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→Rank-2 temperaments: use backslashes for edo steps
 
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{{interwiki
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[[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
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
1: [[27/25]] 133.238 large limma, BP small semitone
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9: [[2/1]] 1200.000 octave
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 {{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.
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)<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's fifth does approximate [[47/32]] to within about 1.2 cents, and remains near enough the boundary of [[perfect fifth]] and [[subfifth]], so it sounds quite dirty but still recognizable. 9 is the first edo to include the [[2L 5s|antidiatonic (2L 5s)]] scale, which this fifth generates as well.


=== Odd harmonics ===
=== Odd harmonics ===
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== Notation ==
== Notation ==
9edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first defines sharp/flat, major/minor and aug/dim in terms of the native antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules. Chord names don't follow diatonic nominals because {{dash|C, E, G|med}} is not {{dash|P1, M3, P5|med}}.
{{Mavila}}
 
The second approach is to essentially pretend 9edo's antidiatonic scale is a normal diatonic, meaning that sharp is lower in pitch than flat (since the "S" step is larger than the "L" step) and major/aug is narrower than minor/dim. This allows music notated in 12edo or another diatonic system to be directly translated to 9edo "on the fly", and it carries over the way interval arithmetic and chord names work from diatonic notation.
 
In this notation, the [[enharmonic unison]] is the augmented 2nd, e.g. E♭ to F♯.
In this notation, the [[enharmonic unison]] is the augmented 2nd, e.g. E♭ to F♯.


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=== Selected just intervals ===
=== 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.]]
=== Zeta peak index ===
{{ZPI
| zpi = 22
| steps = 8.94999197142904
| step size = 134.078332565073
| tempered height = 3.998567
| pure height = 3.622488
| integral = 0.954565
| gap = 13.186387
| octave = 1206.70499308566
| consistent = 8
| distinct = 6
}}


== Regular temperament properties ==
== Regular temperament properties ==
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=== Rank-2 temperaments ===
=== 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").  
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].  


[[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&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.
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 ==
== 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 ==
{{Main|Music in 9edo}}
{{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]].
* [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 ==
== Notes ==
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[[Category:9-tone scales]]
[[Category:9-tone scales]]
[[Category:Pelog]]
[[Category:Pelog]]
{{Todo|add lumatone mapping}}
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