9edo: Difference between revisions

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| Prime factorization = 3<sup>2</sup>
| Prime factorization = 3<sup>2</sup>
| Step size = 133.333¢
| Step size = 133.333¢
| Fifth = 5\9 = 666.667¢
| Fifth = 5\9 (667¢)
| Major 2nd = 1\9 = 133.333¢
| Major 2nd = 1\9 (133¢)
| Consistency = 7
| Consistency = 7
| Monotonicity = 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.
'''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|intervals=odd}}
{{Harmonics in equal|9}}


The 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly. A 7-limit version of 9edo goes
The 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly. A 7-limit version of 9edo goes
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! [[Cent]]s
! [[Cent]]s
! [[Color name]]
! [[Color name]]
! Name(s)
! Name
|-
|-
| 5
| 5
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| 133.24
| 133.24
| Gugu
| Gugu
| Large Limma, Large Semitone, Bug Comma
| Large limma
|-
|-
| 5
| 5
Line 157: Line 156:
| 92.18
| 92.18
| Layobi
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
| Major chroma
|-
|-
| 5
| 5
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| 51.12
| 51.12
| Laquadyo
| Laquadyo
| Negri Comma, Double Augmentation Diesis
| Negri comma
|-
|-
| 5
| 5
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| 41.06
| 41.06
| Trigu
| Trigu
| Diesis, Augmented Comma
| Diesis
|-
|-
| 5
| 5
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| 10.06
| 10.06
| Lasepyo
| Lasepyo
| [[Semicomma]], Fokker Comma
| [[Semicomma]]
|-
|-
| 7
| 7
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| 48.77
| 48.77
| Rugu
| Rugu
| Septimal Quarter Tone
| Septimal quartertone
|-
|-
| 7
| 7
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| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicenna, Avicenna's Enharmonic Diesis
| Avicennma
|-
|-
| 7
| 7
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| 35.70
| 35.70
| Zozo
| Zozo
| Slendro Diesis
| Slendro diesis
|-
|-
| 7
| 7
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| 20.79
| 20.79
| Quadru-ayo
| Quadru-ayo
| Nuwell
| Nuwell comma
|-
|-
| 7
| 7
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| 13.07
| 13.07
| Triru-agu
| Triru-agu
| Orwellisma, Orwell Comma
| Orwellisma
|-
|-
| 7
| 7
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| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
| Septimal Kleisma, Marvel Comma
| Marvel comma
|-
|-
| 7
| 7
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| 5.36
| 5.36
| Sarurutrigu
| Sarurutrigu
| Porwell
| Porwell comma
|-
|-
| 7
| 7
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| 2.35
| 2.35
| Lazoquinyo
| Lazoquinyo
| Horwell
| Horwell comma
|-
|-
| 7
| 7
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| 1.84
| 1.84
| Tritrizo
| Tritrizo
| [[Tritrizo clan|No-fives ennealimma]]
| [[Septimal ennealimma]]
|-
|-
| 11
| 11
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| 2.56
| 2.56
| Bithogu
| Bithogu
| Parizeksma
| Island comma
|}
|}
<references/>
<references/>


== Linear temperaments ==
== Linear temperaments ==
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].
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].
 
== Scales ==
=== Pathological modes ===
2 1 1 1 1 1 1 1 [[1L 7s]] MOS


== JI approximation ==
== JI approximation ==
=== 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.]]


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


[[Category:9edo| ]] <!-- main article -->
[[Category:9edo| ]] <!-- main article -->

Revision as of 12:51, 26 March 2022

← 8edo 9edo 10edo →
Prime factorization 32
Step size 133.333 ¢ 
Fifth 5\9 (666.667 ¢)
Semitones (A1:m2) -1:2 (-133.3 ¢ : 266.7 ¢)
Consistency limit 7
Distinct consistency limit 5

9 equal divisions of the octave (9edo) is the tuning system derived by dividing the octave into 9 equal steps of 133+1/3 cents each precisely. It is also the first odd composite edo.

