22edo
22edo, or 22 equal divisions of the octave, is the equal tuning featuring steps of (1200/22) ~= 54.5 cents, 22 of which stack to the perfect octave 2/1. It is not a meantone system, but it is a functional 11-limit system, with 3 at ~709 cents, 5 at ~382 cents, 7 at ~982 cents, and 11 at ~545 cents.
Note: Interval names in this article use the ADIN system.
Theory
Edostep interpretations
22edo's edostep has the following interpretations in the 2...11.17 subgroup:
- 36/35 (the difference between 5/4 and 9/7)
- 81/80 (the difference between 5/4 and 81/64)
- 25/24 (the difference between 5/4 and 6/5)
- 49/48 (the difference between 8/7 and 7/6)
- 33/32 (the difference between 4/3 and 11/8)
- 34/33 (the difference between 17/16 and 33/32)
JI approximation
22edo is comparable in accuracy in the 7-limit to 12edo in the 5-limit. Because it is not a meantone system, a better diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 22edo as 4-3-2-4-3-4-2, which is 22edo's tempering of Ptolemy's intense diatonic scale. However, 22edo also features a MOS diatonic of 4-4-1-4-4-4-1, which (unlike the Zarlino scale) contains no wolf fifth.
Even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +7.1 | -4.5 | +13.0 | -5.9 | -22.3 | +4.1 | -24.8 | +26.3 | +6.8 | +0.4 |
| Relative (%) | 0.0 | +13.1 | -8.2 | +23.8 | -10.7 | -41.0 | +7.6 | -45.4 | +48.2 | +12.4 | +0.8 | |
| Steps
(reduced) |
22
(0) |
35
(13) |
51
(7) |
62
(18) |
76
(10) |
81
(15) |
90
(2) |
93
(5) |
100
(12) |
107
(19) |
109
(21) | |
Instead of just offering minor and major chords, 22edo offers four types of chords bounded by a perfect fifth and containing a third--subminor, (penta)minor, (penta)major, and supermajor.
| Quality | Subminor | Pentaminor | Pentamajor | Supermajor |
|---|---|---|---|---|
| Cents | 273 | 327 | 382 | 436 |
| Just interpretation | 7/6 | 6/5 | 5/4 | 9/7 |
Thirds available in the diatonic MOS scale are bolded.
Chords
Because it approximates the 7-limit, 22edo supports the harmonic tetrad 4:5:6:7, tuned as [0 7 13 18], and because 5/4 and 7/4 are separated by half an octave, it also supports an alteration shared with any jubilic temperament in which the 5 and 7 are both flattened by a chroma, resulting in the "minor harmonic tetrad" [0 6 13 17], approximating [1/1 6/5 3/2 12/7]. As a consequence, the distance between 5/4 and 6/5 is narrowed, and the distance between 7/4 and 12/7 is widened.
Scales
A scale in 22edo with similar properties to 12edo's diatonic that takes advantage of the important structural role of the semioctave in the aforementioned tetrads is jaric (2L 8s), with the tuning 2-2-2-2-3-2-2-2-2-3. This means that 22edo can be usefully thought of as not just adding more qualities to existing ordinals, but adding three new ordinals with their own qualities, roughly surrounding 8/7, the semioctave, and 7/4. This has the function of giving the simplest 7-limit intervals their own category separate from sixths and sevenths, much as the simplest 5-limit intervals have their own diatonic category in the form of thirds.
From a diatonic perspective, 22edo has four varieties of third: subminor (7/6, 5\22), pentaminor (6/5, 6\22), pentamajor (5/4, 7\22), and supermajor (9/7, 8\22).
Notation
This section provides the standard diatonic notation for 22edo, which takes as a basis the Pythagorean diatonic scale (4-4-1-4-4-4-1) and uses the standard accidentals # and b to raise and lower by 3 edosteps respectively, and the accidentals ^ and v to raise and lower by a single edostep.
