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{{interwiki
{{interwiki
| de =  
| de = 9edo
| en = 9edo
| en = 9edo
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| ja = 9平均律
| ja = 9平均律
}}
}}
'''9 EDO''' divides the octave into nine equal parts each of 133+1/3 [[cent]]s precisely.
{{Infobox ET}}
{{ED intro}}


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


The 9-EDO 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).


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


2: 7/6 266.871 septimal minor third
1: [[27/25]] 133.238 large limma, BP small semitone


3: 63/50 400.108 quasi-equal major third
2: [[7/6]] 266.871 septimal minor third


4: 49/36 533.742 Arabic lute acute fourth
3: [[63/50]] 400.108 quasi-equal major third


5: 72/49 666.258 Arabic lute grave fifth
4: [[49/36]] 533.742 Arabic lute acute fourth


6: 100/63 799.892 quasi-equal minor sixth
5: [[72/49]] 666.258 Arabic lute grave fifth


7: 12/7 933.129 septimal major sixth
6: [[100/63]] 799.892 quasi-equal minor sixth


8: 50/27 1066.762 grave major seventh
7: [[12/7]] 933.129 septimal major sixth


9: 2/1 1200.000 octave
8: [[50/27]] 1066.762 grave major seventh


Here the characterizations are taken from [http://en.wikipedia.org/wiki/Scala_%28program%29 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_subgroups|just intonation subgroup]] 2.27/25.7/3, which is closely related to 9EDO.
9: [[2/1]] 1200.000 octave


== Notation ==
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.
 
=== Odd harmonics ===
{{Harmonics in equal|9}}
 
=== Subsets and supersets ===
9edo is the first odd composite edo, containing [[3edo]] as a subset.


9edo 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.
The [[ennealimmal]] temperament contains 9edo as a subset (splitting 2/1 into 9 equal parts) and is excellent in the 7-limit. However, 9edo by itself tempers out 27/25 by [[Val|patent val]], rather than representing it as 1\9 like in ennealimmal, although the 9bccd val contains both the 27/25 and 7/6 representations above and therefore supports ennealimmal.


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".
== Notation ==
{{Mavila}}
In this notation, the [[enharmonic unison]] is the augmented 2nd, e.g. E♭ to F♯.


