39edo: Difference between revisions

Line 3:

== Theory ==

39edo's [[3/2|perfect fifth]] is 5.8 ~~cents (¢)~~ sharp. Together with its best [[5/4|classical major third]] which is the familiar 400 cents of [[12edo]], we get a system which [[tempering out|tempers out]] the [[diesis]] (128/125) and the [[amity comma]] (1600000/1594323). We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]], which adds [[64/63]] and [[126/125]] to the list. [[Tempering out]] both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine [[11/1|11]], and adding it to consideration we find that the equal temperament tempers out [[99/98]] and [[121/120]] also. This choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}.

39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400 cents of [[12edo]], we get a system which [[tempering out|tempers out]] the [[diesis]] (128/125) and the [[amity comma]] (1600000/1594323). We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]], which adds [[64/63]] and [[126/125]] to the list. [[Tempering out]] both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine [[11/1|11]], and adding it to consideration we find that the equal temperament tempers out [[99/98]] and [[121/120]] also. This choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}.

A particular anecdote with this system was made in the ''Teliochordon'', in 1788 by {{w|Charles Clagget}} (Ireland, 1740?–1820), a little extract [http://ml.oxfordjournals.org/content/76/2/291.extract.jpg here].

Line 439:

== Notation ==

=== Ups and downs notation ===

Using [[~~Helmholtz-Ellis notation|~~Helmholtz–Ellis]] accidentals, 39edo can also be notated using [[ups and downs notation]]:

Using [[Helmholtz–Ellis]] accidentals, 39edo can also be notated using [[ups and downs notation]]:

Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with three arrows.

===Sagittal notation===

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[46edo#Sagittal notation|46-EDO]].

~~====Evo flavor====~~

==== Evo flavor ====

File:39-EDO_Evo_Sagittal.svg

Line 459:

====Revo flavor====

File:39-EDO_Revo_Sagittal.svg

Line 740:

Line 739:

| {{monzo| 62 -39 }}

| {{mapping| 39 62 }}

| ~~−1~~.81

| −1.81

| 1.81

| 5.88

Line 747:

Line 746:

| 128/125, 1594323/1562500

| {{mapping| 39 62 91 }}

| ~~−3~~.17

| −3.17

| 2.42

| 7.89

Line 754:

Line 753:

| 64/63, 126/125, 2430/2401

| {{mapping| 39 62 91 110 }} (39d)

| ~~−3~~.78

| −3.78

| 2.35

| 7.65

Line 761:

Line 760:

| 64/63, 99/98, 121/120, 126/125

| {{mapping| 39 62 91 110 135 }} (39d)

| ~~−3~~.17

| −3.17

| 2.43

| 7.91

Line 786:

Line 785:

| 61.5

| [[Unicorn]] (39d)

| [[1L 18s]], [[19L 1s]]

| [[1L 18s]], [[19L 1s]]

|-

| 1

Line 792:

Line 791:

| 123.1

| [[Negri]] (39c)

| [[1L 8s]], [[9L 1s]], [[10L 9s]], [[10L 19s]]

| [[1L 8s]], [[9L 1s]], [[10L 9s]], [[10L 19s]]

|-

| 1

Line 798:

Line 797:

| 153.8

|

| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]], [[8L 23s]]

| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]], [[8L 23s]]

|-

| 1

Line 804:

Line 803:

| 215.4

| [[Machine]] (39d)

| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[11L 6s]], [[11L 17s]]

| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[11L 6s]], [[11L 17s]]

|-

| 1

Line 810:

Line 809:

| 246.2

| [[Immunity]] (39) / [[immunized]] (39d)

| [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]], [[5L 19s]], [[5L 24s]], [[5L 29s]]

| [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]], [[5L 19s]], [[5L 24s]], [[5L 29s]]

|-

| 1

Line 816:

Line 815:

| 307.7

| [[Familia]] (39df)

| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]], [[4L 27s]], [[4L 31s]]

| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]], [[4L 27s]], [[4L 31s]]

|-

| 1

Line 822:

