17edo: Difference between revisions
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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. | 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. | ||
Since 17edo's perfect eleventh is highly divisible (24 steps), you can do 2\17 x 12, 3\17 x 8, 4\17 x 6, 6\17 x 4, 8\17 x 3, or 12\17 x 2 movements to pump this "comma". | |||
== Tuning theory == | == Tuning theory == | ||
| Line 13: | Line 15: | ||
!Cents | !Cents | ||
!Name (Neutral diatonic / ADIN) | !Name (Neutral diatonic / ADIN) | ||
! | !Notation | ||
!JI interpretation (2.3.25.(11/7).13, 68edo subset) | |||
|- | |- | ||
|0 | |0 | ||
| Line 19: | Line 22: | ||
|Unison | |Unison | ||
|A | |A | ||
|1/1 | |||
|- | |- | ||
|1 | |1 | ||
| Line 24: | Line 28: | ||
|(Far)minor second | |(Far)minor second | ||
|Bb, At | |Bb, At | ||
|25/24 | |||
|- | |- | ||
|2 | |2 | ||
| Line 29: | Line 34: | ||
|Neutral second | |Neutral second | ||
|A#, Bd | |A#, Bd | ||
|13/12 | |||
|- | |- | ||
|3 | |3 | ||
| Line 34: | Line 40: | ||
|(Far)major second | |(Far)major second | ||
|B | |B | ||
|9/8 | |||
|- | |- | ||
|4 | |4 | ||
| Line 39: | Line 46: | ||
|(Far)minor third | |(Far)minor third | ||
|C | |C | ||
|33/28, 32/27 | |||
|- | |- | ||
|5 | |5 | ||
| Line 44: | Line 52: | ||
|Neutral third | |Neutral third | ||
|Db, Ct | |Db, Ct | ||
|16/13 | |||
|- | |- | ||
|6 | |6 | ||
| Line 49: | Line 58: | ||
|(Far)major third | |(Far)major third | ||
|C#, Dd | |C#, Dd | ||
|14/11 | |||
|- | |- | ||
|7 | |7 | ||
| Line 54: | Line 64: | ||
|Perfect fourth | |Perfect fourth | ||
|D | |D | ||
|4/3 | |||
|- | |- | ||
|8 | |8 | ||
| Line 59: | Line 70: | ||
|Semiaugmented/neutral fourth | |Semiaugmented/neutral fourth | ||
|Eb, Dt | |Eb, Dt | ||
|25/18, 18/13 | |||
|- | |- | ||
|9 | |9 | ||
| Line 64: | Line 76: | ||
|Semidiminished/neutral fifth | |Semidiminished/neutral fifth | ||
|D#, Ed | |D#, Ed | ||
|36/25, 13/9 | |||
|- | |- | ||
|10 | |10 | ||
| Line 69: | Line 82: | ||
|Perfect fifth | |Perfect fifth | ||
|E | |E | ||
|3/2 | |||
|- | |- | ||
|11 | |11 | ||
| Line 74: | Line 88: | ||
|(Far)minor sixth | |(Far)minor sixth | ||
|F | |F | ||
|11/7 | |||
|- | |- | ||
|12 | |12 | ||
| Line 79: | Line 94: | ||
|Neutral sixth | |Neutral sixth | ||
|Gb, Ft | |Gb, Ft | ||
|13/8 | |||
|- | |- | ||
|13 | |13 | ||
| Line 84: | Line 100: | ||
|(Far)major sixth | |(Far)major sixth | ||
|F#, Gd | |F#, Gd | ||
|56/33, 27/16 | |||
|- | |- | ||
|14 | |14 | ||
| Line 89: | Line 106: | ||
|(Far)minor seventh | |(Far)minor seventh | ||
|G | |G | ||
|16/9 | |||
|- | |- | ||
|15 | |15 | ||
| Line 94: | Line 112: | ||
|Neutral seventh | |Neutral seventh | ||
|Ab, Gt | |Ab, Gt | ||
|24/13 | |||
|- | |- | ||
|16 | |16 | ||
| Line 99: | Line 118: | ||
|(Far)major seventh | |(Far)major seventh | ||
|G#, Ad | |G#, Ad | ||
|48/25 | |||
|- | |- | ||
|17 | |17 | ||
