8edf
| ← 7edf | 8edf | 9edf → |
8 equal divisions of the fifth (8edf, 8ed3/2) is the tuning system that divides the fifth into 8 steps of 87.7444 cents each, making it very nearly 88cET. It is related to the octacot temperament. 8edf corresponds to about 13.6761edo (similar to every third step of 41edo).
Intervals
8edf can be notated either using native uranian (sesquitave) notation, where the notation repeats every period (i.e. just diatonic fifth), or using double sesquitave (Annapolis) notation, where the notation repeats every two periods (i.e. major diatonic ninth). This interprets 8edf as 16ed9/4, resulting in a tuning for the Natural and Harmonic Minor modes of Annapolis[6L 4s]. It can also be notated using tetratonic 4edf-based notation.
The naturals result from a semiwolf generator (~7/6). For sesquitave notation, letters A-E can be used. For double sesquitave notation, Greek numerals 1-10 can be used (Α,Β,Γ,Δ,Ε,Ϛ/Ϝ,Ζ,Η,Θ,Ι).
| Scale | Cents | Approximate intervals | Uranian | Diatonic | Notation | ||||
|---|---|---|---|---|---|---|---|---|---|
| degree | value | 7-limit | 11-limit | 19-limit | interval | equivalent | Uranian | Annapolis | Tetratonic notation |
| 0 | 0 | 1 | unison | A | Α | C | |||
| 1 | 87.7444 | 21/20 | 22/21 | 20/19, 19/18 | min mos2nd | minor second | A# | Α# | ^C, vD |
| 2 | 175.48875 | 10/9 | 21/19 | maj mos2nd | major second | B | Β | D | |
| 3 | 263.2331 | 7/6 | perf mos3rd | subminor third | C | Γ | ^D, vE | ||
| 4 | 350.9775 | 60/49, 49/40 | 11/9 | aug mos3rd | neutral third | C# | Γ# | E | |
| 5 | 438.7219 | 9/7 | perf mos4th | supermajor third | D | Δ | ^E, vF | ||
| 6 | 526.46625 | 27/20 | 19/14 | min mos5th | wolf fourth | D# | Δ# | F | |
| 7 | 614.2106 | 10/7 | 27/19 | maj mos5th | augmented fourth | E | Ε | ^F, vC | |
| 8 | 701.955 | 3/2 | sesquitave | just fifth | A | Ϛ/Ϝ | C | ||
| 9 | 789.6994 | 63/40 | 11/7 | 30/19 | min mos7th | minor sixth | A# | Ϛ#/Ϝ# | ^C, vD |
| 10 | 877.44375 | 5/3 | 63/38 | maj mos7th | major sixth | B | Ζ | D | |
| 11 | 965.1881 | 7/4 | perf mos8th | subminor seventh | C | Η | ^D, vE | ||
| 12 | 1052.9325 | 90/49, 35/18 | 11/6 | aug mos8th | neutral seventh | C# | Η# | E | |
| 13 | 1140.6769 | 27/14 | perf mos9th | supermajor seventh | D | Θ | ^E, vF | ||
| 14 | 1228.42125 | 81/40 | 57/28 | min mos10th | acute octave | D# | Θ# | F | |
| 15 | 1316.1656 | 15/7 | 81/38 | maj mos10th | minor ninth | E | Ι | ^F, vC | |
| 16 | 1403.91 | 9/4 | double sesquitave | major ninth | A | Α | C | ||
Scale workshop link for a keyboard/MIDI playable 8EDF (with graphical uranian scale, A=220Hz)
Rank 2 temperaments
MOS scales and temperament listed by generator size and period:
| Periods
per octave |
Generator | Scale pattern | Temperaments |
|---|---|---|---|
| 1 | 1\8 | 1L5s, 1L6s | (pathological) |
| 1 | 3\8 | 3L2s (uranian) | Semiwolf |
| 2 | 3\8 | 2L 2s | (augmented-like) |
Images
