15edo/Unque's compositional approach

As of recent, 15edo has been the subject of great debate in the xenharmonic community. Not only are many musicians skeptical of its harmonic content, but even advocates of the system disagree on how to interpret it and use it. On this page, I will present my personal experience with 15edo, and provide a potential framework that others may use to begin their own journeys through this strange and wonderful musical system.

As always, this page will be full of personal touches that may not reflect an objective truth or even wide consensus about how to use 15edo; I encourage learning musicians to experiment with different ideas and develop styles that best suit their own needs, rather than to take my word (or anyone else's for that matter) at face value as a great truth of music.

Intervals

15edo is most commonly interpreted as a subgroup of 11-limit harmony, though the exact intervals being represented is heavily debated.

Intervals of 15edo

Interval	Cents	JI intervals	As a generator	Notes
1\15	80	22/21, 21/20	Valentine	Melodic semitone
2\15	160	12/11, 10/9	Porcupine	One possible choice of whole tone (see below)
3\15	240	8/7	5edo	One possible choice of whole tone (see below)
4\15	320	6/5	Kleismic	Relatively accurate, though noticeably sharper than optimal kleismic
5\15	400	5/4, 14/11	3edo	Same mapping as 12edo
6\15	480	33/25, 4/3	5edo	Highly contentious interpretation; see below
7\15	560	15/11, 11/8	Thuja
8\15	640	16/11, 22/15	Thuja
9\15	720	3/2, 50/33	5edo	Highly contentious interpretation; see below
10\15	800	8/5, 11/7	3edo	Same mapping as 12edo
11\15	880	5/3	Kleismic	Relatively accurate, though noticeably flatter than optimal kleismic
12\15	960	7/4	5edo
13\15	1040	9/5, 11/6	Porcupine
14\15	1120	21/11	Valentine
15\15	1200	2/1		Tuned justly

15edo as a dual-9 system

The intervals 2\15 and 3\15 are both quite distant from a justly-tuned 9/8 interval; as such, some have proposed 15edo as being a "dual nines" system, in which these two intervals are both interpreted as flavors of the whole tone. This interpretation allows for a near-1:1 correspondence between the Left and Right hand versions of Nicetone (see below).

15edo and Carlos Alpha

The Alpha scale created by Wendy Carlos is a dual-octaves equal temperament system. Because the flatter of the two octaves is reached at fifteen steps, many people have offered that 15edo could be treated as a tuning of the Alpha scale that is stretched such that the flat octave is tuned justly. This interpretation provides an explanation for certain peculiarities that composers tend to converge on, such as the usage of [0 5 9 12 15] as an approximation of mode 4 of the Harmonic Series in spite of its high error.

The connection to the Carlos Alpha scale has notably been criticized due to its poor accuracy, and the lack of clear compositional equivalence between the two, especially beyond the first octave. Carlos Alpha in practice emphasizes 9/4 and 18/7 as fundamental consonances, whereas 15edo does not even represent either of these intervals accurately, let alone treat their approximations as fundamental. Additionally, the characteristic quark interval provided by octave-equivalent Gamelismic tunings (those that temper out 1029/1024, as Carlos Alpha does) has been tempered out in 15edo, which leads to extremely heavy error.

15edo and Mode 11

Mode 11 of the Harmonic Series provides another interesting way to interpret intervals of 15edo. Notably, the intervals [0 2 5 7 8 12 13 14 15] can be interpreted as an approximation of the chord 11:12:14:15:16:19:20:21:22. The /11 logic can be extended to supersets of mode 11 to provide interpretations of other intervals, such as mode 33 providing 50/33 as an extremely accurate interpretation of 9\15, and 55/33 as an interpretation of 11\15.

This interpretation may also be criticized due to a lack of accuracy, but it is notably more consistent than the Carlos Alpha interpretations as the difference between the tunings does not accrue per step.

15edo's fifth

The interval at 9\15 is possibly the most contentious interval in the entire xenharmonic community. Some have proposed that is represents 3/2 due to its clear function as a concordant fifth; others argue that 50/33 is more accurate and functions better alongside the other /11 intervals; still others have posited that 97/64 is even more accurate and simpler due to being a rooted overtone.

Notation

Because 15edo does not have a clear diatonic fifth, notation is quite difficult to nail down. Most interpretations treat a certain scale (usually of seven notes) as the nominals, and assign accidentals based on other scale modes.

15edo notations

Interval	Blackwood	Porcupine	Kleismic	Nicetone
0\15	C	C	C	B# = C
1\15	C#	C# = Db	B# = Db	C# = Db
2\15	Db	D	C#	D
3\15	D	D# = Eb	D	D#
4\15	D#	E	Eb	Eb
5\15	Fb	E# = Fb	D# = Fb	E = Fb
6\15	F	F	E	E# = F
7\15	F#	F# = Gb	F	F#
8\15	Gb	G	E# = Gb	Gb
9\15	G	G# = Hb	F# = Ab	G
10\15	G#	H	G	G# = Ab
11\15	Ab	H# = A	A	A
12\15	A	A# = Bb	G# = Bb	A#
13\15	A#	B	A# = Cb	Bb
14\15	Cb	B# = Cb	B	B = Cb
15\15	C	C	C	B# = C

Throughout the rest of this page, wherever notation is used, I will directly specify which notation.

