31edo: Difference between revisions

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== Theory ==

31edo's perfect fifth is flat of the just interval [[3/2]] ~~(over five cents)~~, as befits a tuning [[support]]ing [[meantone]], but the major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of [[13/9]], [[13/11]], [[18/13]], and [[22/13]].

31edo's perfect fifth is flat of the just interval [[3/2]] by 5.2{{c}}, as befits a tuning [[support]]ing [[meantone]], but the major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of [[13/9]], [[13/11]], [[18/13]], and [[22/13]].

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

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One step of 31edo, measuring about 38.7¢, is called a [[diesis]] because it stands in for several intervals called "dieses" (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.

31edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L 3s) [[mos]] accurate 13:17:19 chords.

31edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L 3s) [[mos]] accurate 13:17:19 chords.

=== Prime harmonics ===

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| rowspan="2" | minor

| fourthward wa

| {{monzo|a b}} where {{nowrap|b < ~~−1~~}}

| {{monzo|a b}} where {{nowrap|b < −1}}

| 32/27, 16/9

|-

Line 653:

[[File:31edo CoF semi and sesqui.png|none|thumb|500x500px]]

===Sagittal notation===

=== 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|62-EDO]].

====Evo flavor====

==== Evo flavor ====

File:31-EDO_Evo_Sagittal.svg

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</imagemap>

====Revo flavor====

==== Revo flavor ====

File:31-EDO_Revo_Sagittal.svg

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[[File:31edo Sagittal.png|800px]]

====Evo-SZ flavor====

==== Evo-SZ flavor ====

File:31-EDO_Evo-SZ_Sagittal.svg

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The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains{{clarify}} are also useful:

* 9\31, the neutral third, generates [[ultrasoft]] [[mosh]] and [[superhard]] [[dicotonic]] MOSes.

* 11\31, the supermajor third or diminished fourth, generates a [[TAMNAMS|parahard]] [[sensoid]] scale with resolution from neutral thirds, sixths, and sevenths to perfect fourths, fifths, and octaves, and a [[semihard]] [[3L 8s]] scale with a jagged-but-chromatic feel.

* 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 8s]] scale with a jagged-but-chromatic feel.

* 12\31 generator generates a [[semihard]] [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth close to [[19/13]].

* 12\31 generator generates a [[semihard]] [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth close to [[19/13]].

* A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo.

* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[ultrasoft]] 9-tone [[4L 5s~~|orwelloid (4L 5s)~~]] MOS could be treated as a 9-tone well temperament.

* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[ultrasoft]] 9-tone orwelloid [[4L 5s]] MOS could be treated as a 9-tone well temperament.

* It has close approximations to [[6edf]] (→ [[miracle]]) and [[9edf]] (→ [[Carlos Alpha]]), fifth-equivalent equal divisions that hit many good JI approximations.

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* [[List of 31et rank two temperaments by badness]]

* [[List of edo-distinct 31et rank two temperaments]]

* [[~~Syntonic–31~~ equivalence continuum]]

* [[Syntonic–31 equivalence continuum]]

{| class="wikitable center-1"

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| 2\31

| 77.42

| [[1L 14s]], [[15L 1s]]

| [[1L 14s]], [[15L 1s]]

| [[Valentine]] / [[lupercalia]]

| (P8, P5/9)

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| 3\31

| 116.13

| [[1L 9s]], [[10L 1s]], [[10L 11s]]

| [[1L 9s]], [[10L 1s]], [[10L 11s]]

| [[Mercy]] / [[miracle]]

| (P8, P5/6)

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| 4\31

| 154.84

| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]]

| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]]

| [[Greeley]] / [[nusecond]]

| (P8, P11/11)

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| 5\31

| 193.55

| [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]]

| [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]]

| [[Luna]] / [[didacus]] / [[hemithirds]] / [[hemiwürschmidt]] / [[tutone]]

| (P8, ccP4/15)

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| 6\31

| 232.26

| [[1L 4s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], [[5L 16s]], [[5L 21s]]

| [[1L 4s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], [[5L 16s]], [[5L 21s]]

| [[Mothra]] / [[mosura]] [[Quadrawell]]

| (P8, P5/3)

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| 7\31

| 270.97

| [[1L 3s]], [[4L 1s]], [[4L 5s]], [[9L 4s]], [[9L 13s]]

| [[1L 3s]], [[4L 1s]], [[4L 5s]], [[9L 4s]], [[9L 13s]]

| [[Orson]] / [[orwell]] / [[winston]]

| (P8, P12/7)

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| 8\31

| 309.68

| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]]

| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]]

| [[Myna]] [[Triwell]]

| (P8, ccP5/10)

