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31edo: Difference between revisions

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{{interwiki
{{Interwiki
| en = 31edo
| de = 31-EDO
| de = 31-EDO
| en = 31edo
| es = 31 EDO
| es = 31 EDO
| ja = 31平均律
| ja = 31平均律
| zh = 31平均律
}}
}}
{{Infobox ET}}
{{Infobox ET}}
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Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents. However, intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.
Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents. However, intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.


Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[the Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.
Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.
 
One step of 31edo, measuring about 38.7{{c}}, is called a [[diesis]] because it stands in for several intervals called ''dieses'' (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.


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


=== Prime harmonics ===
=== Prime harmonics ===
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=== As a tuning of other temperaments ===
=== As a tuning of other temperaments ===
Besides meantone, 31edo can be used as a tuning for [[mohajira]], [[mothra]] or less optimally [[miracle]] and [[valentine]]. These temperaments split 31edo's fifth, at 18 steps, into two, three, six, and nine equal parts. Mohajira and its alternative, called [[migration]], merge in 31edo, and create a near-optimal 11-limit meantone structure in one unified system. In fact, 31edo can be defined as the unique temperament that [[tempering out|tempers out]] [[81/80]], [[99/98]], [[121/120]], and [[126/125]].  
Besides meantone, 31edo can be used as a tuning for [[mohajira]], [[mothra]] or less optimally [[miracle]] and [[valentine]]. These temperaments split 31edo's fifth, at 18 steps, into two, three, six, and nine equal parts. In fact, 31edo can be defined as the unique temperament that [[tempering out|tempers out]] [[81/80]], [[99/98]], [[121/120]], and [[126/125]].
 
If we split the meantone [[generator]] of ~3/2 into two neutral thirds, each representing [[11/9]]~[[27/22]], then we get the [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament [[mohaha]], tempering out [[121/120]] and [[243/242]]. We can then map [[7/4]] to the semi-diminished seventh (-13 generators), tempering out [[385/384]], to get the full 11-limit mohajira temperament, which maps 7/6, 6/5, 11/9, 5/4, and 9/7 equidistant from each other. Alternatively, we can use the septimal meantone mapping of 7/4 (+20 generators) to get [[migration]]. Mohajira and [[migration]] merge in 31edo, and create a near-optimal 11-limit meantone structure in one unified system.


