8edo

← 7edo

8edo

9edo →

Prime factorization	23
Step size	150 ¢
Fifth	5\8 (750 ¢)
Semitones (A1:m2)	3:-1 (450 ¢ : -150 ¢)
Consistency limit	5
Distinct consistency limit	3

8 equal divisions of the octave (8edo) is the tuning system derived by dividing the octave into 8 equal steps of 150 cents each, or the eighth root of 2.

Theory

Approximation of odd harmonics in 8edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+48.0	+63.7	-68.8	-53.9	+48.7	+59.5	-38.3	+45.0	+2.5	-20.8	-28.3
	Relative (%)	+32.0	+42.5	-45.9	-35.9	+32.5	+39.6	-25.5	+30.0	+1.7	-13.9	-18.8
Steps (reduced)		13 (5)	19 (3)	22 (6)	25 (1)	28 (4)	30 (6)	31 (7)	33 (1)	34 (2)	35 (3)	36 (4)

A chromatic 8edo scale on C.

8edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system; even so, it does a good job representing the just intonation subgroups 2.11/3.13/5, with good intervals of 13/10 and an excellent version of 11/6. Stacking the 450-cent interval can result in some semi-consonant chords such as 0-3-6 degrees, although these still are quite dissonant compared to standard root-3rd-P5 triads, which are unavailable in 8edo.

Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out.

Notation

8edo can be notated as a subset of 24edo, using ups and downs. It can also be notated as a subset of 16edo, but this is a less intuitive notation.

Edostep	Cents	7mus (hex)	24edo subset notation		16edo subset notation		3L 2s notation (J = 1/1)	Audio
0	0¢	0	P1	D	P1	D	J
1	150	192 (C0)	~2	vE	M2	K
2	300	384 (180)	m3	F	F#		K#, Lb
3	450	576 (240)	^M3 / v4	^F# / vG	d3 / A4	Fb / G#	L
4	600	768 (300)	A4 / d5	G# / Ab	d4 / A5	Gb / A#	M
5	750	960 (3C0)	^5 / vm6	^A / vBb	d5 / A6	Ab / B#	M#, Nb
6	900	1152 (480)	M6	B	m6	Bb	N
7	1050	1344 (540)	~7	^C	m7	C	N#, Jb
8	1200	1536 (600)	P8	D	P8	D	J

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octatonic, and heptatonic 2nd-generated.

Pentatonic 5th-generated: D * E G * A C * D (generator = 5\8 = perfect 5thoid)

D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D

P1 - A1/ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7/d8d - P8d (s = sub-, d = -oid)

pentatonic genchain of 5ths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of 5ths: ...d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1... (s = sub-, d = -oid)

Octatonic: A B C D E F G H A (every interval is a generator)

P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9

Heptatonic 2nd-generated: D E F G * A B C D (generator = 1\8 = perfect 2nd = 150¢)

D - E - F - G - G#/Ab - A -B - C - D

P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8

genchain of 2nds: ...D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db...

genchain of 2nds: ...A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8...

Chord names

Ups and downs can name any 8edo chord. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

8edo chords are very ambiguous, with many chord homonyms. Even the major and minor triads are homonyms. Chord components usually default to M2, M3, P4, P5, M6, m7, M9, P11 and M13. Thus D7 has a M3, P5 and m7. 8-edo chord names using 24edo subset names are greatly simplified by using different defaults: ~2, ^M3, v4, ^5, M6, ~7, ~9, v11 and M13. Thus D7 becomes ^M3, ^5 and ~7.

Chord edosteps	Chord notes	Full name	Abbreviated name	Homonyms
0 – 3 – 5	D ^F♯ ^A	D^(^5)	D	^F♯m or vGm
0 – 2 – 5	D F ^A	Dm(^5)	Dm	^A or vB♭
0 – 3 – 5 – 7	D ^F♯ ^A ^C	D^7(^5)	D7	^F♯m♯11 or vGm♯11
0 – 3 – 5 – 6	D ^F♯ ^A B	D6(^3,^5)	D6	Bm7 and vG,♯9
0 – 2 – 5 – 7	D F ^A ^C	Dm,~7(^5)	Dm7	F6 and vB♭,♯9
0 – 2 – 5 – 6	D F ^A B	Dm6(^5)	Dm6	Bm7(♭5)
0 – 2 – 4 – 7	D F A♭ ^C	Ddim,~7	Dm7(♭5)	Fm6

JI approximation

8ed2-001.svg

Commas

8edo tempers out the following commas. This assumes val ⟨8 13 19 22 28 30].

