9edo

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9edo

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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 EDO divides the octave into nine equal parts each of 133+1/3 cents precisely. It is also the first odd composite EDO.

Theory

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17	prime 19
error	absolute (¢)	0.0	-35.3	+13.7	-35.5	-18.0	-40.5	+28.4	-30.8
	relative (%)	0	-26	+10	-27	-13	-30	+21	-23
nearest EDO-mapping		9	5	3	7	4	6	1	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

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.

Differences between distributionally-even scales and smaller EDOs

N	L-Nedo	s-Nedo
2	66.666¢	-66.666¢
4	100¢	-33.333¢
5	26.667¢	-106.667¢
6	66.666¢	-66.666¢
7	95.238¢	-38.095¢
8	116.667¢	-16.667¢

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	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
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

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

9ed2-001.svg

Images

Instruments

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

Music

• Tenacious Chorale (only movement I is in 9EDO) by Stephen Weigel
• Zones of Lasting Novelty (Un12 2019) by Stephen Weigel; perf. Hans Gunter-Lock, Jacob Barton, and Stephen Weigel
• Gamelan, Origin, Creation by Stephen Weigel (Beo String Quartet, dedicated to Lou Harrison)
• 69 by Tabytha
• 69 Pentangled by Tabytha

• Nocturne in 9EDO by Daniel Wolf
• Prelude in 9ET by Aaron Andrew Hunt
• Improvisation for Electric Guitar in 9EDO by Chris Vaisvil
• Comets Over Flatland 8 by Randy Winchester
• Nine tones per Octave (9-EDO / 9-TET) by Ivor Darreg
• Gerbils at the Wheel of Government by Chris Vaisvil (in 9 and 18 EDOsimultaneously)
• New World by Carlo Serafini (blog entry)
• Interdimensional Train Ride by Santiago Cosentino

Ear Training

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