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{{interwiki
{{interwiki
| de =  
| de = 9edo
| en = 9edo
| en = 9edo
| es =  
| es =  
| ja = 9平均律
| ja = 9平均律
}}
}}
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3<sup>2</sup>
{{ED intro}}
| Step size = 133.333¢
| Fifth = 5\9 (667¢)
| Major 2nd = 1\9 (133¢)
| Consistency = 7
}}
'''9 equal divisions of the octave''' ('''9edo''') is the [[tuning system]] derived by dividing the [[octave]] into 9 equal steps of 133+1/3 [[cent]]s each precisely. It is also the first odd composite edo.


== Theory ==
== Theory ==
{{Harmonics in equal|9}}
[[File:9edo scale.mp3|thumb|A chromatic 9edo scale on C.]]
[[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


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


9: 2/1 1200.000 octave
=== Odd harmonics ===
{{Harmonics in equal|9}}


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.
=== Subsets and supersets ===
9edo is the first odd composite edo, containing [[3edo]] as a subset.  
 
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.


== Notation ==
== Notation ==
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.
{{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"
{| 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
| 0.00
| 0.00
| 1/1
|[[1/1]]
| perfect unison
| perfect unison
| D
| D
Line 63: Line 67:
| 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
Line 72: Line 76:
| 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]]
|[[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
Line 90: Line 94:
| 4
| 4
| 533.33
| 533.33
| 4/3, 11/8
|[[4/3]] (+35.288), [[11/8]] (−17.985)
| perfect 4th
| perfect 4th
| G
| G
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| 5
| 5
| 666.67
| 666.67
| 16/11, 3/2
|[[16/11]] (+17.985), [[3/2]] (−35.288)
| perfect 5th
| perfect 5th
| A
| A
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| 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
Line 117: Line 121:
| 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]]
|[[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
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| 9
| 9
| 1200.00
| 1200.00
| 2/1
|[[2/1]]
| octave
| octave
| D
| D
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|}
|}


== Commas ==
=== Sagittal notation ===
9edo [[tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 9 14 21 25 31 33 }}.)
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.]]
 
== 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
|}
 
=== Uniform maps ===
{{Uniform map|edo=9}}
 
=== Commas ===
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"
{| 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]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
! [[Color name]]
! [[Color name]]
! Name
! Name
|-
| 3
| [[19683/16384]]
| {{monzo| -14 9 }}
| 317.59
| Lawa 2nd
| Pythagorean augmented second
|-
|-
| 5
| 5
Line 160: Line 231:
| 133.24
| 133.24
| Gugu
| Gugu
| Large limma
| Bug comma, large limma
|-
|-
| 5
| 5
Line 167: Line 238:
| 92.18
| 92.18
| Layobi
| Layobi
| Major chroma
| Mavila comma, major chroma
|-
|-
| 5
| 5
Line 181: Line 252:
| 41.06
| 41.06
| Trigu
| Trigu
| Diesis
| Augmented comma, lesser diesis
|-
|-
| 5
| 5
Line 195: Line 266:
| 48.77
| 48.77
| Rugu
| Rugu
| Septimal quartertone
| Mint comma, septimal quarter tone
|-
|-
| 7
| 7
Line 209: Line 280:
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
Line 300: Line 371:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 309: Line 380:
| Island comma
| Island comma
|}
|}
<references/>


== Linear temperaments ==
=== Rank-2 temperaments ===
9edo contains a pentatonic [[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].
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").  


== JI approximation ==
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.
=== Selected just intervals ===
 
[[File:9ed2-001.svg|alt=alt : Your browser has no SVG support.]]
== 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}}.


[[:File:9ed2-001.svg|9ed2-001.svg]]
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 ==
== Diagrams ==
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== 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 9edo with 40 lb. test fishing line (by cenobyte)
* 9edo can be played on the Lumatone, see [[Lumatone mapping for 9edo]]
 
== Music ==
== Music ==
* [https://soundcloud.com/overtoneshock/tencaious-chorale-9-edo-studio-version Tenacious Chorale (only movement I is in 9EDO)] by [[Stephen Weigel]]
{{Main|Music in 9edo}}
* [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 9EDO by [http://home.snafu.de/djwolf/WorksDescriptive.htm Daniel Wolf]
== See also ==
* ''[http://www.h-pi.com/mp3/Prelude9ET.mp3 Prelude in 9ET]''{{dead link}} by [[Aaron Andrew Hunt]]
* ''[http://micro.soonlabel.com/9-edo/daily20110629_fts_e_guit_9et.mp3 Improvisation for Electric Guitar in 9EDO]'' by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/08%20-%208.%209%20octave.mp3 Comets Over Flatland 8]{{dead link}} 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 EDOsimultaneously)
* [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]].
* [https://drive.google.com/a/playgroundsessions.com/folderview?id=0BwsXD8q2VCYUamtVWEgyRFA5alU&usp=sharing#list 9edo ear-training exercises] by [[Alex Ness]].


[[Category:9edo| ]] <!-- main article -->
=== Werntz Nocturne scale ===
{{main|Werntz Nocturne scale}}
 
== Notes ==
<references group="note" />
 
[[Category:9-tone scales]]
[[Category:9-tone scales]]
[[Category:Equal divisions of the octave]]
[[Category:Pelog]]
[[Category:Listen]]
[[Category:Macrotonal]]
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