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 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)

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

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		Audio
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
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

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

9ed2-001.svg

Diagrams

Instruments

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

Music

Santiago Cosentino

• Interdimensional Train Ride (2015)

Ivor Darreg

• "9 Tones Per Octave - Strings", from Detwelvulate! (1994)

Aaron Andrew Hunt

• "Prelude in 9ET", from The Equal-Tempered Keyboard (1999-2022)
• "Fugue a3 in 9ET", from The Equal-Tempered Keyboard (1999-2022) (SoundCloud)

Carlo Serafini

• New World (2013) (details)

Tabytha (Bandcamp)

• "69", from Bad Musick (2020)
• "69 Pentangled", from Bad Musick (2020)

Themnotyou

• "Morgan", from STAFFcirc vol. 7 (2021) (Bandcamp)

Chris Vaisvil

• Improvisation for Electric Guitar in 9EDO (2011)
• Gerbils at the Wheel of Government (2011) (in 9edo and 18edo simultaneously)

Stephen Weigel

• Tenacious Chorale: Movement 1 (2016) (live version) (YouTube)
• Gamelan, Origin, Creation (2017)
• Zones of Lasting Novelty (2015/2019) (formerly In Our Own Little Worlds)

Randy Winchester

• "8. 9 / octave", from Comets Over Flatland (2007)

Daniel Wolf

• Nocturne (2004)

Ear training

• 9edo ear-training exercises by Alex Ness.