9edo: Difference between revisions

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{{interwiki

| de =

| de = 9edo

| en = 9edo

| es =

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}}

== Theory ==

[[File:9edo scale.mp3|thumb|A chromatic 9edo scale on C.]]

~~The~~ 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly. A 7-limit version of 9edo goes

9edo is the most basic tuning which supports an [[antidiatonic]] scale. Its fifth is considerably flatter than just, but still falls into the category of "fifth" despite this. 9edo is also the first edo to have distinct major and minor chords (if 5edo's tendo and arto chords are ignored).

9edo splits the octave into three parts, each representing the major third 5/4, similarly to 12edo, which is of moderate accuracy. A similarly crude approximation of 11/8 (a sharp fourth) is available at the perfect fourth of 4 steps, which means 9edo can be seen as a simple 2.5.11 system. Looking at the intervals in this subgroup, the submajor second 11/10 is tuned to 133 cents (extremely flat) and 25/22 is even worse (but still consistent); the supermajor sixth 55/32 is tuned very accurately at 933 cents (only slightly flat). Overall, 9edo is not a great system for approximating low-complexity JI intervals consistently. However, if we turn to inconsistent representations, we see quite a few options before us. In particular, the 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly, but not the harmonic 7/4 (a subminor seventh) itself (unless [[semaphore]], which equates it with the supermajor sixth 12/7, is taken as an acceptable temperament in this tuning). A 7-limit version of 9edo goes

1: [[27/25]] 133.238 large limma, BP small semitone

Line 31:

Line 33:

9: [[2/1]] 1200.000 octave

~~Here the characterizations are taken from [[Scala]], which also describes the scale itself as "Pelog Nawanada: Sunda".~~ Chords such as 1/1 - 7/6 - 49/36 - 12/7 are therefore natural ones for 9edo. The above scale generates the [[just intonation subgroup]] 2.27/25.7/3, which is closely related to 9edo.

Chords such as {{dash|1/1, 7/6, 49/36, 12/7|med}} are therefore natural ones for 9edo. The above scale generates the [[just intonation subgroup]] 2.27/25.7/3, which is closely related to 9edo.

9edo has very little to offer in terms of accuracy for harmonics. Most other systems that lack good perfect fifths such as [[6edo]], [[11edo]], [[13edo]] and [[18edo]] at least contain a reasonable approximation of 9/8 (or (3/2)2), so they can be viewed as temperaments in subgroups like 2.9.5 and 2.9.7. 9edo, on the other hand, completely misses both 3/2 and 9/8. You could then try to treat 9edo as an entirely no-threes system without any 3/2 or 9/8, in a subgroup like 2.5.7.11, but even here 9edo performs somewhat poorly, because its best [[7/4]] is much closer to 12/7 and is off by 36 cents, while its best [[11/8]] is off by 18 cents. The 13th harmonic is also entirely missed by 9edo. This being said, 9edo does approximate [[47/32]] to within about 1.2 cents.

~~9edo's fifth of 5\9 is near the boundary of "perfect fifth" and "subfifth" so it sounds quite dirty but still recognizable.~~

=== Odd harmonics ===

Line 45:

Line 44:

== Notation ==

~~9edo can be notated with conventional~~ notation, ~~including~~ the ~~staff, note names, relative notation, etc. in two ways. The first preserves~~ the ~~melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat~~, ~~and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works.~~ e.g. ~~M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.~~

In this notation, the [[enharmonic unison]] is the augmented 2nd, e.g. E♭ to F♯.

The second approach preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to ~~9edo "on the fly"~~.