| Line 104: | Line 124: | ||
|Octave | |Octave | ||
|A | |A | ||
|2/1 | |||
|} | |} | ||
The major third is a farmajor third, interpreted as 14/11 either with 17edo's patent val or as a subset of 68edo; 17edo is at the upper edge of [[gentle tuning]]. While 17edo has reasonable approximations of 7 and 11, they are not shared with 68edo, although their ratio 14/11 is. | |||
The edostep is a near-just 25/24, sometimes seen as the optimal size for a leading tone. | |||
=== Prime approximations === | === Prime approximations === | ||
| Line 111: | Line 135: | ||
=== Notation === | === Notation === | ||
Because a sharp is 2 steps in 17edo, it may be notated with neutral [[diatonic notation]] (semisharps and semiflats). | Because a sharp is 2 steps in 17edo, it may be notated with neutral [[diatonic notation]] (semisharps and semiflats). | ||
=== Tuning properties === | |||
== Scales == | == Scales == | ||
Latest revision as of 11:54, 8 June 2026
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.
Since 17edo's perfect eleventh is highly divisible (24 steps), you can do 2\17 x 12, 3\17 x 8, 4\17 x 6, 6\17 x 4, 8\17 x 3, or 12\17 x 2 movements to pump this "comma".
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) | Notation | JI interpretation (2.3.25.(11/7).13, 68edo subset) |
|---|---|---|---|---|
| 0 | 0 | Unison | A | 1/1 |
| 1 | 70.6 | (Far)minor second | Bb, At | 25/24 |
| 2 | 141.2 | Neutral second | A#, Bd | 13/12 |
| 3 | 211.8 | (Far)major second | B | 9/8 |
| 4 | 282.4 | (Far)minor third | C | 33/28, 32/27 |
| 5 | 352.9 | Neutral third | Db, Ct | 16/13 |
| 6 | 423.5 | (Far)major third | C#, Dd | 14/11 |
| 7 | 494.1 | Perfect fourth | D | 4/3 |
| 8 | 564.7 | Semiaugmented/neutral fourth | Eb, Dt | 25/18, 18/13 |
| 9 | 635.3 | Semidiminished/neutral fifth | D#, Ed | 36/25, 13/9 |
| 10 | 705.9 | Perfect fifth | E | 3/2 |
| 11 | 776.5 | (Far)minor sixth | F | 11/7 |
| 12 | 847.1 | Neutral sixth | Gb, Ft | 13/8 |
| 13 | 917.6 | (Far)major sixth | F#, Gd | 56/33, 27/16 |
| 14 | 988.2 | (Far)minor seventh | G | 16/9 |
| 15 | 1058.9 | Neutral seventh | Ab, Gt | 24/13 |
| 16 | 1129.4 | (Far)major seventh | G#, Ad | 48/25 |
| 17 | 1200 | Octave | A | 2/1 |
The major third is a farmajor third, interpreted as 14/11 either with 17edo's patent val or as a subset of 68edo; 17edo is at the upper edge of gentle tuning. While 17edo has reasonable approximations of 7 and 11, they are not shared with 68edo, although their ratio 14/11 is.
The edostep is a near-just 25/24, sometimes seen as the optimal size for a leading tone.
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) | |
Notation
Because a sharp is 2 steps in 17edo, it may be notated with neutral diatonic notation (semisharps and semiflats).
Tuning properties
Scales
Some scales:
- MOS diatonic (5L2s, 3331331)
- Mosh/"malacotonic" (3L4s, 2323232)
- Lissotonic (3223322)
- Trachytonic (3313322)
