8edo

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 (abbreviated 8edo or 8ed2), also called 8-tone equal temperament (8tet) or 8 equal temperament (8et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 8 equal parts of exactly 150.000 ¢ each. Each step represents a frequency ratio of 2^1/8, or the 8th 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
Error	relative (%)	+32	+42	-46	-36	+32	+40	-26	+30	+2	-14	-19
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, containing no good approximation of harmonics 3, 5, 7, 11, 13, and 17; 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.

Intervals

Steps	Cents	Ups and downs notation	Approximate ratios
0	0	D	1/1, 40/39, 55/54, 64/63, 66/65
1	150	F	10/9, 11/10, 12/11, 13/12, 27/25, 35/32, 44/39, 55/52, 72/65
2	300	E	6/5, 11/9, 13/11, 15/13, 16/13, 40/33, 63/52, 65/54, 77/64
3	450	G	4/3, 13/10, 21/16, 33/25, 33/26, 44/35, 50/39, 72/55, 80/63
4	600	^G, vA	11/8, 13/9, 16/11, 18/13, 25/18, 35/24, 36/25, 48/35, 55/39, 63/44, 78/55
5	750	A	3/2, 20/13, 32/21, 35/22, 39/25, 50/33, 52/33, 55/36, 63/40
6	900	C	5/3, 13/8, 18/11, 22/13, 26/15, 33/20
7	1050	B	9/5, 11/6, 20/11, 24/13, 39/22, 50/27, 64/35, 65/36
8	1200	D	2/1, 39/20, 55/27, 63/32, 65/33, 80/39

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	24edo subset notation		16edo subset notation (major narrower than minor)		16edo subset notation (major wider than minor)		3L 2s notation (J = 1/1)
0	0¢	P1	D	P1	D	P1	D	J
1	150	~2	vE	M2	E	m2	E	K
2	300	m3	F	M3	F#	m3	Fb	K#, Lb
3	450	^M3 / v4	^F# / vG	d3 / A4	Fb / G#	A3, D4	F#, Gb	L
4	600	A4, d5	G#, Ab	d4, A5	Gb, A#	A4, D5	G#, Ab	M
5	750	^5, vm6	^A, vBb	d5, A6	Ab, B#	A5, d6	A#, Bb	M#, Nb
6	900	M6	B	m6	Bb	M6	B#	N
7	1050	~7	^C	m7	C	M7	C	N#, Jb
8	1200	P8	D	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

Approximation to JI

8ed2-001.svg

Regular temperament properties

Uniform maps

13-limit uniform maps between 7.5 and 8.5
Min. size	Max. size	Wart notation	Map
7.5000	7.5368	8bcccddeeefff	⟨8 12 17 21 26 28]
7.5368	7.6585	8bcddeeefff	⟨8 12 18 21 26 28]
7.6585	7.6602	8bceeefff	⟨8 12 18 22 26 28]
7.6602	7.7018	8bcefff	⟨8 12 18 22 27 28]
7.7018	7.8866	8bcef	⟨8 12 18 22 27 29]
7.8866	7.9493	8cef	⟨8 13 18 22 27 29]
7.9493	7.9675	8cf	⟨8 13 18 22 28 29]
7.9675	7.9720	8f	⟨8 13 19 22 28 29]
7.9720	8.0147	8	⟨8 13 19 22 28 30]
8.0147	8.2383	8d	⟨8 13 19 23 28 30]
8.2383	8.2423	8dee	⟨8 13 19 23 29 30]
8.2423	8.3709	8deeff	⟨8 13 19 23 29 31]
8.3709	8.3982	8dddeeff	⟨8 13 19 24 29 31]
8.3982	8.5000	8ccdddeeff	⟨8 13 20 24 29 31]

Commas

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

Prime Limit	Ratio^[1]	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

↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Scales

Scala file

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, in the 5-limit 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.