22edo/Unque's compositional approach
NOTE: This page is currently under construction, and will be subject to major expansion in the near future. Come back soon!
22 Equal Divisions of the Octave is arguably the smallest EDO to support the full 11-limit; it is also the intersection of many popular temperaments such as Superpyth, Porcupine, Orwell, and Magic. Additionally, fans of 15edo will likely be drawn to 22edo due to the latter being quite useful as an extension of the former that represents many low-complexity intervals with higher accuracy. On this page, I will present my personal experience with 22edo, and hopefully provide a potential framework that others may use to begin their own journeys through the colorful world of 22 Equal Divisions of the Octave.
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 22edo; 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
22edo is often regarded as a full 11-limit system or some subgroup thereof; I will here prioritize intervals of the 7-limit, as they better describe the harmony of 22edo, but 11-limit intervals will additionally be used where applicable.
| Interval | Cents | JI Intervals | As a generator | Notation | Notes |
|---|---|---|---|---|---|
| 0\22 | 0 | 1/1 | C | ||
| 1\22 | 54.5 | 36/35, 28/27, 25/24 | Escapade | D♭, A𝄪 | |
| 2\22 | 109.1 | 16/15 | 11edo | B♯, E𝄫 | |
| 3\22 | 163.6 | 10/9 | Porcupine | C♯, F𝄫 | |
| 4\22 | 218.2 | 9/8, 8/7 | Wizard, Jubilic | D | |
| 5\22 | 272.7 | 7/6 | Orwell | E♭, B𝄪 | Often considered more dissonant than a true 7/6. |
| 6\22 | 327.3 | 6/5 | Orgone (actually in 11edo) | F♭, C𝄪 | Somewhat contentious JI interpretation (see below). |
| 7\22 | 381.8 | 5/4 | Magic | D♯, G𝄫 | |
| 8\22 | 436.3 | 9/7, 14/11 | Sensamagic | E | |
| 9\22 | 490.9 | 4/3 | Superpyth | F | |
| 10\22 | 545.5 | 15/11, 11/8 | Joan | G♭, D𝄪 | Somewhat contentious JI interpretation. |
| 11\22 | 600.0 | 7/5, 10/7, 45/32, 64/45 | 2edo; period for several temps | E♯, A𝄫 | |
| 12\22 | 654.5 | 16/11, 22/15 | Joan | F♯ | |
| 13\22 | 709.1 | 3/2 | Superpyth | G | |
| 14\22 | 763.6 | 14/9, 11/7 | Sensamagic | A♭, E𝄪 | |
| 15\22 | 818.2 | 8/5 | Magic | B𝄫, F𝄪 | This one specific note is very contentious in notation. |
| 16\22 | 872.7 | 5/3 | Orgone (actually in 11edo) | G♯, C𝄫 | |
| 17\22 | 927.3 | 12/7 | Orwell | A | |
| 18\22 | 981.8 | 7/4, 16/9 | Wizard, Jubilic | B♭ | |
| 19\22 | 1036.3 | 9/5 | Porcupine | C♭, G𝄪 | |
| 20\22 | 1090.9 | 15/8 | 11edo | A♯, D𝄫 | |
| 21\22 | 1145.5 | 48/25, 27/14, 35/18 | Escapade | B | |
| 22\22 | 1200.0 | 2/1 | C |
Ups and downs notation uses the accidentals ^ and v to modify a given note by one step of 22edo
The thirds of 22edo
22edo has two pairs of thirds: a major/minor pair, and a supermajor/subminor pair; despite most often being viewed as an 11-limit system, it lacks clear representation for the neutral thirds that are characteristic of 11-limit harmony.
The subminor third at 5\22 represents 7/6 with moderate accuracy, though it is significantly less consonant than the JI representation. Its fifth complement is the supermajor third at 8\22, which is an excellent representation of 9/7. This interval is perhaps better paired with 14\22 than with 13\22, as the former can be interpreted as 11/7 and thus provides the more consonant otonal 7:9:11 triad.
The minor third at 6\22 is contentious in its interpretation; it is quite sharp as a representation of 6/5, though not sharp enough to constitute a neutral third. Its fifth complement, the major third at 7\22, is a much clearer 5/4, the two being practically indistinguishable to the untrained ear.
