Theory

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17	prime 19
error	absolute (¢)	0	+48.0	+63.7	-68.8	+48.7	+59.5	+45.0	+2.5
error	relative (%)	0	+32	+42	-46	+32	+40	+30	+2
nearest edomapping		8	5	3	6	4	6	1	2
fifthspan		0	+1	-1	-2	+4	-2	-3	+2

8-edo 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.

Another way of looking at 8-EDO 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 12-EDO 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

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

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

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic, 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 8-edo 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).

8-edo 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 24-edo 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 Intervals

8ed2-001.svg

Commas

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

Prime Limit	Ratio^[1]	Monzo	Cents	Color name	Name(s)
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