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| 9\31

| 348.39

| [[3L 1s]], [[3L 4s]], [[7L 3s]], [[7L 10s]], [[7L 17s]]

| [[3L 1s]], [[3L 4s]], [[7L 3s]], [[7L 10s]], [[7L 17s]]

| [[Mohaha]] / [[vicentino]] / [[mohajira]] / [[migration]]

| (P8, P5/2)

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| 10\31

| 387.10

| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]], [[3L 16s]], [[3L 19s]], [[3L 22s]], [[3L 25s]]

| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]], [[3L 16s]], [[3L 19s]], [[3L 22s]], [[3L 25s]]

| [[Würschmidt]] / [[worschmidt]]

| (P8, ccP5/8)

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| 11\31

| 425.81

| [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[14L 3s]]

| [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[14L 3s]]

| [[Squares]] / [[sentinel]]

| (P8, P11/4)

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| 12\31

| 464.52

| [[3L 2s]], [[5L 3s]], [[5L 8s]], [[13L 5s]]

| [[3L 2s]], [[5L 3s]], [[5L 8s]], [[13L 5s]]

| [[A-Team]] [[Semisept]]

| (P8, c5P4/14)

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| 13\31

| 503.23

| [[2L 3s]], [[5L 2s]], [[7L 5s]], [[12L 7s]]

| [[2L 3s]], [[5L 2s]], [[7L 5s]], [[12L 7s]]

| [[Meantone]] / [[meanpop]]

| (P8, P5)

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| 14\31

| 541.94

| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[9L 2s]], [[11L 9s]]

| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[9L 2s]], [[11L 9s]]

| [[Casablanca]] [[Cypress]] [[Oracle]]

| (P8, c5P4/12)

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| 15\31

| 580.65

| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]], [[2L 17s]], [[2L 19s]], [[2L 21s]], [[2L 23s]], [[2L 25s]], [[2L 27s]]

| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]], [[2L 17s]], [[2L 19s]], [[2L 21s]], [[2L 23s]], [[2L 25s]], [[2L 27s]]

| [[Tritonic]] / [[tritoni]]

| (P8, ccP4/5)

|}

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==

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* [[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.

* [[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]]

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=== 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://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]]

=== Software ===

* [http://31et.com/keyboard.php Virtual Piano Keyboard in 31-Tone Equal Temperament]

* [http://www.warmplace.ru/forum/viewtopic.php?f=9&t=4750 31EDO Piano ~~–~~ Mini synthesizer in Pixilang]

* [http://www.warmplace.ru/forum/viewtopic.php?f=9&t=4750 31EDO Piano – Mini synthesizer in Pixilang]

