# 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 (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 ¢ each. Each step represents a frequency ratio of 21/8, 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
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, 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 Approximate Ratios Ups and Downs Notation
0 0 1/1, 40/39, 55/54, 64/63, 66/65 D
1 150 10/9, 11/10, 12/11, 13/12, 27/25, 35/32, 44/39, 55/52, 72/65 F
2 300 6/5, 11/9, 13/11, 15/13, 16/13, 40/33, 63/52, 65/54, 77/64 E
3 450 4/3, 13/10, 21/16, 33/25, 33/26, 44/35, 50/39, 72/55, 80/63 G
4 600 11/8, 13/9, 16/11, 18/13, 25/18, 35/24, 36/25, 48/35, 55/39, 63/44, 78/55 ^G, vA
5 750 3/2, 20/13, 32/21, 35/22, 39/25, 50/33, 52/33, 55/36, 63/40 A
6 900 5/3, 13/8, 18/11, 22/13, 26/15, 33/20 C
7 1050 9/5, 11/6, 20/11, 24/13, 39/22, 50/27, 64/35, 65/36 B
8 1200 2/1, 39/20, 55/27, 63/32, 65/33, 80/39 D

## 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) Audio
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

## 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
1. 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.

## Music

Abnormality
Cenobyte
City of the Asleep
Milan Guštar
Hideya
Aaron Andrew Hunt
NullPointerException Music
Carlo Serafini
Jake Sherman
Ron Sword
Stephen Weigel
Randy Winchester

## Ear training

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