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 | Unnamed Balzano temp | 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 | Unnamed Balzano temp | 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 |
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 - Bb - C | Mixolydian |
| 3|3 | LsLLLsL | C - D - Eb - F - G - A - Bb - C | Dorian |
| 2|4 | LsLLsLL | C - D - Eb - F - G - Ab - Bb - C | Aeolian |
| 1|5 | sLLLsLL | C - Db - Eb - F - G - Ab - Bb - C | Phrygian |
| 0|6 | sLLsLLL | C - Db - Eb - F - Gb - Ab - Bb - C | Locrian |
5L 7s
The 5L 7s scale is an extension of 5L 2s created by continuing the generator sequence. It has twelve unique rotations, which can be categorized as Grave and Acute forms of the seven diatonic 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 is 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 - Ev - F#v - G - A - Bv - C | Lydian | Tritone is precisely the semioctave |
| 5|1 | LsLMsLM | C - D - Eb^ - F^ - G - Ab^ - Bb^ - C | Aeolian | Contains the Wolf fourth |
| 4|2 | LMsLMLs | C - D - Ev - F - G - Av - Bv - C | Ionian | |
| 3|3 | sLMLsLM | C - Db^ - Eb^ - F - G - Ab^ - Bb^ - C | Phrygian | |
| 2|4 | MLsLMsL | C - Dv - Ev - F - G - Av - Bb - C | Mixolydian | |
| 1|5 | sLMsLML | C - Db^ - Eb^ - F - Gb^ - Ab^ - Bb - C | Locrian | Tritone is precisely the semioctave |
| 0|6 | MsLMLsL | C - Dv - Eb - F - Gv - Av - Bb - C | Dorian | Contains the Wolf fifth |
| UDP | Step pattern | Notation | Name | Notes |
|---|---|---|---|---|
| 6|0 | LsLMLsM | C - D - Eb^ - F^ - G - A - Bb^ - C | Dorian | Contains the Wolf fourth |
| 5|1 | LMLsMLs | C - D - Ev - F#v - G - Av - Bv - C | Lydian | Tritone is precisely the semioctave |
| 4|2 | LsMLsLM | C - D - Eb^ - F - G - Ab^ - Bb^ - C | Aeolian | |
| 3|3 | MLsLMLs | C - Dv - Ev - F - G - Av - Bv - C | Ionian | |
| 2|4 | sLMLsML | C - Db^ - Eb^ - F - G - Ab^ - Bb - C | Phrygian | |
| 1|5 | MLsMLsL | C - Dv - Ev - F - Gv - Av - Bb - C | Mixolydian | Contains the Wolf fifth |
| 0|6 | sMLsLML | C - Db^ - Eb - F - Gb^ - Ab^ - Bb - 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 e.g. "Superpyth Ionian" or "Superpyth Aeolian," and the latter as e.g. "Nice Ionian" or "Nice Aeolian."