Theory

Approximation of odd harmonics in 9edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -35.3 +13.7 -35.5 +62.8 -18.0 -40.5 -21.6 +28.4 -30.8 +62.6 +38.4
Relative (%) -26.5 +10.3 -26.6 +47.1 -13.5 -30.4 -16.2 +21.3 -23.1 +46.9 +28.8
Steps
(reduced) 		14
(5) 		21
(3) 		25
(7) 		29
(2) 		31
(4) 		33
(6) 		35
(8) 		37
(1) 		38
(2) 		40
(4) 		41
(5)

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

2: 7/6 266.871 septimal minor third

3: 63/50 400.108 quasi-equal major third

4: 49/36 533.742 Arabic lute acute fourth

5: 72/49 666.258 Arabic lute grave fifth

6: 100/63 799.892 quasi-equal minor sixth

7: 12/7 933.129 septimal major sixth

8: 50/27 1066.762 grave major seventh

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.

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.

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".

degree cents Approximate
Ratios 		Melodic notation
Major wider than minor 		Harmonic notation
Major narrower than minor
0 0.00 1/1 perfect unison D perfect unison D
1 133.33 14/13, 13/12, 12/11 minor 2nd E major 2nd E
2 266.67 7/6 major 2nd, minor 3rd E#, Fb minor 2nd, major 3rd Eb, F#
3 400.00 5/4, 14/11, 9/7 major 3rd F minor 3rd F
4 533.33 4/3, 11/8 perfect 4th G perfect 4th G
5 666.67 16/11, 3/2 perfect 5th A perfect 5th A
6 800.00 14/9, 11/7, 8/5 minor 6th B major 6th B
7 933.33 12/7 major 6th, minor 7th B#, Cb minor 6th, major 7th Bb, C#
8 1066.67 11/6, 13/7 major 7th C minor 7th C
9 1200.00 2/1 octave D octave D

Commas

9edo tempers out the following commas. (Note: This assumes val ⟨9 14 21 25 31 33].)

Prime
Limit 		Ratio[1] Monzo Cents Color name Name
5 27/25 [0 3 -2⟩ 133.24 Gugu Large limma
5 135/128 [-7 3 1⟩ 92.18 Layobi Major chroma
5 16875/16384 [-14 3 4⟩ 51.12 Laquadyo Negri comma
5 128/125 [7 0 -3⟩ 41.06 Trigu Diesis
5 (14 digits) [-21 3 7⟩ 10.06 Lasepyo Semicomma
7 36/35 [2 2 -1 -1⟩ 48.77 Rugu Septimal quartertone
7 525/512 [-9 1 2 1⟩ 43.41 Lazoyoyo Avicennma
7 49/48 [-4 -1 0 2⟩ 35.70 Zozo Slendro diesis
7 686/675 [1 -3 -2 3⟩ 27.99 Trizo-agugu Senga
7 2430/2401 [1 5 1 -4⟩ 20.79 Quadru-ayo Nuwell comma
7 1728/1715 [6 3 -1 -3⟩ 13.07 Triru-agu Orwellisma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Marvel comma
7 6144/6125 [11 1 -3 -2⟩ 5.36 Sarurutrigu Porwell comma
7 65625/65536 [-16 1 5 1⟩ 2.35 Lazoquinyo Horwell comma
7 (16 digits) [-11 -9 0 9⟩ 1.84 Tritrizo Septimal ennealimma
11 99/98 [-1 2 0 -2 1⟩ 17.58 Loruru Mothwellsma
11 121/120 [-3 -1 -1 0 2⟩ 14.37 Lologu Biyatisma
11 176/175 [4 0 -2 -1 1⟩ 9.86 Lorugugu Valinorsma
11 385/384 [-7 -1 1 1 1⟩ 4.50 Lozoyo Keenanisma
11 540/539 [2 3 1 -2 -1⟩ 3.21 Lururuyo Swetisma
13 91/90 [-1 -2 -1 1 0 1⟩ 19.13 Thozogu Superleap
13 676/675 [2 -3 -2 0 0 2⟩ 2.56 Bithogu Island comma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Linear 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 9edo tradition.

JI approximation

Selected just intervals

alt : Your browser has no SVG support.

9ed2-001.svg

Images

9edo wheel.png

Instruments

IMG_2223-800x600.jpg

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

Music

Ear training

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