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 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.

Approximation of prime harmonics in 15edo

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 (9)	35 (5)	42 (12)	52 (7)	56 (11)	61 (1)	64 (4)	68 (8)

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.