Pythagorean notation
In 22edo, the space between each of the notes that is separated by 2 steps in 12edo is instead 4 steps; notes separated by a single step remain that way. As a result, each sharp or flat can be split into three distinct notes. It is important to understand the usage of enharmonic equivalence here; unlike in systems such as 31edo where each note has an easily derivable "canonical" notation, it is important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system).
| D | |
| Eb | ^D |
| ^Eb | vD# |
| vE | D# |
| E | |
Accidentals
22edo's accidentals, as mentioned and demonstrated previously, consist of sharps and flats, as well as up and down accidentals. A sharp moves up three steps, a flat moves down three steps, and as a result ups and downs have a nice sequence to them when combined with multiple sharps and flats. A sharp is one step less than a whole tone, meaning that the diatonic and chromatic semitones straddle the true "semitone" of 22edo.
| Accidental | Steps | Interval |
|---|---|---|
| vbb | -7 | Pentamajor Third |
| bb | -6 | Pentaminor Third |
| ^bb | -5 | Subminor Third |
| vb | -4 | Supermajor Second / Whole Tone |
| b | -3 | Pentamajor Second |
| ^b | -2 | Pentaminor Second |
| v | -1 | Subminor Second |
| Natural | 0 | Unison |
| ^ | 1 | Subminor Second |
| v# | 2 | Pentaminor Second |
| # | 3 | Pentamajor Second |
| ^# | 4 | Supermajor Second / Whole Tone |
| vx | 5 | Subminor Third |
| x | 6 | Pentaminor Third |
| ^x | 7 | Pentamajor Third |
Example: The C Major Scale
The scale consists of diatonic semitones and whole tones. The diatonic semitones are between E and F, as well as between B and C. Meanwhile, the remaining steps in the scale are whole tones.
Note that half of a 22edo whole tone is not the diatonic semitone being used here. The diatonic semitone is the same size as a quarter-tone, and the complementary chromatic semitone is a 3/4-tone. So, in 22edo, a "semitone" may refer to any interval smaller than a whole tone.
The enharmonic equivalents of 22edo are not the same as in 12edo. Eb is lower than D# by half of a whole tone. However, ^Eb and vD# are the same pitch.
Scales
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1.
Pythagorean diatonic
This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of 22edo. This is good for septimal harmony, not so much for 5-limit harmony (to get 5-limit thirds, you have to go 9 steps up!). A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone (in fact, it is enharmonic to the small whole-tone used in the Zarlino system). This can make interval classifications somewhat unintuitive if ups and downs are not used - for instance, a pentamajor triad, 0-7-13, is C-D#-G (meaning that a third-sized interval is technically a second) - and in fact 22edo is the first edo other than 15edo (which is strange in its own right) to require ups and downs to notate all intervals without problems with intuition like this (Edos like 17 and 10 utilize semisharps and semiflats, which 22edo cannot use as it does not have neutral intervals.)
Zarlino diatonic
A faithful way of representing the 5-limit diatonic structure in 22edo (which, unlike its just counterpart, keeps the 7-limit convenient to access) is to use 5/4 as the scale's major third, and likewise 5/3 and 15/8 as the major sixth and seventh respectively. The scale pattern for zarlino is 4-3-2-4-3-4-2. This encounters problems with existing familiarity with notation (for example, one of D-G and G-C must now be a flat 5th, called a "wolf fifth" and representing 16/11 as opposed to 3/2), however it does have one thing going for it: in 22edo in particular (and a family of edos including 15, 29, and 37), the interval which separates Pythagorean intervals from their 5-limit counterparts is the exact same as the interval separating 5-limit major and minor intervals. This means that, if # and b are used to represent this interval as an accidental, no additional accidentals are necessary. This is why the 3-step interval is a "pentamajor second" along with being a chromatic semitone.
Decatonic solutions
One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.
Pajara
Note that the 1-step interval which serves as the 3-limit diatonic semitone is the very same as the 5-limit chromatic semitone, and that the 3-step interval serving as the 3-limit chromatic semitone also happens to map to a certain kind of large 5-limit diatonic semitone (such that it and the chromatic semitone stack to the 4-step whole tone). This suggests that if there was a way to swap the diatonic and chromatic semitones, we could faithfully represent the full 7-limit, including 5.