{| class="wikitable center-all right-1 right-2"

! [[degree]]

|-

! [[cent]]s

![[degree]]

! Approximate Ratios

![[cent]]s

! colspan="2" | ~~Melodic notation~~ Major wider than minor

! Approximate Ratios

! colspan="2" | ~~Harmonic notation~~ Major narrower than minor

! colspan="2" | Antidiatonic Major wider than minor

!Audio

! colspan="2" | Diatonic Major narrower than minor

! Audio

|-

| 0

| 0.00

| [[1/1]]

|[[1/1]]

| perfect unison

| D

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| 1

| 133.33

| [[14/13]] (+5.035), [[13/12]] (-5.239),

|[[14/13]] (+5.035), [[13/12]] (−5.239), [[12/11]] (−17.304)

[[12/11]] (~~-17~~.304)

| minor 2nd

| E

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| 2

| 266.67

| [[7/6]] (-0.204)

|[[7/6]] (−0.204)

| major 2nd, minor 3rd

| E#, Fb

| E♯, F♭

| minor 2nd, major 3rd

| Eb, F#

| E♭, F♯

|[[File:0-266,67 major 2nd, minor 3rd (9-EDO).mp3|frameless]]

|-

| 3

| 400.00

| [[5/4]] (+13.686), [[14/11]] (~~-17~~.508),

|[[5/4]] (+13.686), [[14/11]] (−17.508), [[9/7]] (−35.084)

[[9/7]] (~~-35~~.084)

| major 3rd

| F

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| 4

| 533.33

| [[4/3]] (+35.288), [[11/8]] (~~-17~~.985)

|[[4/3]] (+35.288), [[11/8]] (−17.985)

| perfect 4th

| G

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| 5

| 666.67

| [[16/11]] (+17.985), [[3/2]] (~~-35~~.288)

|[[16/11]] (+17.985), [[3/2]] (−35.288)

| perfect 5th

| A

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| 6

| 800.00

| [[14/9]] (+35.084) [[11/7]] (+17.508),

|[[14/9]] (+35.084) [[11/7]] (+17.508), [[8/5]] (−13.686)

[[8/5]] (~~-13~~.686)

| minor 6th

| B

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| 7

| 933.33

| [[12/7]] (+0.204)

|[[12/7]] (+0.204)

| major 6th, minor 7th

| B#, Cb

| B♯, C♭

| minor 6th, major 7th

| Bb, C#

| B♭, C♯

|[[File:0-933,33 major 6th, minor 7th (9-EDO).mp3|frameless]]

|-

| 8

| 1066.67

| [[11/6]] (+17.304) [[13/7]] (-5.035)

|[[11/6]] (+17.304) [[13/7]] (−5.035)

| major 7th

| C

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| 9

| 1200.00

| [[2/1]]

|[[2/1]]

| octave

| D

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|[[File:0-1200 octave.mp3|frameless]]

|}

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[14edo#Sagittal notation|14-EDO]].

File:9-EDO_Sagittal.svg

desc none

rect 80 0 296 50 [[Sagittal_notation]]

rect 296 0 456 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 296 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation |limma-fraction notation]]

default [[File:9-EDO_Sagittal.svg]]

</imagemap>

== Approximation to JI ==

=== Selected just intervals ===

[[File:9ed2-001.svg|alt=alt : Your browser has no SVG support.]]

~~[[:File:9ed2-001.svg|9ed2-001.svg]]~~

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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=== Uniform maps ===

=== Commas ===

~~9edo~~ [[tempers out]] the following [[comma]]s. ~~(Note:~~ This assumes [[val]] {{val| 9 14 21 25 31 33 }}.)

9et [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 9 14 21 25 31 33 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"

|-

! [[Harmonic limit|Prime limit]]

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref>~~Ratios longer than 10 digits are presented by placeholders with informative hints~~</ref>

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

! [[Cent]]s

Line 373:

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| Island comma

|}

~~<references/>~~

=== Rank-2 temperaments ===

9edo contains a pentatonic [[mos scale]] produced by stacking 4/9 of [[2L 3s]] (1 3 1 3 1) ~~– with~~ a heptatonic extension – [[2L 5s]] (1 1 2 1 1 2 1, sometimes called "mavila" or "antidiatonic"). Indonesian pelog scales sometimes use five-tone subsets of a seven-tone superset in a similar way, and it has been suggested that Indonesian gamelan music stems from a [http://www.neuroscience-of-music.se/pelog%20historical.htm 9edo tradition]. ~~You can also use~~ the 2/9, ~~which generates mos scales~~ of [[~~1L 3s~~]] (3 ~~2 2 2)~~ and [[~~4L 1s~~]] (2 ~~2 2 2 1) and can be interpreted~~ as ~~either an extremely sharp~~ [[~~bug~~]] ~~scale or an extremely flat~~ [[~~orwell~~]] ~~one~~.