Scales
5L 2s
The 5L 2s scale is one of two types of Diatonic scales represented in 22edo, and represents the shade of Diatonic popularized by the Greek mathematician Archytas, which uses the 2.3.7 subgroup of Just Intonation. It is generated by the Circle of Fifths just as in common practice tunings, though the perfect fifth of 22edo is significantly sharper than any tuning attested in common practice, which makes the scale and its chords behave somewhat differently from standard Diatonic functionality. Notably, suspended chords feel less tense and more restful, while typical tertian triads tend to sound tenser and more energetic; this is the opposite of how these chords behave in Meantone tunings, and as such Western music translated into 22edo's Diatonic may sound strange if it is not adjusted to account for its unique properties.
| UDP | Step Pattern | Notation | Name |
|---|---|---|---|
| 6|0 | LLLsLLs | C - D - E - F♯ - G - A - B - C | Lydian |
| 5|1 | LLsLLLs | C - D - E - F - G - A - B - C | Ionian |
| 4|2 | LLsLLsL | C - D - E - F - G - A - B♭ - C | Mixolydian |
| 3|3 | LsLLLsL | C - D - E♭ - F - G - A - B♭ - C | Dorian |
| 2|4 | LsLLsLL | C - D - E♭ - F - G - A♭ - B♭ - C | Aeolian |
| 1|5 | sLLLsLL | C - D♭ - E♭ - F - G - A♭ - B♭ - C | Phrygian |
| 0|6 | sLLsLLL | C - D♭ - E♭ - F - G♭ - A♭ - B♭ - C | Locrian |
5L 7s
The 5L 7s scale is an extension of 5L 2s created by continuing the generator sequence. Because the Circle of Fifths is bidirectional, the seven modes can be extended either by continuing the sequence upwards or downwards; those created by going up the chain are called grave modes, and those extended by going down the chain are called acute modes.
| UDP | Step Pattern | Notation | Name | Notes |
|---|---|---|---|---|
| 11|0 | LsLsLssLsLss | Grave Lydian | Like the seven-note Lydian, lacks a Perfect Fourth over the root. | |
| 10|1 | LsLssLsLsLss | Grave Ionian | ||
| 9|2 | LsLssLsLssLs | Grave Mixolydian | ||
| 8|3 | LssLsLsLssLs | Grave Dorian | ||
| 7|4 | LssLsLssLsLs | Grave Aeolian | ||
| 6|5 | sLsLsLssLsLs | Grave Phrygian | Also accounts for Acute Lydian | |
| 5|6 | sLsLssLsLsLs | Acute Ionian | Also accounts for Grave Locrian | |
| 4|7 | sLsLssLsLssL | Acute Mixolydian | ||
| 3|8 | sLssLsLsLssL | Acute Dorian | ||
| 2|9 | sLssLsLssLsL | Acute Aeolian | ||
| 1|10 | ssLsLsLssLsL | Acute Phrygian | ||
| 0|11 | ssLsLssLsLsL | Acute Locrian | Like the seven-note Locrian, lacks a Perfect Fifth over the root. |
3L 2M 2s
The 3L 2M 2s scale is the other type of Diatonic scale represented in 22edo; this one represents the 5-limit shade of Diatonic popularized by musicians and theorists such as Ptolemy and Zarlino. It is generated by alternating intervals 6\22 and 7\22, and as such yields two different forms ("left-hand" and "right-hand") based on which generator comes first and which comes second.
The 5-limit harmony of this scale is much more reminiscent of the familiar harmony used in common-practice music, so popular music translated into this scale will be much more faithful to the harmony of the original piece than it would be using 5L 2s.
| UDP | Step pattern | Notation | Name | Notes |
|---|---|---|---|---|
| 6|0 | LMLsLMs | C - D - vE - vF♯ - G - A - vB - C | Lydian | Tritone is precisely the semioctave |
| 5|1 | LsLMsLM | C - D - ^E♭ - ^F - G - ^A♭ - ^B♭ - C | Aeolian | Contains the Wolf fourth (^F) |
| 4|2 | LMsLMLs | C - D - vE - F - G - vA - vB - C | Ionian | |
| 3|3 | sLMLsLM | C - ^D♭ - ^E♭ - F - G - vA♭ - ^B♭ - C | Phrygian | |
| 2|4 | MLsLMsL | C - vD - vE - F - G - vA - B♭ - C | Mixolydian | |
| 1|5 | sLMsLML | C - ^D♭ - ^E♭ - F - ^G♭ - ^A♭ - B♭ - C | Locrian | Tritone is precisely the semioctave |
| 0|6 | MsLMLsL | C - vD - E♭ - F - vG - vA - B♭ - C | Dorian | Contains the Wolf fifth (vG) |
| UDP | Step pattern | Notation | Name | Notes |
|---|---|---|---|---|
| 6|0 | LsLMLsM | C - D - ^E♭ - ^F - G - A - ^B♭ - C | Dorian | Contains the Wolf fourth (^F) |
| 5|1 | LMLsMLs | C - D - vE - vF♯ - G - vA - vB - C | Lydian | Tritone is precisely the semioctave |
| 4|2 | LsMLsLM | C - D - ^E♭ - F - G - ^A♭ - ^B♭ - C | Aeolian | |
| 3|3 | MLsLMLs | C - vD - vE - F - G - vA - vB - C | Ionian | |
| 2|4 | sLMLsML | C - ^D♭ - ^E♭ - F - G - ^A♭ - B♭ - C | Phrygian | |
| 1|5 | MLsMLsL | C - vD - vE - F - vG - vA - B♭ - C | Mixolydian | Contains the Wolf fifth (vG) |
| 0|6 | sMLsLML | C - ^D♭ - E♭ - F - ^G♭ - ^A♭ - B♭ - C | Locrian | Tritone is precisely the semioctave |
When disambiguation is needed between the modes of 5L 2s and 3L 2M 2s, the former can be described as "Superpyth" or "Archy," while the latter can be called "Nicetone" or "Zarlino"
5L 2m 3s
The 5L 2m 3s scale functions as an extension to 3L 2M 2s, created by adding a commatic step into each large step of that Diatonic scale. This provides the scale with three new modes in addition to the seven already present. Due to the commatic step, this scale does not distinguish the left-hand from the right-hand variants.
