User:Unque/15edo Composition Theory
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 | 23/22, 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, 38/33 | 5edo, Slendric | 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; triforce period | Same mapping as 12edo |
| 6\15 | 480 | 29/22, 4/3 | 5edo; blacksmith period | Highly contentious interpretation; see below |
| 7\15 | 560 | 15/11, 11/8 | Thuja | |
| 8\15 | 640 | 16/11 | Thuja | |
| 9\15 | 720 | 3/2, 50/33 | 5edo; blackwood period | Highly contentious interpretation; see below |
| 10\15 | 800 | 8/5, 35/22 | 3edo; triforce period | Same mapping as 12edo |
| 11\15 | 880 | 5/3 | Kleismic | Relatively accurate, though noticeably flatter than optimal kleismic |
| 12\15 | 960 | 7/4, 19/11 | 5edo, Slendric | |
| 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).
Where the two types of whole tone need be disambiguated, they can respectively be called the greater and lesser whole tones (after their size) or the Bayati and Slendric seconds (after the structures they generate).
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, alongside its supersets, 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 22 providing 23/22 as an interpretation of 1\15, 29/22 as 6\15, and 35/22 as 10\15; additionally, mode 33 provides 38/33 as 3\15, 50/33 as 9\15, and 55/33 as 11\15.
This interpretation may also be criticized once again 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.
Dual tritones
15edo has two different tritone intervals, each about a quartertone away from the classic semioctave tritone. These tritones may actually be considered consonances in the context of 15edo harmony, as they approximate the 11th harmonic with only approximately 10% relative error. They are quite useful as fully diminished and half diminished fifths respectively, in chords such as the Ptolemismic Triad. Chords containing these tritones are often useful as dominant chords for voice leading and functional harmony (see below)
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 | Nicetone |
|---|---|---|---|---|
| 0\15 | C | C | C | B# = C |
| 1\15 | C# | C# = Db | B# = Db | C# |
| 2\15 | Db | D | C# | Db |
| 3\15 | D | D# = Eb | D | D |
| 4\15 | D# | E | Eb | D# = 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 = Ab | 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 |
The choice of which notation system to use depends heavily on what types of structures are being emphasized. Throughout the rest of this page, wherever notation is used, I will directly specify which type.
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, which can be compared to Major, Minor, and Suspended chords of common-practice Western music.
| Chord Type | Symbol | Formula | Notation (Porcupine) | Name (Unque) | Approximate ratio | Notes |
|---|---|---|---|---|---|---|
| Diminished | c° | 4 + 4 | C - Fbb - G | Patdim 2 | 15:18:22 | Symmetrical |
| Minor (root position) | c | 4 + 5 | C - Fbb - G# | Pat 2 | 10:12:15 | Delta-Rational |
| Major (first inversion) | H/C | 4 + 6 | C - Fbb - Gx | 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 - Ab | Pataug 4 | 15:20:24 | |
| Major (second inversion) | F/C | 6 + 5 | C - F - A | Metdim 4 | 3:4:5 | |
| Quartal | C4 | 6 + 6 | C - F - A# | 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.
| UDP | Step pattern | Notation (Porcupine) | Name (Lynch) | I chords | N/I chords |
|---|---|---|---|---|---|
| 7|0 | LLLLLLLs | C - D - E - F - G - H - A# - B# - C | Octopus | 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 |
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.
| 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 |
3L 2M 2s
The 3L 2M 2s scale is often used as an analog to Diatonic in 15edo, as its step pattern resembles that of the Zarlino scale that was historically used as a ternary version of Diatonic that was considered to have more consonant thirds. Whereas the true Zarlino scale was made by alternating 5/4 and 6/5 as generators, 15edo's 3L 2M 2s scale can be made by alternating 5\15 and 4\15 generators. Rather than tempering out the syntonic comma (the difference between the two types of whole tone) as in common-practice Western music, 15edo tempers the scale such that the syntonic comma is equal to the semitone.
