberg Generalized Diatonic
|-
| | <tt>'''5 2 6 5 2 5 6'''</tt>
| | "Septimal" Natural Minor
|-
| | <tt>'''4 3 4 3 4 3 4 3 3'''</tt>
| | Thirty-one tone Orwell
|-
| | <tt>'''2 5 2 2 5 2 2 2 5 2 2'''</tt>
| | Secor Sentinel
|}


=Music in 31-edo=
2/1: string 7 fret 3
[[31-edo_compositions|An alphabetical list of Tricesimoprimal Compositions]].


[http://archive.org/download/Aire2In31-equalTemperament/Aire2In31.mp3 Aire #2 in 31-equal temperament] by [[Jon_Lyle_Smith|Jon Lyle Smith]]
3/2: string 4 fret 4


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%2031-tET-I%20Stand%20Hopeless%20Before%20the%20Gray%20Sea.mp3 I Stand Hopeless Before the Gray Sea] by Chuckles McGee
5/4: string 4 open


[http://soonlabel.com/xenharmonic/wp-content/uploads/2012/03/Claudi_Meneghin_Chaconne_G_001.mp3 Chaconna en G=, La Padana, ou la septimala (‘The Padanian, or the septimal’)] by [[Claudi_Meneghin|Claudi Meneghin]]
7/4: string 7 open


[[earwig|earwig]] by [[Andrew_Heathwaite|Andrew Heathwaite]]
11/8: string 4 fret 2


[https://www.youtube.com/watch?v=r1mat9f1DZ0 Fanfare and Toccata] by [[Juhani_Nuorvala|Juhani Nuorvala]]
13/8: string 6 fret 1


by Johann alias circular17: [https://www.youtube.com/watch?v=BBC8wiguN1A&index=5&list=PLRE0IICPofOVS-W5X2Rd7d9d3qzhxMnJN Curieuse planète], [https://www.youtube.com/watch?v=8PdJgmDJwu4&index=4&list=PLRE0IICPofOVS-W5X2Rd7d9d3qzhxMnJN Heal], [https://www.youtube.com/watch?v=j5eno0ejH0Y&index=2&list=PLRE0IICPofOVS-W5X2Rd7d9d3qzhxMnJN Wave from the past], [https://www.youtube.com/watch?v=fyPtr24qBd8&index=1&list=PLRE0IICPofOVS-W5X2Rd7d9d3qzhxMnJN Deep but not too much].
17/16: string 1 fret 4  


[https://www.youtube.com/watch?v=zZv-jUCynRU Orphanage of the Dutch Music IX: Sonate no. II in the 31-tone temperament - YouTube]
19/16: string 2 fret 4


[https://soundcloud.com/camtaylor-1/enharmonic-melody-for-guitar Enharmonic melody for guitar] by Cam Taylor
23/16: string 4 fret 3


[https://soundcloud.com/camtaylor-1/what-use-is-a-boy What use is a boy] by Cam Taylor
29/16: string 7 fret 1  


[https://soundcloud.com/camtaylor-1/back-to-31-hyperchromatic Back to 31: Hyperchromatic progression on C^] by Cam Taylor
31/16: string 1 fret 2


==Thirty-one tone pedagogy==
== Music ==
{{Main| 31edo/Music }}
{{Catrel|31edo tracks}}


The [[MicroPedagogyCollective|MicroPedagogyCollective]] is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two [[ThirtyOneToneSinginCamp|ThirtyOneToneSinginCamp]]s as well.
== See also ==
* [[List of 31edo chords]]
* [[Pentachords of 31edo]]
* [[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.
* [[CG-31]]


See also: [[31edo_solfege|31edo solfege]], [[Tricesimoprimal_Tetrachordal_Tesseract|Tricesimoprimal Tetrachordal Tesseract]], [[Pentachords_of_31edo|Pentachords of 31edo]].
== Further reading ==
=== Books ===
*Coates, Bill. ''[https://scribd.com/document/32296502/31-tone-equal-temperament Diesis: An Introduction to the Temperament of 31 Notes to Each Octave]''. Self-published, 1992.
*[[Sword, Ron]]. ''[https://ronsword.bigcartel.com/product/tricesimoprimal-scales-for-guitar Tricesimoprimal Scales for Guitar: Scales for 31-EDO]''. 2009. ([http://www.metatonalmusic.com/books.html Metatonal Music link]) (A comprehensive approach to 31edo and all the families associated for guitar. Features over 300 scale charts/scale examples.)


