31edo: Difference between revisions

(4 intermediate revisions by 3 users not shown)

Line 17:

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

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.

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, which is much more than the two varieties in 12edo.

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 {{nowrap|[[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 {{nowrap|[[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 ===

Line 30:

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.

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]].

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 {{nowrap|[[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 {{nowrap|49/48~64/63}} spacer with [[36/35]], so that {{nowrap|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]] ({{nowrap|26 & 31}}) tempering out [[99/98]], and [[mosura]] ({{nowrap|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]].

[[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.

Another notable temperament it supports is [[myna]], which is generated by the minor third, and sets the intervals [[7/6]], [[6/5]], {{nowrap|11/9~[[16/13]]}}, 5/4, and 9/7 being equidistant. Like mohajira, it creates five interval categories, but with 126/125 tempered out instead of 81/80.

31edo also supports [[squares]], which splits the [[8/3|perfect eleventh]] into four equal parts, each representing [[14/11]]~9/7, two of which make [[18/11]], and four of which make [[8/3]]. The [[2.3.7.11-subgroup|2.3.7.11 subgroup]] version of this temperament is sometimes known as ''skwares'', tempering out 99/98 and 243/242. Then, prime [[5/1|5]] is found by tempering out [[81/80]], completing the 11-limit.

31edo also supports [[squares]], which splits the [[8/3|perfect eleventh]] into four equal parts, each representing {{nowrap|[[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.

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 {{nowrap|14/11~9/7}}, each equidistant from each other.

=== Subsets and supersets ===

Line 58:

! Interval categories

! Approximate ratios<ref group="note">As a 13-limit temperament, with additional ratios of 17, 19, and 23. Inconsistent intervals are in ''italics''.</ref>

! [[Kite's ups and downs notation|Ups and downs notation]]

! colspan="2" |[[Kite's ups and downs notation|Ups and downs notation]]

([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vd2)

|-

| 0

Line 64:

Line 65:

| Unison

| [[1/1]]

|P1

| {{UDnote|step=0}}

|-

Line 70:

Line 72:

| Super-unison

| [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[64/63]], [[128/125]]

|^1, d2

| {{UDnote|step=1}}

|-

Line 76:

Line 79:

| Subminor second

| [[21/20]], [[22/21]], [[23/22]], [[25/24]], [[28/27]]

|A1, vm2

| {{UDnote|step=2}}

|-

Line 82:

Line 86:

| Minor second

| [[14/13]], [[15/14]], [[16/15]]

|m2

| {{UDnote|step=3}}

|-

Line 88:

Line 93:

| Neutral second

| [[11/10]], [[12/11]], [[13/12]], [[35/32]]

|~2

| {{UDnote|step=4}}

|-

Line 94:

Line 100:

| Major second

| [[9/8]], [[10/9]], [[19/17]], [[28/25]]

|M2

| {{UDnote|step=5}}

|-

Line 100:

Line 107:

| Supermajor second

| [[8/7]]

|^M2

| {{UDnote|step=6}}

|-

Line 106:

Line 114:

| Subminor third

| [[7/6]]

|vm3

| {{UDnote|step=7}}

|-

Line 112:

Line 121:

| Minor third

| [[6/5]], [[25/21]], ''[[13/11]]''

|m3

| {{UDnote|step=8}}

|-

Line 118:

Line 128:

| Neutral third

| [[11/9]], [[16/13]]

|~3

| {{UDnote|step=9}}

|-

Line 124:

Line 135:

| Major third

| [[5/4]]

|M3

| {{UDnote|step=10}}

|-

Line 130:

Line 142:

| Supermajor third

| [[9/7]], [[14/11]], [[23/18]], [[32/25]]

|^M3

| {{UDnote|step=11}}

|-

Line 136:

Line 149:

| Subfourth

| [[13/10]], [[17/13]], [[21/16]]

|v4

| {{UDnote|step=12}}

|-

Line 142:

Line 156:

| Perfect fourth

| [[4/3]]

|P4

| {{UDnote|step=13}}

|-

Line 148:

Line 163:

| Superfourth

| [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'', [[48/35]]

|^4, ~4

| {{UDnote|step=14}}

|-

Line 154:

Line 170:

| Augmented fourth

| [[7/5]], [[25/18]], [[45/32]]

|A4, vd5

| {{UDnote|step=15}}

|-

Line 160:

Line 177:

| Diminished fifth

| [[10/7]], [[36/25]], [[64/45]]

|^A4, d5

| {{UDnote|step=16}}

|-

Line 166:

Line 184:

| Subfifth

| [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'', [[35/24]]

|v5, ~5

| {{UDnote|step=17}}

|-

Line 172:

Line 191:

| Perfect fifth

| [[3/2]]

|P5

| {{UDnote|step=18}}

|-

Line 178:

Line 198:

| Superfifth

| [[20/13]], [[26/17]], [[32/21]]

|^5

| {{UDnote|step=19}}

|-

Line 184:

Line 205:

| Subminor sixth

| [[11/7]], [[14/9]], [[25/16]]

|vm6

| {{UDnote|step=20}}

|-

Line 190:

Line 212:

| Minor sixth

| [[8/5]]

|m6

| {{UDnote|step=21}}

|-

Line 196:

Line 219:

| Neutral sixth

| [[13/8]], [[18/11]]

|~6

| {{UDnote|step=22}}

|-

Line 202:

Line 226:

| Major sixth

| [[5/3]], [[42/25]], ''[[22/13]]''

|M6

| {{UDnote|step=23}}

|-

Line 208:

Line 233:

| Supermajor sixth

| [[12/7]]

|^M6

| {{UDnote|step=24}}

|-

Line 214:

Line 240:

| Subminor seventh

| [[7/4]]

|vm7

| {{UDnote|step=25}}

|-

Line 220:

Line 247:

| Minor seventh

| [[9/5]], [[16/9]], [[25/14]], [[34/19]]

|m7

| {{UDnote|step=26}}

|-

Line 226:

Line 254:

| Neutral seventh

| [[11/6]], [[20/11]], [[24/13]], [[64/35]]

|~7

| {{UDnote|step=27}}

|-

Line 232:

Line 261:

| Major seventh

| [[13/7]], [[15/8]], [[28/15]]

|M7

| {{UDnote|step=28}}

|-

Line 238:

Line 268:

| Supermajor seventh

| [[21/11]], [[27/14]], [[40/21]], [[44/23]], [[48/25]]

|^M7

| {{UDnote|step=29}}

|-

Line 244:

Line 275:

| Sub-octave

| [[35/18]], [[49/25]], [[63/32]], [[88/45]], [[96/49]], [[125/64]]

|v8

| {{UDnote|step=30}}

|-

Line 250:

Line 282:

| Octave

| [[2/1]]

|P8

| {{UDnote|step=31}}

|}

Line 258:

Line 291:

{| 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="font-size: 105%; white-space: nowrap;" | Table of proposed interval names and solfèges

|-

! #

Line 713:

Line 746:

| 6:7:9

| {{dash|0, 7, 18|s=hair|d=med}}

|{{dash|C, vE{{flat}}, G|s=hair|d=med}} or {{dash|C, E{{sesquiflat}}, G|s=hair|d=med}}

| {{dash|C, vE{{flat}}, G|s=hair|d=med}} or {{dash|C, E{{sesquiflat}}, G|s=hair|d=med}}

| Cvm

| C downminor

Line 727:

Line 760:

| 18:22:27

| {{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}}

| {{dash|C, vE, G|s=hair|d=med}} or {{dash|C, E{{demiflat}}, G|s=hair|d=med}}

| C~

| C mid

Line 741:

Line 774:

