8edo

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Theory

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 subgroup 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 24-edo subset notation 16-edo subset notation
0 0 P1 D P1 D
1 150 192 (C016) ~2 vE M2 E
2 300 384 (18016) m3 F M3 F#
3 450 576 (24016) ^M3 / v4 ^F# / vG d3 / A4 Fb / G#
4 600 768 (30016) A4 / d5 G# / Ab d4 / A5 Gb / A#
5 750 960 (3C016) ^5 / vm6 ^A / vBb d5 / A6 Ab / B#
6 900 1152 (48016) M6 B m6 Bb
7 1050 1344 (54016) ~7 ^C m7 C
8 1200 1536 (60016) 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

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8ed2-001.svg

Commas

8 EDO tempers out the following commas. (Note: This assumes val < 8 13 19 22 28 30 |).

Ratio Monzo Cents Color Name Name 1 Name 2 Name 3
648/625 | 3 4 -4 > 62.57 Quadgu Major Diesis Diminished Comma
250/243 | 1 -5 3 > 49.17 Triyo Maximal Diesis Porcupine Comma
78732/78125 | 2 9 -7 > 13.40 Sepgu Medium Semi Comma Sensipent Comma
64/63 | 6 -2 0 -1 > 27.26 Ru Septimal Comma Archytas' Comma Leipziger Komma
875/864 | -5 -3 3 1 > 21.90 Zotriyo Keema
321489/320000 | -9 8 -4 2 > 8.04 Labizogugu Varun Isma
6144/6125 | 11 1 -3 -2 > 5.36 Sarurutrigu Cornwell
100/99 | 2 -2 2 0 -1 > 17.40 Luyoyo Ptolemies
121/120 | -3 -1 -1 0 2 > 14.37 Lologu Batista
176/175 | 4 0 -2 -1 1 > 9.86 Lorugugu Valinor Mo
65536/65219 | 16 0 0 -2 -3 > 8.39 Satrilu-aruru Organisms
385/384 | -7 -1 1 1 1 > 4.50 Lozoyo Keenanisma
4000/3993 | 5 -1 3 0 -3 > 3.03 Triluyo Wizard Harry

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 8 EDO 08-EDO.scl

0. 1/1 C Unison

1. 150.000 cents C# (50 cents sharper than C# of 12 tet) Neutral 2nd

2. 300.000 cents D# Minor 3rd (identical to 12-tet)

3. 450.000 cents E Sub-4th, Supermajor 3rd (50 cents sharper than E of 12 tet)

4. 600.000 cents F# Tritone (identical to 12-tet)

5. 750.000 cents G Super 5th, Subminor 6th (50 cents sharper than G of 12 tet)

6. 900.000 cents A Major 6th (identical to 12-tet)

7. 1050.000 cents A# Neutral 7th (50 cents sharper than A# of 12 tet)

8. 2/1 C Octave

Compositions

Tenacious Chorale (only movement II is in 8-EDO) by Stephen Weigel

Fantasia in 8ET[dead link] by Aaron Andrew Hunt

Fugue in 8ET[dead link] by Aaron Andrew Hunt

Špendlíky play by Milan Guštar

Acoustic Improvisation in 8-edo by Ron Sword (dead link)

Comets Over Flatland 7 by Randy Winchester

Malebolge Alison Jones

News To Me Jake Sherman

Tomorrow Night (pop music from 2045) Jake Sherman

FunkEight 1 by Carlo Serafini (blog entry)

8edo-pre-improv.oggPreliminary noodling in 8 EDO with ZynAddSubFX synth.(by Cenobyte)

darkreflections(sketch).ogg8EDO Ukulele (cenobyte)

skiphop(sketch).ogg8EDO Ukulele (cenobyte)

octo-icy-pensive(sketch).ogg octo-icy-pensive-echo(sketch).ogg2 versions of the same song, one with echo, one without. (cenobyte)

Ear Training

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

See also