There are two versions of the 3L 2M 2s scale; the left-hand version results when one begins the sequence on a minor third, and the right-hand version results when one begins the sequence on a major third. Each of these versions has seven unique modes.
| UDP | Step pattern | Notation (Nicetone) | Name | Tonic Chord |
|---|---|---|---|---|
| 6|0 | LMLsLMs | C - D - E - Fx - G - A# - B - C | Lydian | Pat 3 |
| 5|1 | LsLMsLM | C - D - Ebb - F# - G - Ab - Bb - C | Aeolian | Pat 1 |
| 4|2 | LMsLMLs | C - D - E - F - G - A - B - C | Ionian | Pat 3 |
| 3|3 | sLMLsLM | C - Db - Ebb - F - G - Ab - Bb - C | Phrygian | Pat 1 |
| 2|4 | MLsLMsL | C - Db - E - F - G - A - Bbb - C | Mixolydian | Pat 3 |
| 1|5 | sLMsLML | C - Dbb - Eb - F - Gbb - Ab - Bbb - C | Locrian | Susaug 2 |
| 0|6 | MsLMLsL | C - Db - Ebb - F - Gb - A - Bbb - C | Dorian | Patdim 1 |
| UDP | Step pattern | Notation (Nicetone) | Name | Tonic Chord |
|---|---|---|---|---|
| 6|0 | LsLMLsM | C - D - Eb - F# - G - A# - Bb - C | Dorian | Pat 2 |
| 5|1 | LMLsMLs | C - D - E - Fx - G - A - B - C | Lydian | Pat 3 |
| 4|2 | LsMLsLM | C - D - Ebb - F - G - Ab - Bb - C | Aeolian | Pat 1 |
| 3|3 | MLsLMLs | C - Db - E - F - G - A - B - C | Ionian | Pat 3 |
| 2|4 | sLMLsML | C - Db - Ebb - F - G - Ab - Bbb - C | Phrygian | Pat 1 |
| 1|5 | MLsMLsL | C - Db - E - F - Gb - A - Bbb - C | Mixolydian | Patdim 3 |
| 0|6 | sMLsLML | C - Dbb - Ebb - F - Gbb - Ab - Bbb - C | Locrian | Susaug 1 |
Functional Harmony
Useful harmonic progressions may arise in a number of ways depending on the scale being used and depending on what chord the composer wishes to tonicize. Here, I will document some examples of how functional harmonic progressions may be created in the different scales of 15edo, with concepts that can be extended to apply to any scale.
Note that I will be constructing these chord progressions from back to front; this means that we will start with the resolution, then find the dominant chord, and then find a subdominant to precede it.
Elements of Functional Harmony
Just like in common-practice theory, the chords of 15edo have a tendency to rotate about the Circle of Fifths; due to the heavy damage to the 3-limit, however, this tendency is not quite as strong in 15edo as it is in more accurate tunings. Additionally, unlike in common-practice music, the Circle of Fifths in 15edo does not encompass every single note in the tuning; instead, it forms three distinct "rings" of fifths (each one forming 5edo with some kind of offset) that do not share any notes in common.
Because the step size in 15edo is significantly smaller than the typical semitone, leading tones are tenser than in common-practice music. The Bayati second can be used as a useful element of voice leading, though it is not nearly as tense as the semitone. I consider voice leading to be the single most important element of 15edo harmony, because it provides a consistent sense of direction throughout melodies.
Finally, it is important to notice certain tense intervals that have a tendency to voice lead by contrary motion to certain other intervals. Specifically, the Major Tritone at has a tendency to resolve inward and become a Perfect Fourth, and the Minor Tritone has a tendency to resolve outwards and become a Perfect Fifth.
Example: Chord Progression in C Ionian
For this example, I will use the right-hand C Ionian scale (C D E F G A B C in Nicetone notation), and I will treat the Major (Pat 3) Triad as the tonic chord.