| 14:18:21

| {{dash|0, 11, 18|s=hair|d=med}}

|{{dash|C, ^E, G|s=hair|d=med}} or {{dash|C, E{{demisharp}}, G|s=hair|d=med}}

| {{dash|C, ^E, G|s=hair|d=med}} or {{dash|C, E{{demisharp}}, G|s=hair|d=med}}

| C^

| C up, C upmajor

@@ Line 17: / Line 17: @@
 Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents. However, intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. This makes 31edo a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.
-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.
-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, which is much more than the two varieties in 12edo.
+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 {{nowrap|[[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 {{nowrap|[[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 ===
@@ Line 30: / Line 30: @@
 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.
+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]].
+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 {{nowrap|[[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 {{nowrap|49/48~64/63}} spacer with [[36/35]], so that {{nowrap|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]] ({{nowrap|26 & 31}}) tempering out [[99/98]], and [[mosura]] ({{nowrap|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]].
+[[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]].
 edo 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.
+Another notable temperament it supports is [[myna]], which is generated by the minor third, and sets the intervals [[7/6]], [[6/5]], {{nowrap|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.
-edo 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.
+edo also supports [[squares]], which splits the [[8/3|perfect eleventh]] into four equal parts, each representing {{nowrap|[[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.
+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 {{nowrap|14/11~9/7}}, each equidistant from each other.
 === Subsets and supersets ===
@@ Line 58: / Line 58: @@
 ! Interval categories
 ! Approximate ratios<ref group="note">As a 13-limit temperament, with additional ratios of 17, 19, and 23. Inconsistent intervals are in ''italics''.</ref>
-! [[Kite's ups and downs notation|Ups and downs notation]]
+! colspan="2" |[[Kite's ups and downs notation|Ups and downs notation]]
+([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vd2)
 |-
 | 0
@@ Line 64: / Line 65: @@
 | Unison
 | [[1/1]]
+|P1
 | {{UDnote|step=0}}
 |-
@@ Line 70: / Line 72: @@
 | Super-unison
 | [[36/35]], [[45/44]], [[49/48]], [[50/49]], [[64/63]], [[128/125]]
+|^1, d2
 | {{UDnote|step=1}}
 |-
@@ Line 76: / Line 79: @@
 | Subminor second
 | [[21/20]], [[22/21]], [[23/22]], [[25/24]], [[28/27]]
+|A1, vm2
 | {{UDnote|step=2}}
 |-
@@ Line 82: / Line 86: @@
 | Minor second
 | [[14/13]], [[15/14]], [[16/15]]
+|m2
 | {{UDnote|step=3}}
 |-
@@ Line 88: / Line 93: @@
 | Neutral second
 | [[11/10]], [[12/11]], [[13/12]], [[35/32]]
+|~2
 | {{UDnote|step=4}}
 |-
@@ Line 94: / Line 100: @@
 | Major second
 | [[9/8]], [[10/9]], [[19/17]], [[28/25]]
+|M2
 | {{UDnote|step=5}}
 |-
@@ Line 100: / Line 107: @@
 | Supermajor second
 | [[8/7]]
+|^M2
 | {{UDnote|step=6}}
 |-
@@ Line 106: / Line 114: @@
 | Subminor third
 | [[7/6]]
+|vm3
 | {{UDnote|step=7}}
 |-
@@ Line 112: / Line 121: @@
 | Minor third
 | [[6/5]], [[25/21]], ''[[13/11]]''
+|m3
 | {{UDnote|step=8}}