=== Diagrams ===

@@ Line 12: / Line 12: @@
 == Theory ==
-edo's perfect fifth is flat of the just interval [[3/2]] (over five cents), as befits a tuning [[support]]ing [[meantone]], but the major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of [[13/9]], [[13/11]], [[18/13]], and [[22/13]].
+edo's perfect fifth is flat of the just interval [[3/2]] by 5.2{{c}}, as befits a tuning [[support]]ing [[meantone]], but the major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of [[13/9]], [[13/11]], [[18/13]], and [[22/13]].
 Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31edo is relatively quite accurate and is [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo, a zeta peak edo, and a zeta peak integer edo]], meaning it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|strict zeta edo]]. Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the 7-, 9-, and [[11-odd-limit]], which it is [[consistent]] through, and that it is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 11-[[odd prime sum limit|odd-prime-sum-limit]].
@@ Line 18: / Line 18: @@
 One step of 31edo, measuring about 38.7¢, is called a [[diesis]] because it stands in for several intervals called "dieses" (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.
-edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L 3s) [[mos]] accurate 13:17:19 chords.
+edo 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.
 === Prime harmonics ===
@@ Line 473: / Line 473: @@
 | rowspan="2" | minor
 | fourthward wa
-| {{monzo|a b}} where {{nowrap|b &lt; &minus;1}}
+| {{monzo|a b}} where {{nowrap|b &lt; −1}}
 | 32/27, 16/9
 |-
@@ Line 653: / Line 653: @@
 [[File:31edo CoF semi and sesqui.png|none|thumb|500x500px]]
-===Sagittal notation===
+=== 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|62-EDO]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:31-EDO_Evo_Sagittal.svg
@@ Line 666: / Line 665: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:31-EDO_Revo_Sagittal.svg
@@ Line 681: / Line 679: @@
 [[File:31edo Sagittal.png|800px]]
-====Evo-SZ flavor====
+==== Evo-SZ flavor ====
 <imagemap>
 File:31-EDO_Evo-SZ_Sagittal.svg
@@ Line 717: / Line 714: @@
 The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains{{clarify}} are also useful:
 * 9\31, the neutral third, generates [[ultrasoft]] [[mosh]] and [[superhard]] [[dicotonic]] MOSes.
-* 11\31, the supermajor third or diminished fourth, generates a [[TAMNAMS|parahard]] [[sensoid]] scale with resolution from neutral thirds, sixths, and sevenths to perfect fourths, fifths, and octaves, and a [[semihard]] [[3L 8s]] scale with a jagged-but-chromatic feel.
+* 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]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth close to [[19/13]].
+* 12\31 generator generates a [[semihard]] [[oneirotonic]] 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¢) subminor third generator. The [[ultrasoft]] 9-tone [[4L 5s|orwelloid (4L 5s)]] MOS could be treated as a 9-tone well temperament.
+* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[ultrasoft]] 9-tone orwelloid [[4L&nbsp;5s]] MOS could be treated as a 9-tone well temperament.
 * It has close approximations to [[6edf]] (&rarr; [[miracle]]) and [[9edf]] (&rarr; [[Carlos Alpha]]), fifth-equivalent equal divisions that hit many good JI approximations.
@@ Line 1,191: / Line 1,188: @@
 * [[List of 31et rank two temperaments by badness]]
 * [[List of edo-distinct 31et rank two temperaments]]
-* [[Syntonic&ndash;31 equivalence continuum]]
+* [[Syntonic–31 equivalence continuum]]
 {| class="wikitable center-1"
@@ Line 1,210: / Line 1,207: @@
 | 2\31
 | 77.42
-| [[1L 14s]], [[15L 1s]]
+| [[1L&nbsp;14s]], [[15L&nbsp;1s]]
 | [[Valentine]] / [[lupercalia]]
 | (P8, P5/9)
@@ Line 1,216: / Line 1,213: @@
 | 3\31
 | 116.13
-| [[1L 9s]], [[10L 1s]], [[10L 11s]]
+| [[1L&nbsp;9s]], [[10L&nbsp;1s]], [[10L&nbsp;11s]]
 | [[Mercy]] / [[miracle]]
 | (P8, P5/6)
@@ Line 1,222: / Line 1,219: @@
 | 4\31
 | 154.84
-| [[1L 6s]], [[7L 1s]], <br>[[8L 7s]], [[8L 15s]]
+| [[1L&nbsp;6s]], [[7L&nbsp;1s]], <br>[[8L&nbsp;7s]], [[8L&nbsp;15s]]
 | [[Greeley]] / [[nusecond]]
 | (P8, P11/11)
@@ Line 1,228: / Line 1,225: @@
 | 5\31
 | 193.55
-| [[1L 5s]], [[6L 1s]], [[6L 7s]], <br>[[6L 13s]], [[6L 19s]]
+| [[1L&nbsp;5s]], [[6L&nbsp;1s]], [[6L&nbsp;7s]], <br>[[6L&nbsp;13s]], [[6L&nbsp;19s]]