Line 821:

| 338.5

| [[Amity]] (39) / [[accord]] (39d)

| [[3L 1s]], [[4L 3s]], [[7L 4s]], [[7L 11s]], [[7L 18s]], [[7L 25s]]

| [[3L 1s]], [[4L 3s]], [[7L 4s]], [[7L 11s]], [[7L 18s]], [[7L 25s]]

|-

| 1

Line 828:

Line 827:

| 430.8

| [[Hamity]] (39df)

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

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

|-

| 1

Line 834:

Line 833:

| 492.3

| [[Quasisuper]] (39d)

| [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]]

| [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]]

|-

| 1

Line 840:

Line 839:

| 523.1

| [[Mavila]] (39bc)

| [[2L 3s]], [[2L 5s]], [[7L 2s]], [[7L 9s]], [[16L 7s]]

| [[2L 3s]], [[2L 5s]], [[7L 2s]], [[7L 9s]], [[16L 7s]]

|-

| 1

Line 846:

Line 845:

| 584.6

| [[Pluto]] (39d)

| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]] etc. … [[2L 35s]]

| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]] etc. … [[2L 35s]]

|-

| 3

Line 858:

Line 857:

| 61.5

|

| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[3L 12s]], [[3L 15s]], [[18L 3s]]

| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[3L 12s]], [[3L 15s]], [[18L 3s]]

|-

| 3

Line 864:

Line 863:

| 184.6

| [[Terrain]] / [[mirkat]] (39df)

| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15]], [[6L 21s]], [[6L 27s]]

| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15]], [[6L 21s]], [[6L 27s]]

|-

| 3

Line 870:

Line 869:

| 246.2<br />(153.8)

| [[Triforce]] (39)

| [[3L 3s]], [[6L 3s]], [[9L 6s]], [[15L 9s]]

| [[3L 3s]], [[6L 3s]], [[9L 6s]], [[15L 9s]]

|-

| 3

Line 876:

Line 875:

| 492.3<br />(92.3)

| [[Augene]] (39d)

| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], [[12L 15s]]

| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], [[12L 15s]]

|-

| 3

Line 882:

Line 881:

| 523.1<br />(123.0)

| [[Deflated]] (39bd)

| [[3L 3s]], [[3L 6s]], [[9L 3s]], [[9L 12s]], [[9L 21s]]

| [[3L 3s]], [[3L 6s]], [[9L 3s]], [[9L 12s]], [[9L 21s]]

|-

| 13

Line 888:

Line 887:

| 492.3<br />(30.8)

| [[Tridecatonic]]

| [[13L 13s]]

| [[13L 13s]]

|}

<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

== 39edo and world music ==

Line 898:

Line 897:

39edo offers not one, but several different ways to realize the traditional Western diatonic scale. One way is to simply take a [[chain of fifths]] (the diatonic MOS: '''7 7 2 7 7 7 2'''). Because 39edo is a [[superpyth]] rather than a [[meantone]] system, this means that the harmonic quality of its diatonic scale will differ somewhat, since "minor" and "major" triads now approximate 6:7:9 and 14:18:21 respectively, rather than 10:12:15 and 4:5:6 as in meantone diatonic systems. Diatonic compositions translated onto this scale thus acquire a wildly different harmonic character, albeit still pleasing.

Another option is to use a [[MODMOS]], such as '''7 6 3 7 6 7 3'''; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[Wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either '''7 3 3 3 7 3 3 7 3''' (a [[MODMOS]] of type [[3L 6s]]) or '''4 3 6 3 4 3 6 4 3 3.''' There are other MODMOS's that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.

Another option is to use a [[MODMOS]], such as '''7 6 3 7 6 7 3'''; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[Wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either '''7 3 3 3 7 3 3 7 3''' (a [[MODMOS]] of type [[3L 6s]]) or '''4 3 6 3 4 3 6 4 3 3.''' There are other MODMOS's that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.

The MOS and the MODMOS's all have smaller-than-usual semitones, which makes them more effective for melody than their counterparts in 12edo or meantone systems.

Line 905:

Line 904:

=== Indian ===

A similar situation arises with [[Indian music]] since the sruti system, like the Western system, also has multiple possible mappings in 39edo. Many of these are modified versions of the [[17L 5s]] MOS (where the generator is a perfect fifth).