And as it turns out, there is a way to do that. We start with the pentatonic scale, not only because it's closer to even but because the interval between its small and large steps is exactly the 1-step interval we want to function as our chroma. Then, we add another pentatonic scale that is offset by a tritone from the first one. This results in the decatonic "pajara" scale (3-2-2-2-2-3-2-2-2-2), where every note has a corresponding note a tritone apart. It solves the problem of representing intervals of both 5 and 7 by introducing three new ordinal classes to provide space for 7-limit intervals to fit on their own degrees of the scale. That way, 7/4 isn't a subminor seventh, it's a major version of the pajara 8-step. One can even define a notation system for pajara, wherein the notes are numbered 0-9 and # and b represent alterations by a single step. Pajara retains the property where most notes have a fifth over them.
To extend to the 11-limit, pajara[12] (1-2-2-2-2-2-1-2-2-2-2-2) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step.
Blackdye
Blackdye is a rank-3 scale similar to zarlino diatonic, which attempts to compromise between zarlino and Pythagorean diatonic in a way, by including a few intervals from both at once. 22edo can do this because the difference between 3- and 5-limit intervals is exaggerated so much. Blackdye can be thought of as dividing each 4-step whole tone into a 3-step tone and a single step called an aberrisma, which separates zarlino and Pythagorean intervals. The blackdye scale pattern is 3-1-3-2-3-1-3-1-3-2. The most useful way to use blackdye is to essentially treat it as multiple overlapping diatonics, which you modulate between based on necessity. You can somewhat see it as the 22edo counterpart of the two forms of B in old choir music, or of the raised seventh degree in the 12edo minor scale. The best notation for blackdye in 22edo in particular is just the same as the Zarlino diatonic notation, where a # or b represents an aberrisma as well as a chromatic semitone. (In the general case, you'd notate the aberrisma with a + or - sign.)
| Accidental | Steps | Interval |
|---|---|---|
| bb | -2 | Pentaminor Second |
| b | -1 | Subminor Second |
| Natural | 0 | Unison |
| # | 1 | Subminor Second |
| x | 2 | Pentaminor Second |
Comparison of notation systems
| Edostep | Cents | Interval name (ADIN) | Just Intonation | Pajara | Mosdiatonic | Blackdye/Zarlino |
|---|---|---|---|---|---|---|
| 0 | 0 | Perfect 1sn | 1/1 | 0 | C | C |
| 1 | 55 | Subminor 2nd | 36/35, 25/24 | 1b | Db | C# |
| 2 | 109 | Pentaminor 2nd | 16/15 | 1 | vC#, ^Db | Db |
| 3 | 164 | Pentamajor 2nd | 10/9 | 1# | C# | D |
| 4 | 218 | Supermajor 2nd | 9/8, 8/7 | 2 | D | D# |
| 5 | 273 | Subminor 3rd | 7/6 | 2# | Eb | Ebb / Dx |
| 6 | 327 | Pentaminor 3rd | 6/5 | 3b | vD#, ^Eb | Eb |
| 7 | 382 | Pentamajor 3rd | 5/4 | 3 | D# | E |
| 8 | 436 | Supermajor 3rd | 9/7 | 4b | E | E# |
| 9 | 491 | Perfect 4th | 4/3 | 4 | F | F |
| 10 | 545 | Penta4th | 11/8 | 4# | Gb | F# |
| 11 | 600 | Tritone | 7/5, 10/7 | 5 | vF#, ^Gb | Gbb / Fx |
| 12 | 655 | Penta5th | 16/11 | 6b | F# | Gb |
| 13 | 709 | Perfect 5th | 3/2 | 6 | G | G |
| 14 | 764 | Subminor 6th | 14/9 | 6# | Ab | G# |
| 15 | 818 | Pentaminor 6th | 8/5 | 7 | vG#, ^Ab | Ab |