{| class="wikitable center-all right-1 right-2"
{| 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
! colspan="2" | Diatonic<br />Major narrower than minor
! Audio
|-
|-
| 0
| 0
| 0.00
| 0.00
| 1/1
|[[1/1]]
| perfect unison
| perfect unison
| D
| D
| perfect unison
| perfect unison
| D
| D
|[[File:0-0 unison.mp3|frameless]]
|-
|-
| 1
| 1
| 133.33
| 133.33
| 14/13, 13/12, 12/11
|[[14/13]] (+5.035), [[13/12]] (−5.239),<br />[[12/11]] (−17.304)
| minor 2nd
| minor 2nd
| E
| E
| major 2nd
| major 2nd
| E
| E
|[[File:0-133,33 minor second (9-EDO).mp3|frameless]]
|-
|-
| 2
| 2
| 266.67
| 266.67
| 7/6
|[[7/6]] (−0.204)
| major 2nd, minor 3rd
| major 2nd, minor 3rd
| E#, Fb
| E♯, F♭
| minor 2nd, major 3rd
| minor 2nd, major 3rd
| Eb, F#
| E♭, F♯
|[[File:0-266,67 major 2nd, minor 3rd (9-EDO).mp3|frameless]]
|-
|-
| 3
| 3
| 400.00
| 400.00
| 5/4, 14/11, 9/7
|[[5/4]] (+13.686), [[14/11]] (−17.508),<br />[[9/7]] (−35.084)
| major 3rd
| major 3rd
| F
| F
| minor 3rd
| minor 3rd
| F
| F
|[[File:0-400 major third (12-EDO).mp3|frameless]]
|-
|-
| 4
| 4
| 533.33
| 533.33
| 4/3, 11/8
|[[4/3]] (+35.288), [[11/8]] (−17.985)
| perfect 4th
| perfect 4th
| G
| G
| perfect 4th
| perfect 4th
| G
| G
|[[File:0-533,33 fourth (9-EDO).mp3|frameless]]
|-
|-
| 5
| 5
| 666.67
| 666.67
| 16/11, 3/2
|[[16/11]] (+17.985), [[3/2]] (−35.288)
| perfect 5th
| perfect 5th
| A
| A
| perfect 5th
| perfect 5th
| A
| A
|[[File:0-666,67 fifth (9-EDO).mp3|frameless]]
|-
|-
| 6
| 6
| 800.00
| 800.00
| 14/9, 11/7, 8/5
|[[14/9]] (+35.084) [[11/7]] (+17.508),<br />[[8/5]] (−13.686)
| minor 6th
| minor 6th
| B
| B
| major 6th
| major 6th
| B
| B
|[[File:0-800 minor sixth (12-EDO).mp3|frameless]]
|-
|-
| 7
| 7
| 933.33
| 933.33
| 12/7
|[[12/7]] (+0.204)
| major 6th, minor 7th
| major 6th, minor 7th
| B#, Cb
| B♯, C♭
| minor 6th, major 7th
| minor 6th, major 7th
| Bb, C#
| B♭, C♯
|[[File:0-933,33 major 6th, minor 7th (9-EDO).mp3|frameless]]
|-
|-
| 8
| 8
| 1066.67
| 1066.67
| 11/6, 13/7
|[[11/6]] (+17.304) [[13/7]] (−5.035)
| major 7th
| major 7th
| C
| C
| minor 7th
| minor 7th
| C
| C
|[[File:0-1066,67 major seventh (9-EDO).mp3|frameless]]
|-
|-
| 9
| 9
| 1200.00
| 1200.00
| 2/1
|[[2/1]]
| octave
| octave
| D
| D
| octave
| octave
| D
| D
|[[File:0-1200 octave.mp3|frameless]]
|}
|}


9EDO contains a pentatonic [[MOSScales|MOS scale]] -- 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].
=== 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|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 (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -14 9 }}
| {{mapping| 9 14 }}
| +11.13
| 11.24
| 8.35
|-
| 2.3.5
| 27/25, 128/125
| {{mapping| 9 14 21 }}
| +5.36
| 12.18
| 9.10
|-
| 2.3.5.7
| 21/20, 36/35, 49/48
| {{mapping| 9 14 21 25 }}
| +7.20
| 11.02
| 8.21
|-
| 2.3.5.7.11
| 21/20, 33/32, 36/35, 45/44
| {{mapping| 9 14 21 25 31 }}
| +6.80
| 9.89
| 7.37
|}


== Commas ==
=== Uniform maps ===
{{Uniform map|edo=9}}


9 EDO tempers out the following commas. (Note: This assumes val &lt; 9 14 21 25 31 33 |.)
=== Commas ===
9et [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 9 14 21 25 31 33 }}.