Prime Limit	Ratio	Monzo	Cents	Color name	Name(s)
3	8192/6561	[13 -8⟩	384.35	sawa 4th	Pythagorean diminished fourth
5	16/15	[4 -1 -1⟩	111.73	gu 2nd	Classic minor second
5	648/625	[3 4 -4⟩	62.57	Quadgu	Major diesis, diminished comma
5	250/243	[1 -5 3⟩	49.17	Triyo	Maximal diesis, porcupine comma
5	78732/78125	[2 9 -7⟩	13.40	Sepgu	Medium semicomma, sensipent comma
7	64/63	[6 -2 0 -1⟩	27.26	Ru	Septimal comma, Archytas' comma, Leipziger Komma
7	875/864	[-5 -3 3 1⟩	21.90	Zotriyo	Keema
7	(12 digits)	[-9 8 -4 2⟩	8.04	Labizogugu	Varunisma
7	6144/6125	[11 1 -3 -2⟩	5.36	Sarurutrigu	Porwell
11	100/99	[2 -2 2 0 -1⟩	17.40	Luyoyo	Ptolemisma
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	65536/65219	[16 0 0 -2 -3⟩	8.39	Satrilu-aruru	Orgonisma
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
11	4000/3993	[5 -1 3 0 -3⟩	3.03	Triluyo	Wizardharry
13	40/39	[3 -1 1 0 0 -1⟩	43.83	Thuyo unison	tridecimal minor diesis

Scales

Scala scale file

For those who use the new tuner Lingot, which accepts scala files, or for anyone else, here is a .scl file of 8edo 08-edo.scl

! 08-EDO.scl
!
8 EDO
 8
!
 150.00
 300.00
 450.00
 600.00
 750.00
 900.00
 1050.00
 2/1

Temperaments

8edo is fairly composite, so the only step that generates a mos scale that covers every interval other than the 1 is the 3, producing scales of 332 and 21212. In terms of temperaments, this is best interpreted as Father, as 8edo is the highest edo that tempers out the diatonic semitone in it's patent val, merging 5/4 and 4/3 into a single interval, which is also the generator. This means major and minor chords are rotations of each other, making them inaccurate but very simple, with even the 5 note mos having 3 of both and providing a functional skeleton of 5-limit harmony, albeit with some very strange enharmonic equivalences. In terms of 7-limit extensions things get even more inaccurate, as the patent val supports Mother, but the ideal tuning for that is much closer to 5edo. The d val supports septimal father and Pater, and is much closer to the ideal tuning for both, as the extremely sharp 7 works better with the 3&5. In terms of multi-period temperaments, it makes for a near perfect Walid or a much less accurate Diminished scale.

Music

Cenobyte

• 8edo-pre-improv.ogg (2011)
• darkreflections(sketch).ogg (2011)
• skiphop(sketch).ogg (2011)
• octo-icy-pensive(sketch).ogg, octo-icy-pensive-echo(sketch).ogg (2011)

City of the Asleep

• "Malebolge", from Transcendissonance (2011)

Milan Guštar

• Špendlíky (1988-2005)

Aaron Andrew Hunt

• "Fantasia & Fugue a4 in 8ET", from The Equal-Tempered Keyboard (1999-2022)

Carlo Serafini

• FunkEight 1 (2013) (details)

Jake Sherman

• News To Me (2011)
• Tomorrow Night (pop music from 2045) (2011)

Ron Sword

• Acoustic Improvisation in 8EDO^{[dead link]}

Stephen Weigel

• Tenacious Chorale: Movement 2 (2016)

Randy Winchester

• "7. 8 / octave", from Comets Over Flatland (2007)

Ear training

8edo ear-training exercises by Alex Ness available here.

See also

• Octatonic scale - a scale based on alternating whole and half steps