| 16 | 873 | Pentamajor 6th | 5/3 | 7# | G# | A |
| 17 | 927 | Supermajor 6th | 12/7 | 8b | A | A# |
| 18 | 982 | Subminor 7th | 7/4, 16/9 | 8 | Bb | Bbb / Ax |
| 19 | 1036 | Pentaminor 7th | 9/5 | 9b | vA#, ^Bb | Bb |
| 20 | 1091 | Pentamajor 7th | 15/8 | 9 | A# | B |
| 21 | 1145 | Supermajor 7th | 48/25, 35/18 | 9# | B | Cb |
| 22 | 1200 | Octave | 2/1 | 10 | C | C |
Triads and tetrads
Triads bounded by P5
| Name | 1 | 2 | Bounding interval | Edostep | Chart |
|---|---|---|---|---|---|
| Sus4 triad | Perfect 4th | Supermajor 2nd | Perfect 5th | [0 9 13] | ├────────┴───┴────────┐ |
| Supermajor triad | Supermajor 3rd | Subminor 3rd | Perfect 5th | [0 8 13] | ├───────┴────┴────────┐ |
| Pentamajor triad | Pentamajor 3rd | Pentaminor 3rd | Perfect 5th | [0 7 13] | ├──────┴─────┴────────┐ |
| Pentaminor triad | Pentaminor 3rd | Pentamajor 3rd | Perfect 5th | [0 6 13] | ├─────┴──────┴────────┐ |
| Subminor triad | Subminor 3rd | Supermajor 3rd | Perfect 5th | [0 5 13] | ├────┴───────┴────────┐ |
| Sus2 triad | Supermajor 2nd | Perfect 4th | Perfect 5th | [0 4 13] | ├───┴────────┴────────┐ |
Tetrads with P5th
Harmonic tetrads
These represent the conventional structure of a 4:5:6:7 harmonic tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect median above the middle note. This is the characteristic kind of chord found in pajara temperament, and can also be thought of the stacking of a fifth-bounded triad and a fourth-bounded triad.
| Name | 1 | 2 | 3 | 4 | Bounding interval 1 | Bounding interval 2 | Bounding interval 3 | Edostep | Chart |
|---|---|---|---|---|---|---|---|---|---|
| Pentamajor harmonic tetrad | Pentamajor 3rd | Pentaminor 3rd | Subminor 3rd | Supermajor 2nd | Perfect 5th | Subminor 7th | Perfect 8ve | [0 7 13 18] | ├──────┴─────┴────┴───┤ |
| Pentaminor harmonic tetrad | Pentaminor 3rd | Pentamajor 3rd | Supermajor 2nd | Subminor 3rd | Perfect 5th | Supermajor 6th | Perfect 8ve | [0 6 13 17] | ├─────┴──────┴───┴────┤ |
Diatonic tetrads
These represent the conventional structure of an 8:10:12:15 diatonic major tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect fifth above the middle note. The sus4 diatonic tetrad is the same as the triad version up to inversion.
| Name | 1 | 2 | 3 | Bounding interval 1 | Bounding interval 2 | Edostep | Chart |
|---|---|---|---|---|---|---|---|
| Supermajor diatonic tetrad | Supermajor 3rd | Subminor 3rd | Supermajor 3rd | Perfect 5th | Supermajor 7th | [0 8 13 21] | ├───────┴────┴───────┴┤ |
| Pentamajor diatonic tetrad | Pentamajor 3rd | Pentaminor 3rd | Pentamajor 3rd | Perfect 5th | Pentamajor 7th | [0 7 13 20] | ├──────┴─────┴──────┴─┤ |
| Pentaminor diatonic tetrad | Pentaminor 3rd | Pentamajor 3rd | Pentaminor 3rd | Perfect 5th | Pentaminor 7th | [0 6 13 19] | ├─────┴──────┴─────┴──┤ |
| Subminor diatonic tetrad | Subminor 3rd | Supermajor 3rd | Subminor 3rd | Perfect 5th | Subminor 7th | [0 5 13 18] | ├────┴───────┴────┴───┤ |
| Sus2 diatonic tetrad | Supermajor 2nd | Perfect 4th | Supermajor 2nd | Perfect 5th | Supermajor 6th | [0 4 13 17] | ├───┴────────┴───┴────┤ |
Scales
Symmetric scale
One possible scale of 22edo, as mentioned previously, is the pajara[10] decatonic scale, represented as ├─┴─┴──┴─┴─┴─┴─┴──┴─┴─┤. The modes of this scale (in order of both brightness and rotation) are as follows.
