15edo
15edo, or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave 2/1. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.
Theory
JI approximation
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its fifth being 720 cents. Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | +18.0 | +13.7 | -8.8 | +8.7 | +39.5 | -25.0 | +22.5 | +11.7 |
| Relative (%) | +0.0 | +22.6 | +17.1 | -11.0 | +10.9 | +49.3 | -31.2 | +28.1 | +14.7 | |
| Steps
(reduced) |
15
(0) |
24
(8) |
35
(4) |
42
(12) |
52
(6) |
56
(10) |
61
(1) |
64
(3) |
68
(7) | |
Chords
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).
Scales
15edo contains a large number of useful scales. Among them are onyx tuned to 2-2-2-2-2-2-3, the aforementioned Zarlino diatonic, and pentawood tuned to 2-1-2-1-2-1-2-1-2-1, which splits each 5edo-step into alternating large and small steps and contains the Zarlino diatonic as a subset. Pentawood is notable in that there is a perfect fifth on every note, which is distinct from even mosdiatonic where there is one diminished fifth, and the triads alternate between major and minor, with a harmonic seventh available on every root. This is distinct from 12edo's diminished scale (which follows a similar pattern, splitting 4edo) in which half of the notes lack a perfect fifth above them entirely.
The perfect fourth halves to 8/7 and doubles to 7/4.
Regular temperaments
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 24edo, and Blackwood with 10edo.
Notation
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for ups and downs notation, and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often KISS notation based on onyx or pentawood, or notation based on the Zarlino diatonic scale.
