17edo
17edo, or 17 equal divisions of the octave, is the equal tuning featuring steps of (1200/17) ~= 70.6 cents, 17 of which stack to the octave 2/1.
17edo is the smallest edo that has a sharper-than-just diatonic fifth (705.8c, compared to just 3/2 = 702.0c), not counting the degenerate case 5edo. 17edo has discordant diatonic major thirds and provides no good approximation of 5/4, which to some extent forces one to use nonfunctional and modal harmony instead of standard functional harmony. It is also notable for having neutral intervals.
Tuning theory
Intervals
17edo can be notated with diatonic notation with #/b = 2\17. Since the sharp can be split in two, half-sharps and half-flats for 1\17 alterations can be used too.
| Edostep | Cents | Name (Neutral diatonic / ADIN) | Diatonic notation with half sharps/flats |
|---|---|---|---|
| 0 | 0 | Unison | A |
| 1 | 70.6 | (Far)minor second | Bb, At |
| 2 | 141.2 | Neutral second | A#, Bd |
| 3 | 211.8 | (Far)major second | B |
| 4 | 282.4 | (Far)minor third | C |
| 5 | 352.9 | Neutral third | Db, Ct |
| 6 | 423.5 | (Far)major third | C#, Dd |
| 7 | 494.1 | Perfect fourth | D |
| 8 | 564.7 | Semiaugmented/neutral fourth | Eb, Dt |
| 9 | 635.3 | Semidiminished/neutral fifth | D#, Ed |
| 10 | 705.9 | Perfect fifth | E |
| 11 | 776.5 | (Far)minor sixth | F |
| 12 | 847.1 | Neutral sixth | Gb, Ft |
| 13 | 917.6 | (Far)major sixth | F#, Gd |
| 14 | 988.2 | (Far)minor seventh | G |
| 15 | 1058.9 | Neutral seventh | Ab, Gt |
| 16 | 1129.4 | (Far)major seventh | G#, Ad |
| 17 | 1200 | Octave | A |
Prime approximations
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +3.9 | -33.4 | +19.4 | +13.4 | +6.5 | -34.4 | -15.2 | +7.0 |
| Relative (%) | 0.0 | +5.6 | -47.3 | +27.5 | +19.0 | +9.3 | -48.7 | -21.5 | +9.9 | |
| Steps
(reduced) |
17
(0) |
27
(10) |
39
(5) |
48
(14) |
59
(8) |
63
(12) |
69
(1) |
72
(4) |
77
(9) | |
Scales
Some scales:
- MOS diatonic (5L2s, 3331331)
- Mosh/"malacotonic" (3L4s, 2323232)
- Lissotonic (3223322)
- Trachytonic (3313322)
| View • Talk • EditEqual temperaments | |
|---|---|
| EDOs | |
| Macrotonal | 5 • 7 • 8 • 9 • 10 • 11 |
| 12-23 | 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 |
| 24-35 | 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35 |
| 36-47 | 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47 |
| 48-59 | 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58 |
| 60-71 | 60 • 63 • 64 • 65 • 67 • 68 • 70 |
| 72-83 | 72 • 77 • 80 • 81 |
| 84-95 | 84 • 87 • 89 • 90 • 93 • 94 |
| Large EDOs | 99 • 104 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665 |
| Nonoctave equal temperaments | |
| Tritave | 4 • 9 • 13 • 17 • 26 • 39 |
| Fifth | 8 • 9 • 11 • 20 |
| Other | |