Chords

Many attempts have been made to categorize the chords that can be made in 15edo. I have my own chord organization that is most useful for Blackwood structures, but other methods of organization also exist for various other purposes. The choice between different categorization methods for chords in 15edo depends primarily on which structure you are focusing on, and what kind of modal harmony complements that structure.

Chords of Porcupine

In the Porcupine scales, chords are made by stacking intervals of 4, 5, and 6 steps; this provides a 3x3 contrast for chord types, comparable to the three way distinction between Major, Minor, and Suspended in common-practice Western music.

Chords of Porcupine

Chord Type	Symbol	Formula	Notation (Porcupine)	Name (Unque)	Approximate ratio	Notes
Diminished	c°	4 + 4	C - Fbb - Hbb	Patdim 2	15:18:22	Symmetrical
Minor (root position)	c	4 + 5	C - Fbb - Hb	Pat 2	10:12:15	Delta-Rational
Major (first inversion)	H/C	4 + 6	C - Fbb - H	Pataug 2	5:6:8	Delta-Rational
Major (root posiion)	C	5 + 4	C - Fb - Hb	Pat 3	4:5:6	Delta-Rational
Augmented	C+	5 + 5	C - Fb - H	Pataug 3	16:20:25	Symmetrical
Minor (first inversion)	h#/c	5 + 6	C - Fb - H#	Metdim 3	12:15:20	Delta-Rational
Minor (second inversion)	f/c	6 + 4	C - F - H	Pataug 4	15:20:24
Major (second inversion)	F/C	6 + 5	C - F - H#	Metdim 4	3:4:5
Quartal	C4	6 + 6	C - F - Hx	Met 4	9:12:16	Symmetrical

Note that each of these chords is either symmetrical or DR in at least one inversion; no chord has both properties.

Scales

15edo supports a plethora of useful scales, each with its own unique character. While I of course can't detail every scale in the tuning, I will document the ones that I find to be the most important to its structure.

7L 1s

The 7L 1s scale is one of the most popular scales in 15edo, and it is much more versatile than its seeming uniformity would suggest. William Lynch suggests names for the modes based on sea creatures, with octopus being relevant to the number 8.

Modes of 7L1s

UDP	Step pattern	Notation (Porcupine)	Name (Lynch)	I chords	N/I chords
7|0	LLLLLLLs	C - D - E - F - G - H - A# - B# - C	Octupus	c°, C4	F4/C, A#4/C
6|1	LLLLLLsL	C - D - E - F - G - H - A# - B - C	Mantis	c°, C4	F4/C, A#4/C
5|2	LLLLLsLL	C - D - E - F - G - H - A - B - C	Dolphin	c°	F/C
4|3	LLLLsLLL	C - D - E - F - G - Hb - A - B - C	Crab	c°	F/C
3|4	LLLsLLLL	C - D - E - F - Gb - Hb - A - B - C	Tuna	c	F/C
2|5	LLsLLLLL	C - D - E - Fb - Gb - Hb - A - B - C	Salmon	c, C	a/c
1|6	LsLLLLLL	C - D - Eb - Fb - Gb - Hb - A - B - C	Starfish	C	a/c
0|7	sLLLLLLL	C - Db - Eb - Fb - Gb - Hb - A - B - C	Whale	C	a/c

5L 5s

The 5L 5s scale is another extremely popular scale, due to its relative simplicity and incorporation of useful melodic and harmonic ideas. It is an example of a mode of limited transposition, which means that certain modes of the scale are just transpositions of the same mode into different keys; for instance, the Bright Mode in the key of C has the same notes as the bright mode in the key of D, but they start on different root pitches.

Modes of 5L5s

Mode	Step Pattern	Notation (Blackwood)
Bright	LsLsLsLsLs	C - Db - D - Fb - F - Gb - G - Ab - A - Cb - C
Dark	sLsLsLsLsL	C - C# - D - D# - F - F# - G - G# - A - A# - C

4L 3s

The 4L 3s scale, while not nearly as commonly used as the previous two, is another important structural scale. In this scale, the large step is three steps of 15edo, rather than two as in the previous scales; as such, the different modes provide much more contrast with one another than in the previous scales. Ayceman offers to name the seven modes in relation to the Almsivi in Morrowmind (from the Elder Scrolls). The tonic chord can be made by taking degrees I-III-V, I-III-VI, or I-IV-VI of the scale.

Modes of 4L3s

UDP	Step Pattern	Notation (Kleismic)	Name (Ayceman)	I-III-V chord	I-III-VI chord	I-IV-VI chord
6|0	LLsLsLs	C - D - E - F - G - A - B - C	Nerevarine	Pataug 4	Metdim 4	Metdim 5
5|1	LsLLsLs	C - D - Eb - F - G - A - B - C	Vivecan	Pataug 2	Metdim 2	Metdim 5
4|2	LsLsLLs	C - D - Eb - F - Gb - A - Bb - C	Lorkhanic	Patdim2	Metdim 2	Metdim 5
3|3	LsLsLsL	C - D - Eb - F - Gb - A - Bb - C	Sothic	Patdim2	Metdim 2	Metdim 5
2|4	sLLsLsL	C - Db - Eb - F - Gb - A - Bb - C	Kagrenacan	Patdim2	Metdim 2	Metdim 5
1|5	sLsLLsL	C - Db - Eb - Fb - Gb - A - Bb - C	Almalexian	Patdim2	Metdim 2	Metdim 3
0|6	sLsLsLL	C - Db - Eb - Fb - Gb - Ab - Bb - C	Dagothic	Patdim2	Pat 2	Pat 3