=Practical Theory / Books=
=== Articles ===
* [http://www.huygens-fokker.org/docs/beerart.html ''The Development of 31-tone Music''] [https://www.webcitation.org/5xeFzBM9b Permalink] by [[Anton de Beer]]
* [http://www.huygens-fokker.org/docs/fokkerorg.html ''Equal Temperament and the Thirty-one-keyed organ''] [https://www.webcitation.org/5xeG6Tmli Permalink] by [[Adriaan Daniël Fokker]]
* ''New Music with 31 Notes'' by Adriaan Daniël Fokker, translated by Leigh Gerdine
* [http://www.huygens-fokker.org/docs/rap31.html ''About 31-tone Equal Temperament''] [https://www.webcitation.org/5xeGH4uBH Permalink] by [[Paul Rapoport]]
* [http://www.huygens-fokker.org/docs/terp31.html ''Toward a Theory of Meantone (and 31-et) Harmony''] [https://www.webcitation.org/5xeGMeCMd Permalink] by [[Siemen Terpstra]]
* [http://tonalsoft.com/enc/number/31edo.aspx 31-ed2 / 31-edo / 31-ET / 31-tone equal-temperament] [https://www.webcitation.org/5xeGYj7QU Permalink] on [[Tonalsoft Encyclopedia]]
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Harmonic-Resources-31Et-EMT-31EBMT.pdf ''Harmonic Resources of 31Et EMT and 31EBMT''] by [[Juhan Puhm]] (2016)


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== External links ==
External image: http://ronsword.com/images/TSG_sm.jpg<br>
=== Websites ===
: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>
* [https://www.31edo.com/ 31edo.com] by [[User:KingHyperio | Alex Racz]]
: <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
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[[Category:IMPORTDEBUG - Change External Images]]


[http://www.ronsword.com/books.html Sword, Ronald. "Tricesimoprimal Scales for Guitar." IAAA Press, UK-USA. First Ed: March 2009.] - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.
=== Videos ===
* [https://youtu.be/E_VD3tqwCAM ''Quarter sharps and flats in the same diatonic key signature'' – Youtube] by [[Stephen Weigel]] – a list of diatonic key signatures and major scales in 31edo (including semi- and sesqui-sharps); and docs in its description.
* [https://www.youtube.com/watch?v=7cv-nuSjbY4&list=PLiWv7dE90L6CsQmQySVdAiRSIIDaAymiJ&pp=iAQB Playlist of 31edo music theory videos on YouTube] by [[Zhea Erose]]


=Other Articles=
=== Software ===
<ul><li><span style="">[http://www.huygens-fokker.org/docs/beerart.html de Beer, Anton, ''The Development of 31-tone Music'']</span> [http://www.webcitation.org/5xeFzBM9b Permalink]</li><li><span style="">[http://www.huygens-fokker.org/docs/fokkerorg.html Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ'']</span> [http://www.webcitation.org/5xeG6Tmli Permalink]</li><li>Fokker, A.D., "New Music with 31 Notes" translated by Leigh Gerdine</li><li><span style="">[http://www.huygens-fokker.org/docs/rap31.html Rapoport, Paul, ''About 31-tone Equal Temperament'']</span> [http://www.webcitation.org/5xeGH4uBH Permalink]</li><li><span style="">[http://www.huygens-fokker.org/docs/terp31.html Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony'']</span> [http://www.webcitation.org/5xeGMeCMd Permalink]</li><li><span style="">[http://tonalsoft.com/enc/number/31edo.aspx Tonalsoft Encyclopedia article]</span> [http://www.webcitation.org/5xeGYj7QU Permalink]</li></ul>
* [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]


[[File:31-edo.svg|alt=alt : Your browser has no SVG support.]]
=== Diagrams ===
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keys-and-Modes-of-31Et.pdf ''Keys and Modes of 31Et''] by Juhan Puhm (2016)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keyboard-Mapping-for-31Et.pdf ''Keyboard Mapping for 31Et''] by Juhan Puhm (2017)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Mapping-Range-for-31Et.pdf ''Mapping Range for 31Et''] by Juhan Puhm (2017)


[[:File:31-edo.svg|31-edo.svg]]     
[[Category:Golden meantone]]
[[Category:31edo]]
[[Category:Historical]]
[[Category:books]]
[[Category:Meantone]]
[[Category:edo]]
[[Category:Oneirotonic]]
[[Category:golden]]
[[Category:Orwell]]
[[Category:listen]]
[[Category:Semicomma]]
[[Category:meantone]]
[[Category:Valentine]]
[[Category:prime_edo]]
[[Category:Würschmidt]]
[[Category:semicomma]]
[[Category:zeta]]
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