{| class="wikitable centerl-all left-2 right-3"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Ratios|Ratio]]
! [[Harmonic limit|Prime<br />limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
! [[Color notation/Temperament Names|Color Name]]
! [[Color name]]
! Name 1
! Name
! Name 2
|-
! Name 3
| 3
| [[19683/16384]]
| {{monzo| -14 9 }}
| 317.59
| Lawa 2nd
| Pythagorean augmented second
|-
|-
| 27/25
| 5
| {{Monzo| 0 3 -2 }}
| [[27/25]]
| {{monzo| 0 3 -2 }}
| 133.24
| 133.24
| Gugu
| Gugu
| Large Limma
| Bug comma, large limma
| Large Semitone
| Bug Comma
|-
|-
| 135/128
| 5
| {{Monzo| -7 3 1 }}
| [[135/128]]
| {{monzo| -7 3 1 }}
| 92.18
| 92.18
| Layobi
| Layobi
| Major Chroma
| Mavila comma, major chroma
| Major Limma
| Pelogic Comma
|-
|-
| 16875/16384
| 5
| {{Monzo| -14 3 4 }}
| [[16875/16384]]
| {{monzo| -14 3 4 }}
| 51.12
| 51.12
| Laquadyo
| Laquadyo
| Negri Comma
| Negri comma
| Double Augmentation Diesis
|
|-
|-
| 128/125
| 5
| {{Monzo| 7 0 -3 }}
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| 41.06
| Trigu
| Trigu
| Diesis
| Augmented comma, lesser diesis
| Augmented Comma
|
|-
|-
| 2109375/2097152
| 5
| {{Monzo| -21 3 7 }}
| [[2109375/2097152|(14 digits)]]
| {{monzo| -21 3 7 }}
| 10.06
| 10.06
| Lasepyo
| Lasepyo
| Semicomma
| [[Semicomma]]
| Fokker Comma
|
|-
|-
| 36/35
| 7
| {{Monzo| 2 2 -1 -1 }}
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| 48.77
| Rugu
| Rugu
| Septimal Quarter Tone
| Mint comma, septimal quarter tone
|
|
|-
|-
| 525/512
| 7
| {{Monzo| -9 1 2 1 }}
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicenna
| Avicennma
| Avicenna's Enharmonic Diesis
|
|-
|-
| 7
| 49/48
| 49/48
| {{Monzo| -4 -1 0 2 }}
| {{monzo| -4 -1 0 2 }}
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro Diesis
| Semaphoresma, slendro diesis
|
|
|-
|-
| 686/675
| 7
| {{Monzo| 1 -3 -2 3 }}
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| 27.99
| Trizo-agugu
| Trizo-agugu
| Senga
| Senga
|
|
|-
|-
| 2430/2401
| 7
| {{Monzo| 1 5 1 -4 }}
| [[2430/2401]]
| {{monzo| 1 5 1 -4 }}
| 20.79
| 20.79
| Quadru-ayo
| Quadru-ayo
| Nuwell
| Nuwell comma
|
|
|-
|-
| 1728/1715
| 7
| {{Monzo| 6 3 -1 -3 }}
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| 13.07
| 13.07
| Triru-agu
| Triru-agu
| Orwellisma
| Orwellisma
| Orwell Comma
|
|-
|-
| 225/224
| 7
| {{Monzo| -5 2 2 -1 }}
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
| Septimal Kleisma
| Marvel comma
| Marvel Comma
|
|-
|-
| 6144/6125
| 7
| {{Monzo| 11 1 -3 -2 }}
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| 5.36
| Sarurutrigu
| Sarurutrigu
| Porwell
| Porwell comma
|
|
|-
|-
| 65625/65536
| 7
| {{Monzo| -16 1 5 1 }}
| [[65625/65536]]
| {{monzo| -16 1 5 1 }}
| 2.35
| 2.35
| Lazoquinyo
| Lazoquinyo
| Horwell
| Horwell comma
|
|
|-
|-
| 99/98
| 7
| {{Monzo| -1 2 0 -2 1 }}
| <abbr title="40353607/40310784">(16 digits)</abbr>
| {{monzo| -11 -9 0 9 }}
| 1.84
| Tritrizo
| [[Septimal ennealimma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruru
| Mothwellsma
| Mothwellsma
|
|
|-
|-
| 121/120
| 11
| {{Monzo| -3 -1 -1 0 2 }}
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
| Lologu
| Lologu
| Biyatisma
| Biyatisma
|
|
|-
|-
| 176/175
| 11
| {{Monzo| 4 0 -2 -1 1 }}
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Lorugugu
| Valinorsma
| Valinorsma
|
|
|-
|-
| 385/384
| 11
| {{Monzo| -7 -1 1 1 1 }}
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|
|
|-
|-
| 540/539
| 11
| {{Monzo| 2 3 1 -2 -1 }}
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| 3.21
| Lururuyo
| Lururuyo
| Swetisma
| Swetisma
|
|
|-
|-
| 91/90
| 13
| {{Monzo| -1 -2 -1 1 0 1 }}
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|
|
|-
|-
| 676/675
| 13
| {{Monzo| 2 -3 -2 0 0 2 }}
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| 2.56
| Bithogu
| Bithogu
| Parizeksma
| Island comma
|
|
|}
|}