 | [[Luna]] / [[didacus]] / [[hemithirds]] /<br>[[hemiwürschmidt]] / [[tutone]]
 | (P8, ccP4/15)
@@ Line 1,234: / Line 1,231: @@
 | 6\31
 | 232.26
-| [[1L 4s]], [[5L 1s]], [[5L 6s]], <br>[[5L 11s]], [[5L 16s]], [[5L 21s]]
+| [[1L&nbsp;4s]], [[5L&nbsp;1s]], [[5L&nbsp;6s]], <br>[[5L&nbsp;11s]], [[5L&nbsp;16s]], [[5L&nbsp;21s]]
 | [[Mothra]] / [[mosura]]<br>[[Quadrawell]]
 | (P8, P5/3)
@@ Line 1,240: / Line 1,237: @@
 | 7\31
 | 270.97
-| [[1L 3s]], [[4L 1s]], [[4L 5s]], <br>[[9L 4s]], [[9L 13s]]
+| [[1L&nbsp;3s]], [[4L&nbsp;1s]], [[4L&nbsp;5s]], <br>[[9L&nbsp;4s]], [[9L&nbsp;13s]]
 | [[Orson]] / [[orwell]] / [[winston]]
 | (P8, P12/7)
@@ Line 1,246: / Line 1,243: @@
 | 8\31
 | 309.68
-| [[3L 1s]], [[4L 3s]], [[4L 7s]], <br>[[4L 11s]], [[4L 15s]], [[4L 19s]], <br>[[4L 23s]]
+| [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[4L&nbsp;7s]], <br>[[4L&nbsp;11s]], [[4L&nbsp;15s]], [[4L&nbsp;19s]], <br>[[4L&nbsp;23s]]
 | [[Myna]]<br>[[Triwell]]
 | (P8, ccP5/10)
@@ Line 1,252: / Line 1,249: @@
 | 9\31
 | 348.39
-| [[3L 1s]], [[3L 4s]], [[7L 3s]], <br>[[7L 10s]], [[7L 17s]]
+| [[3L&nbsp;1s]], [[3L&nbsp;4s]], [[7L&nbsp;3s]], <br>[[7L&nbsp;10s]], [[7L&nbsp;17s]]
 | [[Mohaha]] / [[vicentino]] /<br>[[mohajira]] / [[migration]]
 | (P8, P5/2)
@@ Line 1,258: / Line 1,255: @@
 | 10\31
 | 387.10
-| [[3L 1s]], [[3L 4s]], [[3L 7s]], <br>[[3L 10s]], [[3L 13s]], [[3L 16s]], <br>[[3L 19s]], [[3L 22s]], [[3L 25s]]
+| [[3L&nbsp;1s]], [[3L&nbsp;4s]], [[3L&nbsp;7s]], <br>[[3L&nbsp;10s]], [[3L&nbsp;13s]], [[3L&nbsp;16s]], <br>[[3L&nbsp;19s]], [[3L&nbsp;22s]], [[3L&nbsp;25s]]
 | [[Würschmidt]] / [[worschmidt]]
 | (P8, ccP5/8)
@@ Line 1,264: / Line 1,261: @@
 | 11\31
 | 425.81
-| [[3L 2s]], [[3L 5s]], [[3L 8s]], <br>[[3L 11s]], [[14L 3s]]
+| [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[3L&nbsp;8s]], <br>[[3L&nbsp;11s]], [[14L&nbsp;3s]]
 | [[Squares]] / [[sentinel]]
 | (P8, P11/4)
@@ Line 1,270: / Line 1,267: @@
 | 12\31
 | 464.52
-| [[3L 2s]], [[5L 3s]], <br>[[5L 8s]], [[13L 5s]]
+| [[3L&nbsp;2s]], [[5L&nbsp;3s]], <br>[[5L&nbsp;8s]], [[13L&nbsp;5s]]
 | [[A-Team]]<br>[[Semisept]]
 | (P8, c<sup>5</sup>P4/14)
@@ Line 1,276: / Line 1,273: @@
 | 13\31
 | 503.23
-| [[2L 3s]], [[5L 2s]], <br>[[7L 5s]], [[12L 7s]]
+| [[2L&nbsp;3s]], [[5L&nbsp;2s]], <br>[[7L&nbsp;5s]], [[12L&nbsp;7s]]
 | [[Meantone]] / [[meanpop]]
 | (P8, P5)
@@ Line 1,282: / Line 1,279: @@
 | 14\31
 | 541.94
-| [[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[9L 2s]], [[11L 9s]]
+| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[9L&nbsp;2s]], [[11L&nbsp;9s]]
 | [[Casablanca]]<br>[[Cypress]]<br>[[Oracle]]
 | (P8, c<sup>5</sup>P4/12)
@@ Line 1,288: / Line 1,285: @@
 | 15\31
 | 580.65
-| [[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[2L 9s]], [[2L 11s]], [[2L 13s]], <br>[[2L 15s]], [[2L 17s]], [[2L 19s]], <br>[[2L 21s]], [[2L 23s]], [[2L 25s]], <br>[[2L 27s]]
+| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]], <br>[[2L&nbsp;15s]], [[2L&nbsp;17s]], [[2L&nbsp;19s]], <br>[[2L&nbsp;21s]], [[2L&nbsp;23s]], [[2L&nbsp;25s]], <br>[[2L&nbsp;27s]]
 | [[Tritonic]] / [[tritoni]]
 | (P8, ccP4/5)
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Instruments ==
@@ Line 1,315: / Line 1,312: @@
 * [[Pentachords of 31edo]]
 * [[Tricesimoprimal Tetrachordal Tesseract]]
-* [[MicroPedagogyCollective]] &ndash; is at work (as of 2012) producing demonstrative material which will encourage and enable more people to learn this system. There have been two [[ThirtyOneToneSinginCamp]]s as well.
+* [[MicroPedagogyCollective]] – is at work (as of 2012) producing demonstrative material which will encourage and enable more people to learn this system. There have been two [[ThirtyOneToneSinginCamp]]s as well.
 * [[CG-31]]
@@ Line 1,340: / Line 1,337: @@
 === Videos ===
-* [https://youtu.be/E_VD3tqwCAM ''Quarter sharps and flats in the same diatonic key signature'' &ndash; Youtube] by [[Stephen Weigel]] &ndash; a list of diatonic key signatures and major scales in 31edo (including semi- and sesqui-sharps); and docs in its description.
+* [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]]
 === Software ===
 * [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 &ndash; Mini synthesizer in Pixilang]
+* [http://www.warmplace.ru/forum/viewtopic.php?f=9&t=4750 31EDO Piano – Mini synthesizer in Pixilang]
 === Diagrams ===