9edo contains a pentatonic [[mos scale]] produced by stacking 4\9 of [[2L 3s]] (1 3 1 3 1), which has a heptatonic extension, [[2L 5s]] (1 1 2 1 1 2 1, sometimes called "mavila" or "antidiatonic").

You can also use 2\9, which generates mos scales of [[1L 3s]] (3 2 2 2) and [[4L 1s]] (2 2 2 2 1) and can be interpreted as either an extremely sharp [[bug]] scale or an extremely flat [[orwell]] one.

== Historical (and other) relevance ==

[[Indonesian]] pelog scales sometimes use five-tone subsets of a seven-tone superset in a similar way as the 5-tone and 7-tone mavila scale (see [[#Rank-2 temperaments|Rank-2 temperaments]]), and it has been suggested that Indonesian gamelan music stems from a [http://www.neuroscience-of-music.se/pelog%20historical.htm 9edo tradition].

As a division of the octave into 32 parts, i. e. a dominant position of the number 3, 9edo also has some suitability as base tuning for [https://en.wikipedia.org/wiki/Klingon Klingon] music (since the tradtional Klingon number system is also based on 3). See, for this:

[http://%5B%5Bhttps://www.youtube.com/watch?v=1LjcBv-OWtQ%5D%5D Levi McClain, Klingon music theory is weird]

== Octave stretch or compression ==

9edo's [[prime]]s 3, 7, 11 and 13 are all tuned flat, so it can benefit from [[octave stretching]].

Pure-octaves 9edo makes a decent 2.5.11 tuning, approximating all those three primes within 18{{c}}.

9edo with octaves stretched about 5{{c}}, as in [[zpi|22zpi]], makes a decent 2.7.11.13 tuning, approximating all those four primes within 17{{c}}.

9edo with octaves stretched about 10{{c}}, as in [[ed12|32ed12]], makes a decent 2.3.7.11.13 tuning, approximating all those five primes within 20{{c}}.

== Diagrams ==

Line 383:

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== Instruments ==

[[File:IMG_2223-800x600.jpg|alt=IMG_2223-800x600.jpg|400px|IMG_2223-800x600.jpg]]

* Ukulele (MicroUke 1.2) set to 9edo with 40 lb. test fishing line (by cenobyte)

~~Ukulele (MicroUke 1.2) set to~~ 9edo ~~with 40 lb. test fishing line (by cenobyte)~~

* 9edo can be played on the Lumatone, see [[Lumatone mapping for 9edo]]

== Music ==

== Ear training ==

== See also ==

=== Ear training ===

* [https://drive.google.com/a/playgroundsessions.com/folderview?id=0BwsXD8q2VCYUamtVWEgyRFA5alU&usp=sharing#list 9edo ear-training exercises] by [[Alex Ness]].