| Name | Chart | 2 | 3 | 4 | 6 | 8 | 9 | Mode on fifth | Mode on fourth |
|---|---|---|---|---|---|---|---|---|---|
| ├─┴─┴─┴─┴──┴─┴─┴─┴─┴──┤ | minor | minor | dim | perfect | minor | minor | |||
| ├─┴─┴─┴──┴─┴─┴─┴─┴──┴─┤ | minor | minor | perfect | perfect | minor | major | |||
| ├─┴─┴──┴─┴─┴─┴─┴──┴─┴─┤ | minor | major | perfect | perfect | major | major | |||
| ├─┴──┴─┴─┴─┴─┴──┴─┴─┴─┤ | major | major | perfect | perfect | major | major | |||
| ├──┴─┴─┴─┴─┴──┴─┴─┴─┴─┤ | major | major | perfect | aug | major | major |
Note that minor and major are swapped compared to standard heptatonic modes.
Pentachordal scale
This scale is constructed from two identical "pentachords" and the semioctave. Unlike the symmetric scale, there are ten distinct modes of the pentachordal scale. Ordered by rotation, they are as follows:
| Name | Chart | 2 | 3 | 4 | 6 | 8 | 9 | Mode on fifth | Mode on fourth |
|---|---|---|---|---|---|---|---|---|---|
| Bediyic | ├─┴─┴─┴─┴─┴──┴─┴─┴─┴──┤ | minor | minor | dim | perfect | minor | minor | Hininic | - |
| Skoronic | ├─┴─┴─┴─┴──┴─┴─┴─┴──┴─┤ | minor | minor | dim | perfect | minor | major | Aujalic | - |
| Moriolic | ├─┴─┴─┴──┴─┴─┴─┴──┴─┴─┤ | minor | minor | perfect | perfect | major | major | Mielauic | Hininic |
| Staimosic | ├─┴─┴──┴─┴─┴─┴──┴─┴─┴─┤ | minor | major | perfect | perfect | major | major | Prathuic | Aujalic |
| Sebaic | ├─┴──┴─┴─┴─┴──┴─┴─┴─┴─┤ | major | major | perfect | aug | major | major | - | Mielauic |
| Awanic | ├──┴─┴─┴─┴──┴─┴─┴─┴─┴─┤ | major | major | perfect | aug | major | major | - | Prathuic |
| Hininic | ├─┴─┴─┴──┴─┴─┴─┴─┴─┴──┤ | minor | minor | perfect | perfect | minor | minor | Moriolic | Bediyic |
| Aujalic | ├─┴─┴──┴─┴─┴─┴─┴─┴──┴─┤ | minor | major | perfect | perfect | minor | major | Staimosic | Skoronic |
| Mielauic | ├─┴──┴─┴─┴─┴─┴─┴──┴─┴─┤ | major | major | perfect | perfect | major | major | Sebaic | Moriolic |
| Prathuic | ├──┴─┴─┴─┴─┴─┴──┴─┴─┴─┤ | major | major | perfect | perfect | major | major | Awanic | Staimosic |
The mode names are taken from ten of Leriendil's eleven Neiran tribes, as they are derivative of mode names for the Tellurian ├─┴─┴─┴┴─┴─┴─┴─┴┴─┴─┴─┤ scale, which was inspired by Leriendil's music theory.
Other scales
| Name | Chart | Notes |
|---|---|---|
| Onyx | ├──┴──┴──┴───┴──┴──┴──┤ | Greek scale (equable diatonic), onyx, basic MOS of porcupine. |
| Zarlino diatonic | ├─┴───┴──┴───┴─┴───┴──┤ | Greek scale (intense diatonic). Zarlino rank-3 diatonic. |
| Mosdiatonic | ├┴───┴───┴───┴┴───┴───┤ | Greek scale (Pythagorean or Archytas diatonic). Basic MOS of superpyth |
| Zarlino pentatonic | ├─────┴──┴───┴─────┴──┤ | One possible pentatonic analog to the Zarlino diatonic. |
| Blackdye | ├─┴──┴┴──┴┴──┴─┴──┴┴──┤ | 22edo analog of 15edo porcupine blackwood scale |
| Kee'ra | ├─┴┴──┴──┴─┴─┴─┴┴─┴───┤ | The Chair of Mr. Bob. |
| Pentic | ├────┴───┴───┴────┴───┤ | Basic MOS of superpyth |
