9edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Sintel (talk | contribs)
autopatrolled, emailconfirmed
1,637 edits
remove table "Differences between distributionally-even scales and smaller edos"
Used new primes template, lowercase edo, marked 2 dead links, misc. edits
Line 13: Line 13:
| Monotonicity = 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 ==
{| class="wikitable"
{{primes in equal|9}}
! colspan="2" | <!-- empty cell -->
 
! prime 2
The 9edo scale has the peculiar property of representing certain [[7-limit]] intervals almost exactly. A 7-limit version of 9edo goes
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -35.3
| +13.7
| -35.5
| -18.0
| -40.5
| +28.4
| -30.8
|-
! [[Relative error|relative]] (%)
| 0
| -26
| +10
| -27
| -13
| -30
| +21
| -23
|-
! colspan="2" | [[Patent val|nearest EDO-mapping]]
| 9
| 5
| 3
| 7
| 4
| 6
| 1
| 2
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +3
| +2
| +4
|}
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
1: 27/25 133.238 large limma, BP small semitone
Line 88: Line 38:
9: 2/1 1200.000 octave
9: 2/1 1200.000 octave


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.
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.


== 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. 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 <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".
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"
Line 184: Line 134:


== Commas ==
== Commas ==
9EDO [[tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 9 14 21 25 31 33 }}.)
9edo [[tempers out]] the following [[comma]]s. (Note: 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"
Line 352: Line 302:


== 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 [[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].


== Pathological Modes==
== Pathological Modes==
Line 368: Line 318:
[[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)


== Music ==
== Music ==
Line 378: Line 328:


* Nocturne in 9EDO by [http://home.snafu.de/djwolf/WorksDescriptive.htm Daniel Wolf]
* Nocturne in 9EDO 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://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/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] by [[Randy Winchester]]
* [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://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://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)
Line 387: Line 337:


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


[[Category:9-tone]]
[[Category:9edo| ]] <!-- main article -->
[[Category:9edo| ]]
[[Category:9-tone scales]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Macrotonal]]

Revision as of 01:16, 15 January 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

Template:Primes in equal

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(s)
5 27/25 [0 3 -2⟩ 133.24 Gugu Large Limma, Large Semitone, Bug Comma
5 135/128 [-7 3 1⟩ 92.18 Layobi Major Chroma, Major Limma, Pelogic Comma
5 16875/16384 [-14 3 4⟩ 51.12 Laquadyo Negri Comma, Double Augmentation Diesis
5 128/125 [7 0 -3⟩ 41.06 Trigu Diesis, Augmented Comma
5 (14 digits) [-21 3 7⟩ 10.06 Lasepyo Semicomma, Fokker Comma
7 36/35 [2 2 -1 -1⟩ 48.77 Rugu Septimal Quarter Tone
7 525/512 [-9 1 2 1⟩ 43.41 Lazoyoyo Avicenna, Avicenna's Enharmonic Diesis
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
7 1728/1715 [6 3 -1 -3⟩ 13.07 Triru-agu Orwellisma, Orwell Comma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Septimal Kleisma, Marvel Comma
7 6144/6125 [11 1 -3 -2⟩ 5.36 Sarurutrigu Porwell
7 65625/65536 [-16 1 5 1⟩ 2.35 Lazoquinyo Horwell
7 (16 digits) [-11 -9 0 9⟩ 1.84 Tritrizo No-fives 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 Parizeksma
  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.

Pathological Modes

2 1 1 1 1 1 1 1 1L 7s MOS

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

9edo ear-training exercises by Alex Ness available here.

Retrieved from "https://en.xen.wiki/index.php?title=9edo&oldid=84016"