A similar situation arises with [[Indian music]] since the sruti system, like the Western system, also has multiple possible mappings in 39edo. Many of these are modified versions of the [[17L 5s]] MOS (where the generator is a perfect fifth).

=== Arabic, Turkish, Iranian ===

Line 927:

Line 926:

39edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator.

It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] MOS (which is '''9 7 7 9 7'''). [[Slendro]] can be approximated using this scale or using something like the [[quasi-equal]] '''8 8 8 8 7'''. A more expressive [[pentatonic]] scale is the oneirotonic subset '''9 6 9 9 6'''. Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] MOS (which is '''9 7 7 9 7'''). [[Slendro]] can be approximated using this scale or using something like the [[quasi-equal]] '''8 8 8 8 7'''. A more expressive [[pentatonic]] scale is the oneirotonic subset '''9 6 9 9 6'''. Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

== Instruments ==

@@ Line 3: / Line 3: @@
 == Theory ==
-edo's [[3/2|perfect fifth]] is 5.8 cents (¢) sharp. Together with its best [[5/4|classical major third]] which is the familiar 400 cents of [[12edo]], we get a system which [[tempering out|tempers out]] the [[diesis]] (128/125) and the [[amity comma]] (1600000/1594323). We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]], which adds [[64/63]] and [[126/125]] to the list. [[Tempering out]] both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine [[11/1|11]], and adding it to consideration we find that the equal temperament tempers out [[99/98]] and [[121/120]] also. This choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}.
+edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400 cents of [[12edo]], we get a system which [[tempering out|tempers out]] the [[diesis]] (128/125) and the [[amity comma]] (1600000/1594323). We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]], which adds [[64/63]] and [[126/125]] to the list. [[Tempering out]] both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine [[11/1|11]], and adding it to consideration we find that the equal temperament tempers out [[99/98]] and [[121/120]] also. This choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}.
 A particular anecdote with this system was made in the ''Teliochordon'', in 1788 by {{w|Charles Clagget}} (Ireland, 1740?–1820), a little extract [http://ml.oxfordjournals.org/content/76/2/291.extract.jpg here].
@@ Line 439: / Line 439: @@
 == Notation ==
 === Ups and downs notation ===
-Using [[Helmholtz-Ellis notation|Helmholtz–Ellis]] accidentals, 39edo can also be notated using [[ups and downs notation]]:
+Using [[Helmholtz–Ellis]] accidentals, 39edo can also be notated using [[ups and downs notation]]:
 {{Sharpness-sharp5}}
 Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with three arrows.
-===Sagittal notation===
+=== Sagittal notation ===
 This notation uses the same sagittal sequence as [[46edo#Sagittal notation|46-EDO]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:39-EDO_Evo_Sagittal.svg
@@ Line 459: / Line 459: @@
 ====Revo flavor====
 <imagemap>
 File:39-EDO_Revo_Sagittal.svg
@@ Line 740: / Line 739: @@
 | {{monzo| 62 -39 }}
 | {{mapping| 39 62 }}
-| &minus;1.81
+| −1.81
 | 1.81
 | 5.88
@@ Line 747: / Line 746: @@
 | 128/125, 1594323/1562500
 | {{mapping| 39 62 91 }}
-| &minus;3.17
+| −3.17
 | 2.42
 | 7.89
@@ Line 754: / Line 753: @@
 | 64/63, 126/125, 2430/2401
 | {{mapping| 39 62 91 110 }} (39d)
-| &minus;3.78
+| −3.78
 | 2.35
 | 7.65