Because the 3L 2M 2s scale is reminiscent of the common-practice Diatonic scale, elements of functional harmony from classical traditions may carry over into 15edo; however, it is important to note that 15edo's inaccuracy in the 5-limit may cause some amount of distortion in the functions of certain chords. Notably, the Wolf Fifth that usually occurs in the Dorian mode is no longer a fifth at all, but rather a major tritone.
A recognizable Major (Pat 3) Triads occurs on the fifth degree of the scale, providing a familiar circle-of-fifths resolution as well as a leading tone from the B of the V chord into the C of the tonic chord. The subminor (Pat 1) triad on the third degree provides an interesting voice leading into the V chord if voiced correctly (with the notes E, G, and B respectively leading to D, G, and B). Finally, the major (Pat 3) triad on the fourth degree provides a leading tone from F to E and from C to B.
Ultimately, our four-chord progression looks like C - F - em - G, or I - IV - iii - V. This progression prioritizes voice leading to create a coherent and flowing sound, and provides a great framework for melodies to be written over top.
Example: Chord Progression in C Starfish
For this next example, I will use the C Starfish scale (C D Eb Fb Gb Hb A B C in Porcupine notation), and I will once again treat the Major (Pat 3) Triad as the tonic chord.
First, notice that the small step occurs between D and Eb in this mode; this step is the most important place to note in the scale, as it plays a major role in voice leading. Here, neither D nor Eb is present in the chord that we want to tonicize, which means that we won't be able to rely on it as a leading tone. Instead, we might choose to rely on the Circle of Fifths pull that can be established by moving the root by intervals of 5edo. In this case, we can use the diminished chord on Hb for dominant function.
Because the Hb chord contains D, the subdominant chord that precedes it may use Eb as a leading tone. In this case, I will use the major chord rooted on B; the note B carries over from the B chord to the Hb chord, and the note Eb in the B chord leads into the D of the Hb chord.
Finally, we can select a nondominant function that emerges from the tonic at the beginning of the progression. I will use the minor chord on A, because it sounds unresolved without sounding too tense.
Ultimately, our four-chord progression is C - am - B - hb°, or I - vii - VIII - vi°. This progression uses a combination of voice leading, circle of fifths movement, and tension and release to achieve a useful and functional sound, and similar principles can be applied to other scales to create similar functional progressions.
Superstructures and Modulation
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Todo: expand |
Due to its plethora of useful structures with so few notes per octave, 15edo compositions can make great use of modulation from one structure to another; if used well, this modulation may be less comparable to Western key changes, and more so to Jins changes in Maqam traditions.
What are Superstructures?
I will here be using the term "superstructure" to describe any singular overarching structure that contains multiple constituent structures within it. For instance, a scale that contains a mode of 7L 1s over a given tonic, plus a second copy of that mode with its tonic a Perfect Fifth above the first, would provide a useful superstructure that allows for modulation between the two keys.
Superstructures may contain multiple copies of the same structure, multiple entirely different structures, or some combination of both. 15edo itself may additionally be taken as a single superstructure that contains all possible constituent structures over all possible roots.
5L 5s as a Superstructure
The 5L 5s scale in 15edo contains 3L 2M 2s as a constituent structure; each note of 5L 5s is the root of several 3L 2M 2s modes, which means that the 5L 5s scale can be used as a means by which to modulate from one key of 3L 2M 2s to another.
In the bright mode of 5L 5s (C Db D Fb F Gb G A A# Cb C in Nicetone notation), we can see that the Ionian mode of 3L 2M 2s exists over the first degree (C) as well as the eighth degree (A#). Because the C Ionian scale does not contain the note A#, we would normally not be able to modulate directly from one to the other without passing through at least one other scale; however, since 5L 5s is acting as a superstructure, we can easily use the superstructure to move smoothly from the key of C to the key of A# without needing to introduce other structures in passing.
An extension of the chord progression from before (C - F - fbm - G) may be expanded to move through the 5L 5s structure and resolve to A rather than C; for instance, we might use the major chord on Fb as a transitional chord that leads from G into A, since it promotes circle of fifths movement and has clear, smooth voice leading.