=== Werntz Nocturne scale ===

== Notes ==

[[Category:9-tone scales]]

[[Category:~~Macrotonal~~]]

[[Category:Pelog]]

@@ Line 1: / Line 1: @@
 {{interwiki
-| de =
+| de = 9edo
 | en = 9edo
 | es =
@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|9}}
+{{ED intro}}
 == Theory ==
 [[File:9edo scale.mp3|thumb|A chromatic 9edo scale on C.]]
-The 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly. A 7-limit version of 9edo goes
+edo is the most basic tuning which supports an [[antidiatonic]] scale. Its fifth is considerably flatter than just, but still falls into the category of "fifth" despite this. 9edo is also the first edo to have distinct major and minor chords (if 5edo's tendo and arto chords are ignored).
+edo splits the octave into three parts, each representing the major third 5/4, similarly to 12edo, which is of moderate accuracy. A similarly crude approximation of 11/8 (a sharp fourth) is available at the perfect fourth of 4 steps, which means 9edo can be seen as a simple 2.5.11 system. Looking at the intervals in this subgroup, the submajor second 11/10 is tuned to 133 cents (extremely flat) and 25/22 is even worse (but still consistent); the supermajor sixth 55/32 is tuned very accurately at 933 cents (only slightly flat). Overall, 9edo is not a great system for approximating low-complexity JI intervals consistently. However, if we turn to inconsistent representations, we see quite a few options before us. In particular, the 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly, but not the harmonic 7/4 (a subminor seventh) itself (unless [[semaphore]], which equates it with the supermajor sixth 12/7, is taken as an acceptable temperament in this tuning). A 7-limit version of 9edo goes
 : [[27/25]] 133.238 large limma, BP small semitone
@@ Line 31: / Line 33: @@
 : [[2/1]] 1200.000 octave
-Here the characterizations are taken from [[Scala]], which also describes the scale itself as "Pelog Nawanada: Sunda". Chords such as 1/1 - 7/6 - 49/36 - 12/7 are therefore natural ones for 9edo. The above scale generates the [[just intonation subgroup]] 2.27/25.7/3, which is closely related to 9edo.
+Chords such as {{dash|1/1, 7/6, 49/36, 12/7|med}} are therefore natural ones for 9edo. The above scale generates the [[just intonation subgroup]] 2.27/25.7/3, which is closely related to 9edo.
-edo has very little to offer in terms of accuracy for harmonics. Most other systems that lack good perfect fifths such as [[6edo]], [[11edo]], [[13edo]] and [[18edo]] at least contain a reasonable approximation of 9/8 (or (3/2)<sup>2</sup>), so they can be viewed as temperaments in subgroups like 2.9.5 and 2.9.7. 9edo, on the other hand, completely misses both 3/2 and 9/8. You could then try to treat 9edo as an entirely no-threes system without any 3/2 or 9/8, in a subgroup like 2.5.7.11, but even here 9edo performs somewhat poorly, because its best [[7/4]] is much closer to 12/7 and is off by 36 cents, while its best [[11/8]] is off by 18 cents. The 13th harmonic is also entirely missed by 9edo. This being said, 9edo does approximate [[47/32]] to within about 1.2 cents.
-edo's fifth of 5\9 is near the boundary of "perfect fifth" and "subfifth" so it sounds quite dirty but still recognizable.
 === Odd harmonics ===
 {{Harmonics in equal|9}}
@@ Line 45: / Line 44: @@
 == Notation ==
-edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
+{{Mavila}}
+In this notation, the [[enharmonic unison]] is the augmented 2nd, e.g. E♭ to F♯.
-The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 9edo "on the fly".
 {| class="wikitable center-all right-1 right-2"
-! [[degree]]
+|-
-! [[cent]]s
+![[degree]]
-! Approximate <br>Ratios
+![[cent]]s
-! colspan="2" | Melodic notation <br> Major wider than minor
+! Approximate<br />Ratios
-! colspan="2" | Harmonic notation <br> Major narrower than minor
+! colspan="2" | Antidiatonic<br />Major wider than minor
-!Audio
+! colspan="2" | Diatonic<br />Major narrower than minor
+! Audio
 |-
 | 0
 | 0.00
-| [[1/1]]
+|[[1/1]]
 | perfect unison
 | D
@@ Line 68: / Line 67: @@
 | 1
 | 133.33
-| [[14/13]] (+5.035), [[13/12]] (-5.239),
+|[[14/13]] (+5.035), [[13/12]] (−5.239),<br />[[12/11]] (−17.304)
-[[12/11]] (-17.304)
 | minor 2nd
 | E
@@ Line 78: / Line 76: @@
 | 2
 | 266.67
-| [[7/6]] (-0.204)
+|[[7/6]] (−0.204)
 | major 2nd, minor 3rd
-| E#, Fb
+| E♯, F♭
 | minor 2nd, major 3rd
-| Eb, F#
+| E♭, F♯
 |[[File:0-266,67 major 2nd, minor 3rd (9-EDO).mp3|frameless]]
 |-
 | 3
 | 400.00
-| [[5/4]] (+13.686), [[14/11]] (-17.508),
+|[[5/4]] (+13.686), [[14/11]] (−17.508),<br />[[9/7]] (−35.084)
-[[9/7]] (-35.084)
 | major 3rd
 | F
@@ Line 97: / Line 94: @@
 | 4
 | 533.33
-| [[4/3]] (+35.288), [[11/8]] (-17.985)
+|[[4/3]] (+35.288), [[11/8]] (−17.985)
 | perfect 4th
 | G
@@ Line 106: / Line 103: @@
 | 5
 | 666.67
-| [[16/11]] (+17.985), [[3/2]] (-35.288)
+|[[16/11]] (+17.985), [[3/2]] (−35.288)
 | perfect 5th
 | A
@@ Line 115: / Line 112: @@
 | 6
 | 800.00
-| [[14/9]] (+35.084) [[11/7]] (+17.508),
+|[[14/9]] (+35.084) [[11/7]] (+17.508),<br />[[8/5]] (−13.686)
-[[8/5]] (-13.686)
 | minor 6th
 | B
@@ Line 125: / Line 121: @@
 | 7
 | 933.33
-| [[12/7]] (+0.204)
+|[[12/7]] (+0.204)
 | major 6th, minor 7th
-| B#, Cb
+| B♯, C♭
 | minor 6th, major 7th
-| Bb, C#
+| B♭, C♯
 |[[File:0-933,33 major 6th, minor 7th (9-EDO).mp3|frameless]]
 |-
 | 8
 | 1066.67
-| [[11/6]] (+17.304) [[13/7]] (-5.035)
+|[[11/6]] (+17.304) [[13/7]] (−5.035)
 | major 7th
 | C
@@ Line 143: / Line 139: @@
 | 9
 | 1200.00
-| [[2/1]]
+|[[2/1]]
 | octave
 | D
@@ Line 150: / Line 146: @@
 |[[File:0-1200 octave.mp3|frameless]]
 |}
+=== Sagittal notation ===
+This notation uses the same sagittal sequence as [[14edo#Sagittal notation|14-EDO]].
+<imagemap>
+File:9-EDO_Sagittal.svg
+desc none
+rect 80 0 296 50 [[Sagittal_notation]]
+rect 296 0 456 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 296 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation |limma-fraction notation]]
+default [[File:9-EDO_Sagittal.svg]]
+</imagemap>
 == Approximation to JI ==
 === Selected just intervals ===
 [[File:9ed2-001.svg|alt=alt : Your browser has no SVG support.]]
-[[:File:9ed2-001.svg|9ed2-001.svg]]
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 198: / Line 205: @@
 === Uniform maps ===
-{{Uniform map|13|8.5|9.5}}
+{{Uniform map|edo=9}}
 === Commas ===
-edo [[tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 9 14 21 25 31 33 }}.)
+et [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 9 14 21 25 31 33 }}.
 {| class="commatable wikitable center-all left-3 right-4 left-6"
 |-
-! [[Harmonic limit|Prime<br>limit]]
+! [[Harmonic limit|Prime<br />limit]]
-! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