@@ Line 761: / Line 760: @@
 | 64/63, 99/98, 121/120, 126/125
 | {{mapping| 39 62 91 110 135 }} (39d)
-| &minus;3.17
+| −3.17
 | 2.43
 | 7.91
@@ Line 786: / Line 785: @@
 | 61.5
 | [[Unicorn]] (39d)
-| [[1L 18s]], [[19L 1s]]
+| [[1L&nbsp;18s]], [[19L&nbsp;1s]]
 |-
 | 1
@@ Line 792: / Line 791: @@
 | 123.1
 | [[Negri]] (39c)
-| [[1L 8s]], [[9L 1s]], [[10L 9s]], [[10L 19s]]
+| [[1L&nbsp;8s]], [[9L&nbsp;1s]], [[10L&nbsp;9s]], [[10L&nbsp;19s]]
 |-
 | 1
@@ Line 798: / Line 797: @@
 | 153.8
 |
-| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]], [[8L 23s]]
+| [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[8L&nbsp;7s]], [[8L&nbsp;15s]], [[8L&nbsp;23s]]
 |-
 | 1
@@ Line 804: / Line 803: @@
 | 215.4
 | [[Machine]] (39d)
-| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[11L 6s]], [[11L 17s]]
+| [[1L&nbsp;4s]], [[5L&nbsp;1s]], [[6L&nbsp;5s]], [[11L&nbsp;6s]], [[11L&nbsp;17s]]
 |-
 | 1
@@ Line 810: / Line 809: @@
 | 246.2
 | [[Immunity]] (39) / [[immunized]] (39d)
-| [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]], [[5L 19s]], [[5L 24s]], [[5L 29s]]
+| [[4L&nbsp;1s]], [[5L&nbsp;4s]], [[5L&nbsp;9s]], [[5L&nbsp;14s]], [[5L&nbsp;19s]], [[5L&nbsp;24s]], [[5L&nbsp;29s]]
 |-
 | 1
@@ Line 816: / Line 815: @@
 | 307.7
 | [[Familia]] (39df)
-| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]], [[4L 27s]], [[4L 31s]]
+| [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[4L&nbsp;7s]], [[4L&nbsp;11s]], [[4L&nbsp;15s]], [[4L&nbsp;19s]], [[4L&nbsp;23s]], [[4L&nbsp;27s]], [[4L&nbsp;31s]]
 |-
 | 1
@@ Line 822: / Line 821: @@
 | 338.5
 | [[Amity]] (39) / [[accord]] (39d)
-| [[3L 1s]], [[4L 3s]], [[7L 4s]], [[7L 11s]], [[7L 18s]], [[7L 25s]]
+| [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[7L&nbsp;4s]], [[7L&nbsp;11s]], [[7L&nbsp;18s]], [[7L&nbsp;25s]]
 |-
 | 1
@@ Line 828: / Line 827: @@
 | 430.8
 | [[Hamity]] (39df)
-| [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[14L 11s]]
+| [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[3L&nbsp;8s]], [[11L&nbsp;3s]], [[14L&nbsp;11s]]
 |-
 | 1
@@ Line 834: / Line 833: @@
 | 492.3
 | [[Quasisuper]] (39d)
-| [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]]
+| [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[5L&nbsp;12s]], [[17L&nbsp;5s]]
 |-
 | 1
@@ Line 840: / Line 839: @@
 | 523.1
 | [[Mavila]] (39bc)
-| [[2L 3s]], [[2L 5s]], [[7L 2s]], [[7L 9s]], [[16L 7s]]
+| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[7L&nbsp;2s]], [[7L&nbsp;9s]], [[16L&nbsp;7s]]
 |-
 | 1
@@ Line 846: / Line 845: @@
 | 584.6
 | [[Pluto]] (39d)
-| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]] etc. … [[2L 35s]]
+| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], [[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]] etc. … [[2L&nbsp;35s]]
 |-
 | 3
@@ Line 858: / Line 857: @@
 | 61.5
 |
-| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[3L 12s]], [[3L 15s]], [[18L 3s]]
+| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[3L&nbsp;9s]], [[3L&nbsp;12s]], [[3L&nbsp;15s]], [[18L&nbsp;3s]]
 |-
 | 3
@@ Line 864: / Line 863: @@
 | 184.6
 | [[Terrain]] / [[mirkat]] (39df)
-| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15]], [[6L 21s]], [[6L 27s]]
+| [[3L&nbsp;3s]], [[6L&nbsp;3s]], [[6L&nbsp;9s]], [[6L 15]], [[6L&nbsp;21s]], [[6L&nbsp;27s]]
 |-
 | 3
@@ Line 870: / Line 869: @@
 | 246.2<br />(153.8)
 | [[Triforce]] (39)
-| [[3L 3s]], [[6L 3s]], [[9L 6s]], [[15L 9s]]
+| [[3L&nbsp;3s]], [[6L&nbsp;3s]], [[9L&nbsp;6s]], [[15L&nbsp;9s]]
 |-
 | 3
@@ Line 876: / Line 875: @@
 | 492.3<br />(92.3)