== Images ==
=== Rank-2 temperaments ===
9edo 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 ==
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 ==
[[File:9edo_wheel.png|alt=9edo wheel.png|385x385px|9edo wheel.png]]
[[File:9edo_wheel.png|alt=9edo wheel.png|385x385px|9edo wheel.png]]


== Instruments ==
== Instruments ==
[[File:IMG_2223-800x600.jpg|alt=IMG_2223-800x600.jpg|400px|IMG_2223-800x600.jpg]]
[[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 9 EDO with 40 lb. test fishing line (by cenobyte)
* 9edo can be played on the Lumatone, see [[Lumatone mapping for 9edo]]
 
== Music ==
== Music ==
{{Main|Music in 9edo}}


* [https://soundcloud.com/overtoneshock/tencaious-chorale-9-edo-studio-version Tenacious Chorale (only movement I is in 9-EDO)] by [[Stephen Weigel]]
== See also ==
* [https://soundcloud.com/overtoneshock/in-our-own-little-worlds-9-edo Zones of Lasting Novelty] (Un12 2019) by [[Stephen Weigel]]; perf. [[Hans Gunter-Lock]], [[Jacob Barton]], and Stephen Weigel
* [https://soundcloud.com/overtoneshock/gamelan-genesis-and-birth-9-edo Gamelan, Origin, Creation] by Stephen Weigel ([http://www.beostringquartet.com/ Beo String Quartet], dedicated to [[wikipedia:Lou_Harrison|Lou Harrison]])
* [https://tabytha.bandcamp.com/track/69 69] by [https://tabytha.bandcamp.com/album/bad-musick Tabytha]
* [https://tabytha.bandcamp.com/track/69-pentangled 69 Pentangled] by [https://tabytha.bandcamp.com/album/bad-musick Tabytha]
 
* Nocturne in 9tet by [http://home.snafu.de/djwolf/WorksDescriptive.htm Daniel Wolf]
* ''[http://www.h-pi.com/mp3/Prelude9ET.mp3 Prelude in 9ET]'' by [[Aaron Andrew Hunt]]
* ''[http://micro.soonlabel.com/9-edo/daily20110629_fts_e_guit_9et.mp3 Improvisation for Electric Guitar in 9 edo]'' by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/08%20-%208.%209%20octave.mp3 Comets Over Flatland 8] by [[Randy Winchester]]
* [http://www.youtube.com/watch?v=bDFCsCoaUO4 Nine tones per Octave (9-EDO / 9-TET)] by [[Ivor Darreg]]
* [http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3 Gerbils at the Wheel of Government] by [[Chris Vaisvil]] (in 9 and 18 edo simultaneously)
* [http://www.seraph.it/dep/det/NewWorld.mp3 New World] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/f533be803cb9ed1efc23fc9e2db10c6f-167.html blog entry])
* [https://soundcloud.com/santiagocosentino/interdimensional-train-ride Interdimensional Train Ride by Santiago Cosentino]


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


9 EDO ear-training exercises by Alex Ness available [https://drive.google.com/a/playgroundsessions.com/folderview?id=0BwsXD8q2VCYUamtVWEgyRFA5alU&usp=sharing#list here].
=== Werntz Nocturne scale ===
{{main|Werntz Nocturne scale}}


[[Category:9-tone]]
== Notes ==
[[Category:9edo]]
<references group="note" />
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]


[[Category:Todo:unify precision]]
[[Category:9-tone scales]]
[[Category:Pelog]]
Retrieved from "https://en.xen.wiki/w/9edo"