+! [[Ratio]]<ref group="note">{{rd}}</ref>
 ! [[Monzo]]
 ! [[Cent]]s
@@ Line 373: / Line 380: @@
 | Island comma
 |}
-<references/>
 === Rank-2 temperaments ===
-edo contains a pentatonic [[mos scale]] produced by stacking 4/9 of [[2L 3s]] (1 3 1 3 1) – with a heptatonic extension – [[2L 5s]] (1 1 2 1 1 2 1, sometimes called "mavila" or "antidiatonic"). Indonesian pelog scales sometimes use five-tone subsets of a seven-tone superset in a similar way, and it has been suggested that Indonesian gamelan music stems from a [http://www.neuroscience-of-music.se/pelog%20historical.htm 9edo tradition]. You can also use the 2/9, which generates mos scales of [[1L 3s]] (3 2 2 2) and [[4L 1s]] (2 2 2 2 1) and can be interpreted as either an extremely sharp [[bug]] scale or an extremely flat [[orwell]] one.
+edo contains a pentatonic [[mos scale]] produced by stacking 4\9 of [[2L&nbsp;3s]] (1 3 1 3 1), which has a heptatonic extension, [[2L&nbsp;5s]] (1 1 2 1 1 2 1, sometimes called "mavila" or "antidiatonic").
+You can also use 2\9, which generates mos scales of [[1L&nbsp;3s]] (3 2 2 2) and [[4L&nbsp;1s]] (2 2 2 2 1) and can be interpreted as either an extremely sharp [[bug]] scale or an extremely flat [[orwell]] one.
+== Historical (and other) relevance ==
+[[Indonesian]] pelog scales sometimes use five-tone subsets of a seven-tone superset in a similar way as the 5-tone and 7-tone mavila scale (see [[#Rank-2 temperaments|Rank-2 temperaments]]), and it has been suggested that Indonesian gamelan music stems from a [http://www.neuroscience-of-music.se/pelog%20historical.htm 9edo tradition].
+As a division of the octave into 3<sup>2</sup> parts, i. e. a dominant position of the number 3, 9edo also has some suitability as base tuning for [https://en.wikipedia.org/wiki/Klingon Klingon] music (since the tradtional Klingon number system is also based on 3). See, for this:
+[http://%5B%5Bhttps://www.youtube.com/watch?v=1LjcBv-OWtQ%5D%5D Levi McClain, Klingon music theory is weird]
+== Octave stretch or compression ==
+edo's [[prime]]s 3, 7, 11 and 13 are all tuned flat, so it can benefit from [[octave stretching]].
+Pure-octaves 9edo makes a decent 2.5.11 tuning, approximating all those three primes within 18{{c}}.
+edo with octaves stretched about 5{{c}}, as in [[zpi|22zpi]], makes a decent 2.7.11.13 tuning, approximating all those four primes within 17{{c}}.
+edo with octaves stretched about 10{{c}}, as in [[ed12|32ed12]], makes a decent 2.3.7.11.13 tuning, approximating all those five primes within 20{{c}}.
 == Diagrams ==
@@ Line 383: / Line 408: @@
 == Instruments ==
 [[File:IMG_2223-800x600.jpg|alt=IMG_2223-800x600.jpg|400px|IMG_2223-800x600.jpg]]
+* Ukulele (MicroUke 1.2) set to 9edo with 40 lb. test fishing line (by cenobyte)
-Ukulele (MicroUke 1.2) set to 9edo with 40 lb. test fishing line (by cenobyte)
+* 9edo can be played on the Lumatone, see [[Lumatone mapping for 9edo]]
 == Music ==
 {{Main|Music in 9edo}}
-== Ear training ==
+== See also ==
+=== Ear training ===
 * [https://drive.google.com/a/playgroundsessions.com/folderview?id=0BwsXD8q2VCYUamtVWEgyRFA5alU&usp=sharing#list 9edo ear-training exercises] by [[Alex Ness]].
+=== Werntz Nocturne scale ===
+{{main|Werntz Nocturne scale}}
+== Notes ==
+<references group="note" />
 [[Category:9-tone scales]]
-[[Category:Macrotonal]]
+[[Category:Pelog]]