While there is no singular generator sequence used to create this scale, it can be thought of as two pentatonic scales offset by a neutral second, which is coincidentally the amount by which a sharp or flat alters a note; this makes the scale easy to notate as a pentatonic scale and its sharp/flat counterpart. This splits the modes into five "grave" modes and five "acute" modes, where the grave modes place the root on the flattened pentatonic, and the acute modes place the root on the sharpened one.
| UDP | Step pattern | Notation | Name | Notes |
|---|---|---|---|---|
| 4|0 | LmLsLmLsLs | C - C♯ - E♭ - E - F - F# - A♭ - A - B♭ - B - C | Aeolian | Contains the Wolf fifth (A♭) |
| 3|1 | LmLsLsLmLs | C - C♯ - E♭ - E - F - F# - G - G♯ - B♭ - B - C | Dorian | |
| 2|2 | LsLmLsLmLs | C - C♯ - D - D♯ - F - F# - G - G♯ - B♭ - B - C | Mixolydian | |
| 1|3 | LsLmLsLsLm | C - C♯ - D - D♯ - F - F# - G - G♯ - A - A♯ - C | Ionian | |
| 0|4 | LsLsLmLsLm | C - C♯ - D - D♯ - E - E♯ - G - G♯ - A - A♯ - C | Lydian | Brightness is reversed - Lydian is the darkest grave mode! |
| UDP | Step pattern | Notation | Name | Notes |
|---|---|---|---|---|
| 4|0 | mLsLmLsLsL | C - E𝄫 - E♭ - F♭ - F - A𝄫 - A♭ - B𝄫 - B♭ - C♭ - C | Locrian | Brightness is reversed - Locrian is the brightest acute mode! |
| 3|1 | mLsLsLmLsL | C - E𝄫 - E♭ - F♭ - F - G♭ - G - B𝄫 - B♭ - C♭ - C | Phrygian | |
| 2|2 | sLmLsLmLsL | C - D♭ - D - F♭ - F - G♭ - G - B𝄫 - B♭ - C♭ - C | Aeolian | |
| 1|3 | sLmLsLsLmL | C - D♭ - D - F♭ - F - G♭ - G - A♭ - A - C♭ - C | Dorian | |
| 0|4 | sLsLmLsLmL | C - D♭ - D - E♭ - E - G♭ - G - A♭ - A - C♭ - C | Mixolydian | E is used as a Wolf fourth, not a Supermajor Third |
Where disambiguation is needed between the modes of 5L 7s and 5L 2M 2s, the former can be described as "chromatic," and the latter as "blackdye."
2L 8s
The 2L 8s scale is another characteristic scale of 22edo, which was discovered independently by Paul Erlich, Gene Ward Smith, and Olivier Messiaen (though the latter theorist documented the scale's analog in 12edo rather than in 22).
This scale is a mode of limited transposition, which means that it has some amount of rotational symmetry. In this case, the semioctave is the axis of symmetry, and transposing a mode up by 11 steps of 22edo (or 6 steps of 12edo in the case of Messiaen's works) yields the same set of pitches as the mode on which you started.
Erlich divides four of the five unique rotations into a 2x2 grid, categorized by how many thirds it contains (one third is "static," two is "dynamic,") and which type of third falls on the fourth degree of the scale (major or minor). The brightest mode does not appear to have been considered by Erlich, due to lacking a perfect fifth above the root.
| UDP | Step pattern | Notation | Name | Notes |
|---|---|---|---|---|
| 8|0 | LssssLssss | Wolf Minor | Contains the wolf fifth | |
| 6|2 | sLssssLsss | Dynamic Major | Dual thirds are subminor and major | |
| 4|4 | ssLssssLss | Static Major | Laterally symmetrical (flipping the scale upside-down yields the same pitches) | |
| 2|6 | sssLssssLs | Static Minor | Messiaen treated this rotation as the "default" | |
| 0|8 | ssssLssssL | Dynamic Minor | Dual thirds are minor and supermajor |