 | [[Augene]] (39d)
-| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], [[12L 15s]]
+| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[3L&nbsp;9s]], [[12L&nbsp;3s]], [[12L&nbsp;15s]]
 |-
 | 3
@@ Line 882: / Line 881: @@
 | 523.1<br />(123.0)
 | [[Deflated]] (39bd)
-| [[3L 3s]], [[3L 6s]], [[9L 3s]], [[9L 12s]], [[9L 21s]]
+| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[9L&nbsp;3s]], [[9L&nbsp;12s]], [[9L&nbsp;21s]]
 |-
 | 13
@@ Line 888: / Line 887: @@
 | 492.3<br />(30.8)
 | [[Tridecatonic]]
-| [[13L 13s]]
+| [[13L&nbsp;13s]]
 |}
-<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
 == 39edo and world music ==
@@ Line 898: / Line 897: @@
 edo offers not one, but several different ways to realize the traditional Western diatonic scale. One way is to simply take a [[chain of fifths]] (the diatonic MOS: '''7 7 2 7 7 7 2'''). Because 39edo is a [[superpyth]] rather than a [[meantone]] system, this means that the harmonic quality of its diatonic scale will differ somewhat, since "minor" and "major" triads now approximate 6:7:9 and 14:18:21 respectively, rather than 10:12:15 and 4:5:6 as in meantone diatonic systems. Diatonic compositions translated onto this scale thus acquire a wildly different harmonic character, albeit still pleasing.
-Another option is to use a [[MODMOS]], such as '''7 6 3 7 6 7 3'''; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[Wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either '''7 3 3 3 7 3 3 7 3''' (a [[MODMOS]] of type [[3L 6s]]) or '''4 3 6 3 4 3 6 4 3 3.''' There are other MODMOS's that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.
+Another option is to use a [[MODMOS]], such as '''7 6 3 7 6 7 3'''; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[Wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either '''7 3 3 3 7 3 3 7 3''' (a [[MODMOS]] of type [[3L&nbsp;6s]]) or '''4 3 6 3 4 3 6 4 3 3.''' There are other MODMOS's that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.
 The MOS and the MODMOS's all have smaller-than-usual semitones, which makes them more effective for melody than their counterparts in 12edo or meantone systems.
@@ Line 905: / Line 904: @@
 === Indian ===
-A similar situation arises with [[Indian music]] since the sruti system, like the Western system, also has multiple possible mappings in 39edo. Many of these are modified versions of the [[17L 5s]] MOS (where the generator is a perfect fifth).
+A similar situation arises with [[Indian music]] since the sruti system, like the Western system, also has multiple possible mappings in 39edo. Many of these are modified versions of the [[17L&nbsp;5s]] MOS (where the generator is a perfect fifth).
 === Arabic, Turkish, Iranian ===
@@ Line 927: / Line 926: @@
 edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator.
-It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] MOS (which is '''9 7 7 9 7'''). [[Slendro]] can be approximated using this scale or using something like the [[quasi-equal]] '''8 8 8 8 7'''. A more expressive [[pentatonic]] scale is the oneirotonic subset '''9 6 9 9 6'''. Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.
+It also offers ''many'' possible [[pentatonic]] scales, including the [[2L&nbsp;3s]] MOS (which is '''9 7 7 9 7'''). [[Slendro]] can be approximated using this scale or using something like the [[quasi-equal]] '''8 8 8 8 7'''. A more expressive [[pentatonic]] scale is the oneirotonic subset '''9 6 9 9 6'''. Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.
 == Instruments ==