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.
| Interval | Cents | JI intervals | As a generator | Notes |
|---|---|---|---|---|
| 1\15 | 80 | 22/21, 21/20 | Valentine | May act as a stretched version of Carlos Alpha (see below) |
| 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 | |
| 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 and Carlos Alpha
The Alpha scale created by Wendy Carlos is a dual-octaves equal temperament system. Because the flat octave 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 octave is tuned justly.
Treating 15edo as a stretched version of Carlos Alpha provides an interesting lens with which to view its intervals. While some object to this view on the grounds of accuracy, it is undeniable that this interpretation helps explain many peculiarities about 15edo composition that composers tend to converge on, such as the usage of [0 5 9 12 15] as an approximation of the 4::8 segment.
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 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.
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.
Where do I stand in this debate? I think all of these interpretations are useful in their own right, and succeed in explaining various properties of the interval - ultimately, it is all of them and none of them.
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.
| Interval | Blackwood | Porcupine | Kleismic |
|---|---|---|---|
| 0\15 | C | C | C |
| 1\15 | C# | C# = Db | B# = Db |
| 2\15 | Db | D | C# |
| 3\15 | D | D# = Eb | D |
| 4\15 | D# | E | Eb |
| 5\15 | Fb | E# = Fb | D# = Fb |
| 6\15 | F | F | E |
| 7\15 | F# | F# = Gb | F |
| 8\15 | Gb | G | E# = Gb |
| 9\15 | G | G# = Hb | F# = Ab |
| 10\15 | G# | H | G |
| 11\15 | Ab | H# = A | A |
| 12\15 | A | A# = Bb | G# = Bb |
| 13\15 | A# | B | A# = Cb |
| 14\15 | Cb | B# = Cb | B |
| 15\15 | C | C | 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.
| 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 | Isodifferential |
| Major (first inversion) | H/C | 4 + 6 | C - Fbb - H | Pataug 2 | 5:6:8 | Isodifferential |
| Major (root posiion) | C | 5 + 4 | C - Fb - Hb | Pat 3 | 4:5:6 | Isodifferential |
| 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 | Isodifferential |
| 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 isodifferential in at least one inversion; neither 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.
| 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.
| 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 |