8edo
Theory[edit]
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.
Intervals[edit]
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 | 24-edo subset notation | 16-edo subset notation | ||
|---|---|---|---|---|---|
| 0 | 0¢ | P1 | D | P1 | D |
| 1 | 150 | ~2 | Ev | M2 | E |
| 2 | 300 | m3 | F | M3 | F# |
| 3 | 450 | ^M3 / v4 | F#^ / Gv | d3 / A4 | Fb / G# |
| 4 | 600 | A4 / d5 | G# / Ab | d4 / A5 | Gb / A# |
| 5 | 750 | ^5 / vm6 | A^ / Bbv | d5 / A6 | Ab / B# |
| 6 | 900 | M6 | B | m6 | Bb |
| 7 | 1050 | ~7 | C^ | m7 | C |
| 8 | 1200 | P8 | D | P8 | D |
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 | Proper name | Abbreviated name | Homonyms |
|---|---|---|---|---|
| 0 – 3 – 5 | D F♯^ A^ | D.^(^5) | D | F♯^.m or Gv.m |
| 0 – 2 – 5 | D F A^ | Dm(^5) | Dm | A^ or B♭v |
| 0 – 3 – 5 – 7 | D F♯^ A^ C^ | D.^7(^5) | D7 | F♯^.m♯11 or Gv.m♯11 |
| 0 – 3 – 5 – 6 | D F♯^ A^ B | D6(^3,^5) | D6 | Bm7 and Gv,♯9 |
| 0 – 2 – 5 – 7 | D F A^ C^ | Dm~7(^5) | Dm7 | F6 and B♭v,♯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 |
Scala scale file[edit]
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[edit]
Tenacious Chorale (only movement II is in 8-EDO) by Stephen Weigel
Fantasia in 8ET by Aaron Andrew Hunt
Fugue in 8ET 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[edit]
8 EDO ear-training exercises by Alex Ness available here.
See also[edit]
- Octatonic scale - a scale based on alternating whole and half steps
Commas[edit]
8 EDO tempers out the following commas. (Note: This assumes val < 8 13 19 22 28 30 |). ||~ Comma ||~ Monzo ||~ Cents ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
| 648/625 | | 3 4 -4 > | 62.57 | Major Diesis | Diminished Comma | |
| 250/243 | | 1 -5 3 > | 49.17 | Maximal Diesis | Porcupine Comma | |
| 78732/78125 | | 2 9 -7 > | 13.40 | Medium Semi Comma | Sensipent Comma | |
| 64/63 | | 6 -2 0 -1 > | 27.26 | Septimal Comma | Archytas' Comma | Leipziger Komma |
| 875/864 | | -5 -3 3 1 > | 21.90 | Keema | ||
| 321489/320000 | | -9 8 -4 2 > | 8.04 | Varun Isma | ||
| 6144/6125 | | 11 1 -3 -2 > | 5.36 | Cornwell | ||
| 100/99 | | 2 -2 2 0 -1 > | 17.40 | Ptolemies | ||
| 121/120 | | -3 -1 -1 0 2 > | 14.37 | Batista | ||
| 176/175 | | 4 0 -2 -1 1 > | 9.86 | Valinor Mo | ||
| 65536/65219 | | 16 0 0 -2 -3 > | 8.39 | Organisms | ||
| 385/384 | | -7 -1 1 1 1 > | 4.50 | Keenanisma | ||
| 4000/3993 | | 5 -1 3 0 -3 > | 3.03 | Wizard Harry |