 |-
@@ Line 118: / Line 128: @@
 | Neutral third
 | [[11/9]], [[16/13]]
+|~3
 | {{UDnote|step=9}}
 |-
@@ Line 124: / Line 135: @@
 | Major third
 | [[5/4]]
+|M3
 | {{UDnote|step=10}}
 |-
@@ Line 130: / Line 142: @@
 | Supermajor third
 | [[9/7]], [[14/11]], [[23/18]], [[32/25]]
+|^M3
 | {{UDnote|step=11}}
 |-
@@ Line 136: / Line 149: @@
 | Subfourth
 | [[13/10]], [[17/13]], [[21/16]]
+|v4
 | {{UDnote|step=12}}
 |-
@@ Line 142: / Line 156: @@
 | Perfect fourth
 | [[4/3]]
+|P4
 | {{UDnote|step=13}}
 |-
@@ Line 148: / Line 163: @@
 | Superfourth
 | [[11/8]], [[15/11]], [[26/19]], ''[[18/13]]'', [[48/35]]
+|^4, ~4
 | {{UDnote|step=14}}
 |-
@@ Line 154: / Line 170: @@
 | Augmented fourth
 | [[7/5]], [[25/18]], [[45/32]]
+|A4, vd5
 | {{UDnote|step=15}}
 |-
@@ Line 160: / Line 177: @@
 | Diminished fifth
 | [[10/7]], [[36/25]], [[64/45]]
+|^A4, d5
 | {{UDnote|step=16}}
 |-
@@ Line 166: / Line 184: @@
 | Subfifth
 | [[16/11]], [[19/13]], [[22/15]], ''[[13/9]]'', [[35/24]]
+|v5, ~5
 | {{UDnote|step=17}}
 |-
@@ Line 172: / Line 191: @@
 | Perfect fifth
 | [[3/2]]
+|P5
 | {{UDnote|step=18}}
 |-
@@ Line 178: / Line 198: @@
 | Superfifth
 | [[20/13]], [[26/17]], [[32/21]]
+|^5
 | {{UDnote|step=19}}
 |-
@@ Line 184: / Line 205: @@
 | Subminor sixth
 | [[11/7]], [[14/9]], [[25/16]]
+|vm6
 | {{UDnote|step=20}}
 |-
@@ Line 190: / Line 212: @@
 | Minor sixth
 | [[8/5]]
+|m6
 | {{UDnote|step=21}}
 |-
@@ Line 196: / Line 219: @@
 | Neutral sixth
 | [[13/8]], [[18/11]]
+|~6
 | {{UDnote|step=22}}
 |-
@@ Line 202: / Line 226: @@
 | Major sixth
 | [[5/3]], [[42/25]], ''[[22/13]]''
+|M6
 | {{UDnote|step=23}}
 |-
@@ Line 208: / Line 233: @@
 | Supermajor sixth
 | [[12/7]]
+|^M6
 | {{UDnote|step=24}}
 |-
@@ Line 214: / Line 240: @@
 | Subminor seventh
 | [[7/4]]
+|vm7
 | {{UDnote|step=25}}
 |-
@@ Line 220: / Line 247: @@
 | Minor seventh
 | [[9/5]], [[16/9]], [[25/14]], [[34/19]]
+|m7
 | {{UDnote|step=26}}
 |-
@@ Line 226: / Line 254: @@
 | Neutral seventh
 | [[11/6]], [[20/11]], [[24/13]], [[64/35]]
+|~7
 | {{UDnote|step=27}}
 |-
@@ Line 232: / Line 261: @@
 | Major seventh
 | [[13/7]], [[15/8]], [[28/15]]
+|M7
 | {{UDnote|step=28}}
 |-
@@ Line 238: / Line 268: @@
 | Supermajor seventh
 | [[21/11]], [[27/14]], [[40/21]], [[44/23]], [[48/25]]
+|^M7
 | {{UDnote|step=29}}
 |-
@@ Line 244: / Line 275: @@
 | Sub-octave
 | [[35/18]], [[49/25]], [[63/32]], [[88/45]], [[96/49]], [[125/64]]
+|v8
 | {{UDnote|step=30}}
 |-
@@ Line 250: / Line 282: @@
 | Octave
 | [[2/1]]
+|P8
 | {{UDnote|step=31}}
 |}
@@ Line 258: / Line 291: @@
 {| 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="font-size: 105%; white-space: nowrap;" | Table of proposed interval names and solfèges
 |-
 ! #
@@ Line 713: / Line 746: @@
 | 6:7:9
 | {{dash|0, 7, 18|s=hair|d=med}}
-|{{dash|C, vE{{flat}}, G|s=hair|d=med}} or {{dash|C, E{{sesquiflat}}, G|s=hair|d=med}}
+| {{dash|C, vE{{flat}}, G|s=hair|d=med}} or {{dash|C, E{{sesquiflat}}, G|s=hair|d=med}}
 | Cvm
 | C downminor
@@ Line 727: / Line 760: @@
 | 18:22:27
 | {{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}}
+| {{dash|C, vE, G|s=hair|d=med}} or {{dash|C, E{{demiflat}}, G|s=hair|d=med}}
 | C~
 | C mid
@@ Line 741: / Line 774: @@
 | 14:18:21
 | {{dash|0, 11, 18|s=hair|d=med}}
-|{{dash|C, ^E, G|s=hair|d=med}} or {{dash|C, E{{demisharp}}, G|s=hair|d=med}}
+| {{dash|C, ^E, G|s=hair|d=med}} or {{dash|C, E{{demisharp}}, G|s=hair|d=med}}
 | C^
 | C up, C upmajor