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9edo's fifth of 5\9 is near the boundary of "perfect fifth" and "subfifth" so it sounds quite dirty but still recognizable.
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 ===
=== Odd harmonics ===
{{Harmonics in equal|9}}
{{Harmonics in equal|9}}
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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 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.
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. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Chord names are different because {{dash|C, E, G|med}} isn't {{dash|P1, M3, P5}}.


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". In this notation, the [[enharmonic unison]] is the augmented 2nd, e.g. Eb 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". In this notation, the [[enharmonic unison]] is the augmented 2nd, e.g. Eb to F#.


{| class="wikitable center-all right-1 right-2"
{| class="wikitable center-all right-1 right-2"
|-
! [[degree]]
! [[degree]]
! [[cent]]s
! [[cent]]s
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! colspan="2" | Melodic notation <br> Major wider than minor
! colspan="2" | Melodic notation <br> Major wider than minor
! colspan="2" | Harmonic notation <br> Major narrower than minor
! colspan="2" | Harmonic notation <br> Major narrower than minor
!Audio
! Audio
|-
|-
| 0
| 0
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| perfect unison
| perfect unison
| D
| D
|[[File:0-0 unison.mp3|frameless]]
| [[File:0-0 unison.mp3|frameless]]
|-
|-
| 1
| 1
| 133.33
| 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
| minor 2nd
| E
| E
| major 2nd
| major 2nd
| E
| E
|[[File:0-133,33 minor second (9-EDO).mp3|frameless]]
| [[File:0-133,33 minor second (9-EDO).mp3|frameless]]
|-
|-
| 2
| 2
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| minor 2nd, major 3rd
| minor 2nd, major 3rd
| Eb, F#
| Eb, 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]] (+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
| major 3rd
| F
| F
| minor 3rd
| minor 3rd
| F
| F
|[[File:0-400 major third (12-EDO).mp3|frameless]]
| [[File:0-400 major third (12-EDO).mp3|frameless]]
|-
|-
| 4
| 4
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| perfect 4th
| perfect 4th
| G
| G
|[[File:0-533,33 fourth (9-EDO).mp3|frameless]]
| [[File:0-533,33 fourth (9-EDO).mp3|frameless]]
|-
|-
| 5
| 5
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| perfect 5th
| perfect 5th
| A
| A
|[[File:0-666,67 fifth (9-EDO).mp3|frameless]]
| [[File:0-666,67 fifth (9-EDO).mp3|frameless]]
|-
|-
| 6
| 6
| 800.00
| 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
| minor 6th
| B
| B
| major 6th
| major 6th
| B
| B
|[[File:0-800 minor sixth (12-EDO).mp3|frameless]]
| [[File:0-800 minor sixth (12-EDO).mp3|frameless]]
|-
|-
| 7
| 7
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| minor 6th, major 7th
| minor 6th, major 7th
| Bb, C#
| Bb, 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
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| minor 7th
| minor 7th
| C
| C
|[[File:0-1066,67 major seventh (9-EDO).mp3|frameless]]
| [[File:0-1066,67 major seventh (9-EDO).mp3|frameless]]
|-
|-
| 9
| 9
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| octave
| octave
| D
| D
|[[File:0-1200 octave.mp3|frameless]]
| [[File:0-1200 octave.mp3|frameless]]
|}
|}


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


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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
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|-
|-
! [[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
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| Island comma
| Island comma
|}
|}
<references/>


=== Rank-2 temperaments ===
=== 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&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"). 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.


== Diagrams ==
== Diagrams ==
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== 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]].
== Notes ==
<references group="note" />


[[Category:9-tone scales]]
[[Category:9-tone scales]]
[[Category:Macrotonal]]
[[Category:Macrotonal]]
[[Category:Pelog]]
{{todo|add lumatone mapping}}
{{todo|add lumatone mapping}}
Retrieved from "https://en.xen.wiki/w/9edo"