17edo: Difference between revisions
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!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 20: | ||
|Unison | |Unison | ||
|A | |A | ||
|1/1 | |||
|- | |- | ||
|1 | |1 | ||
| Line 24: | Line 26: | ||
|(Far)minor second | |(Far)minor second | ||
|Bb, At | |Bb, At | ||
|25/24 | |||
|- | |- | ||
|2 | |2 | ||
| Line 29: | Line 32: | ||
|Neutral second | |Neutral second | ||
|A#, Bd | |A#, Bd | ||
|13/12 | |||
|- | |- | ||
|3 | |3 | ||
| Line 34: | Line 38: | ||
|(Far)major second | |(Far)major second | ||
|B | |B | ||
|9/8 | |||
|- | |- | ||
|4 | |4 | ||
| Line 39: | Line 44: | ||
|(Far)minor third | |(Far)minor third | ||
|C | |C | ||
|33/28, 32/27 | |||
|- | |- | ||
|5 | |5 | ||
| Line 44: | Line 50: | ||
|Neutral third | |Neutral third | ||
|Db, Ct | |Db, Ct | ||
|16/13 | |||
|- | |- | ||
|6 | |6 | ||
| Line 49: | Line 56: | ||
|(Far)major third | |(Far)major third | ||
|C#, Dd | |C#, Dd | ||
|14/11 | |||
|- | |- | ||
|7 | |7 | ||
| Line 54: | Line 62: | ||
|Perfect fourth | |Perfect fourth | ||
|D | |D | ||
|4/3 | |||
|- | |- | ||
|8 | |8 | ||
| Line 59: | Line 68: | ||
|Semiaugmented/neutral fourth | |Semiaugmented/neutral fourth | ||
|Eb, Dt | |Eb, Dt | ||
|25/18, 18/13 | |||
|- | |- | ||
|9 | |9 | ||
| Line 64: | Line 74: | ||
|Semidiminished/neutral fifth | |Semidiminished/neutral fifth | ||
|D#, Ed | |D#, Ed | ||
|36/25, 13/9 | |||
|- | |- | ||
|10 | |10 | ||
| Line 69: | Line 80: | ||
|Perfect fifth | |Perfect fifth | ||
|E | |E | ||
|3/2 | |||
|- | |- | ||
|11 | |11 | ||
| Line 74: | Line 86: | ||
|(Far)minor sixth | |(Far)minor sixth | ||
|F | |F | ||
|11/7 | |||
|- | |- | ||
|12 | |12 | ||
| Line 79: | Line 92: | ||
|Neutral sixth | |Neutral sixth | ||
|Gb, Ft | |Gb, Ft | ||
|13/8 | |||
|- | |- | ||
|13 | |13 | ||
| Line 84: | Line 98: | ||
|(Far)major sixth | |(Far)major sixth | ||
|F#, Gd | |F#, Gd | ||
|56/33, 27/16 | |||
|- | |- | ||
|14 | |14 | ||
| Line 89: | Line 104: | ||
|(Far)minor seventh | |(Far)minor seventh | ||
|G | |G | ||
|16/9 | |||
|- | |- | ||
|15 | |15 | ||
| Line 94: | Line 110: | ||
|Neutral seventh | |Neutral seventh | ||
|Ab, Gt | |Ab, Gt | ||
|24/13 | |||
|- | |- | ||
|16 | |16 | ||
| Line 99: | Line 116: | ||
|(Far)major seventh | |(Far)major seventh | ||
|G#, Ad | |G#, Ad | ||
|48/25 | |||
|- | |- | ||
|17 | |17 | ||
| Line 104: | Line 122: | ||
|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 133: | ||
=== 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 == | ||
Revision as of 00:03, 20 April 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.
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)