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


=== Subsets and supersets ===
=== Subsets and supersets ===
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! Cents
! Cents
! Interval categories
! Interval categories
! Approximate ratios<ref group="note">{{sg|limit=23-limit}} Inconsistent intervals are in ''italics''.</ref>
! 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]]
! [[Kite's ups and downs notation|Ups and downs notation]]
|-
|-
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| 38.7
| 38.7
| Super-unison
| Super-unison
| [[45/44]], [[49/48]], [[46/45]], [[128/125]], [[36/35]]
| [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[64/63]], [[128/125]]
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|-
|-
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| 77.4
| 77.4
| Subminor second
| Subminor second
| [[25/24]], [[21/20]], [[22/21]], [[23/22]]
| [[21/20]], [[22/21]], [[23/22]], [[25/24]], [[28/27]]
| {{UDnote|step=2}}
| {{UDnote|step=2}}
|-
|-
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| 116.1
| 116.1
| Minor second
| Minor second
| [[15/14]], [[16/15]]
| [[14/13]], [[15/14]], [[16/15]]
| {{UDnote|step=3}}
| {{UDnote|step=3}}
|-
|-
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| 154.8
| 154.8
| Neutral second
| Neutral second
| [[12/11]], [[11/10]], [[35/32]]
| [[11/10]], [[12/11]], [[13/12]], [[35/32]]
| {{UDnote|step=4}}
| {{UDnote|step=4}}
|-
|-
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| 232.3
| 232.3
| Supermajor second
| Supermajor second
| [[8/7]], [[144/125]]
| [[8/7]]
| {{UDnote|step=6}}
| {{UDnote|step=6}}
|-
|-
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| 271.0
| 271.0
| Subminor third
| Subminor third
| [[7/6]], [[75/64]]
| [[7/6]]
| {{UDnote|step=7}}
| {{UDnote|step=7}}
|-
|-
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| 309.7
| 309.7
| Minor third
| Minor third
| [[6/5]], ''[[13/11]]'', [[25/21]]
| [[6/5]], [[25/21]], ''[[13/11]]''
| {{UDnote|step=8}}
| {{UDnote|step=8}}
|-
|-
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| 348.4
| 348.4
| Neutral third
| Neutral third
| [[11/9]], [[27/22]], [[16/13]], [[60/49]], [[49/40]]
| [[11/9]], [[16/13]]
| {{UDnote|step=9}}
| {{UDnote|step=9}}
|-
|-
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| 464.5
| 464.5
| Subfourth
| Subfourth
| [[21/16]], [[64/49]], [[13/10]], [[17/13]], [[125/96]]
| [[13/10]], [[17/13]], [[21/16]]
| {{UDnote|step=12}}
| {{UDnote|step=12}}
|-
|-
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| 541.9
| 541.9
| Superfourth
| Superfourth
| [[175/128]], [[11/8]], [[15/11]], ''[[18/13]]'', [[26/19]]
| [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'', [[48/35]]
| {{UDnote|step=14}}
| {{UDnote|step=14}}
|-
|-
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| 580.6
| 580.6
| Augmented fourth
| Augmented fourth
| [[7/5]], [[45/32]], [[25/18]]
| [[7/5]], [[25/18]], [[45/32]]
| {{UDnote|step=15}}
| {{UDnote|step=15}}
|-
|-
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| 619.4
| 619.4
| Diminished fifth
| Diminished fifth
| [[10/7]], [[64/45]], [[36/25]]
| [[10/7]], [[36/25]], [[64/45]]
| {{UDnote|step=16}}
| {{UDnote|step=16}}
|-
|-
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| 658.1
| 658.1
| Subfifth
| Subfifth
| [[256/175]], ''[[13/9]]'', [[16/11]], [[22/15]], [[19/13]]
| [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'', [[35/24]]
| {{UDnote|step=17}}
| {{UDnote|step=17}}
|-
|-
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| 735.5
| 735.5
| Superfifth
| Superfifth
| [[32/21]], [[49/32]], [[20/13]], [[26/17]], [[192/125]]
| [[20/13]], [[26/17]], [[32/21]]
| {{UDnote|step=19}}
| {{UDnote|step=19}}
|-
|-
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| 774.2
| 774.2
| Subminor sixth
| Subminor sixth
| [[14/9]], [[11/7]], [[25/16]]
| [[11/7]], [[14/9]], [[25/16]]
| {{UDnote|step=20}}
| {{UDnote|step=20}}
|-
|-
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| 851.6
| 851.6
| Neutral sixth
| Neutral sixth
| [[18/11]], [[44/27]], [[13/8]], [[49/30]], [[80/49]]
| [[13/8]], [[18/11]]
| {{UDnote|step=22}}
| {{UDnote|step=22}}
|-
|-
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| 890.3
| 890.3
| Major sixth
| Major sixth
| [[5/3]], [[42/25]]
| [[5/3]], [[42/25]], ''[[22/13]]''
| {{UDnote|step=23}}
| {{UDnote|step=23}}
|-
|-
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| 929.0
| 929.0
| Supermajor sixth
| Supermajor sixth
| [[12/7]], [[128/75]]
| [[12/7]]
| {{UDnote|step=24}}
| {{UDnote|step=24}}
|-
|-
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| 967.7
| 967.7
| Subminor seventh
| Subminor seventh
| [[7/4]], [[125/72]]
| [[7/4]]
| {{UDnote|step=25}}
| {{UDnote|step=25}}
|-
|-
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| 1006.5
| 1006.5
| Minor seventh
| Minor seventh
| [[16/9]], [[9/5]], [[34/19]], [[25/14]]
| [[9/5]], [[16/9]], [[25/14]], [[34/19]]
| {{UDnote|step=26}}
| {{UDnote|step=26}}
|-
|-
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| 1045.2
| 1045.2
| Neutral seventh
| Neutral seventh
| [[11/6]], [[20/11]], [[64/35]]
| [[11/6]], [[20/11]], [[24/13]], [[64/35]]
| {{UDnote|step=27}}
| {{UDnote|step=27}}
|-
|-
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| 1083.9
| 1083.9
| Major seventh
| Major seventh
| [[28/15]], [[15/8]]
| [[13/7]], [[15/8]], [[28/15]]
| {{UDnote|step=28}}
| {{UDnote|step=28}}
|-
|-
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| 1122.6
| 1122.6
| Supermajor seventh
| Supermajor seventh
| [[48/25]], [[40/21]], [[21/11]], [[44/23]]
| [[21/11]], [[27/14]], [[40/21]], [[44/23]], [[48/25]]
| {{UDnote|step=29}}
| {{UDnote|step=29}}
|-
|-
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| 1161.3
| 1161.3
| Sub-octave
| Sub-octave
| [[88/45]], [[96/49]], [[45/23]], [[125/64]], [[35/18]]
| [[35/18]], [[49/25]], [[63/32]], [[88/45]], [[96/49]], [[125/64]]
| {{UDnote|step=30}}
| {{UDnote|step=30}}
|-
|-
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| 3.584
| 3.584
|}
|}
* 31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next equal temperaments doing better in those subgroups are 72, 72, 41, and 46, respectively.
* 31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next equal temperaments doing better in those subgroups are [[72edo|72]], 72, [[41edo|41]], and [[46edo|46]], respectively.
* 31et excels in the 2.5.7 subgroup (the JI chord 4:5:7 is represented highly [[consistent]]ly: to [[consistency #Consistency to distance d|distance]] 10.36). In 2.5.7 it tempers out the didacus comma [[3136/3125]] and the quince comma [[823543/819200]], thus also tempering out the very small [[rainy comma]], the simplest 2.5.7 comma tempered out by the 7-limit microtemperament [[171edo]].  
* 31et excels in the [[2.5.7 subgroup]] (the JI chord [[4:5:7]] is represented highly [[consistent]]ly: to [[consistency #Consistency to distance d|distance]] 10.36). In 2.5.7 it tempers out the didacus comma [[3136/3125]] and the quince comma [[823543/819200]], thus also tempering out the very small [[rainy comma]], the simplest 2.5.7 comma tempered out by the 7-limit microtemperament [[171edo]].  
* In the 17-limit it tempers out [[120/119]], equating the otonal tetrad of [[4:5:6:7]] and the inversion of the [[10:12:15:17]] minor tetrad.
* In the [[17-limit]] it tempers out [[120/119]], equating the otonal tetrad of [[4:5:6:7]] and the inversion of the [[10:12:15:17]] minor tetrad.


=== Uniform maps ===
=== Uniform maps ===
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| 0.42
| 0.42
| Sathurugu
| Sathurugu
| Schismina
| Minisma
|}
|}
<references group="note" />
<references group="note" />
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| (P8, ccP4/5)
| (P8, ccP4/5)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


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


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


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


'''Skip fretting system 31 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]].
'''Skip fretting system 31 2 5''' is another skip fretting system for 31edo. All examples on this page are for 7-string [[guitar]].
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