22edo
22edo, or 22 equal divisions of the octave (sometimes called 22-TET or 22-tone equal temperament), is the equal tuning with a step size of 1200/22 ~= 54.5 cents, dividing 2/1 into 22 steps.
22edo is the fourth-smallest EDO with a diatonic (5L 2s) MOS scale formed by a chain of fifths, which has a hardness of 4:1. It achieves this with a perfect fifth tuned sharpward (~709¢) so that the same interval comprises 9/8 and 8/7. Its logic is therefore that of Archy (or Superpyth) temperament, rather than Meantone: that is, the minor and major thirds available in the diatonic MOS approximate the septal thirds, 7/6 and 9/7, often called "subminor" and "supermajor" (including in the ADIN system for melodic qualities, which will be used in the remainder of this article).
As an even EDO, 22edo includes the 600¢ tritone familiar from 12edo, but it divides neither the perfect fourth nor fifth in half, meaning that it does not include semifourths or neutral thirds. It divides the perfect fourth (9\22) in three, however, implying that a tetrachord of three equal intervals is possible in 22edo. 22edo also includes 11edo as a subset, and similarly to 6edo (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
22edo distinguishes its native subminor and supermajor thirds from approximations to 5-limit intervals, 6/5 and 5/4 (called "nearminor" and "nearmajor" thirds in ADIN). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full 7-limit harmony, as it is the first to distinctly (and consistently) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the 11th harmonic, although many 11-limit intervals are not distinguished from 5-limit intervals (e.g. 11/9 is mapped to the same interval as 6/5), as well as the 17th.
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of keemic temperaments such as porcupine. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.
General theory
Derivation of 22edo
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.
Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.
From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.
JI approximation
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that 25/24, the chroma separating the classical major triad 4:5:6 and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, 49/48, the chroma separating 6:7:8 from its complement, is exaggerated, in fact to the same size as 25/24. This gives 7/5 the most damage out of the 7-odd-limit, tuning it (and thus 10/7) to the semioctave at 600¢. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3¢ sharp, approximating quarter-comma archy tuning.
22edo also approximates the interval 11/10 to within 1.4¢, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and 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. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13).
Among the higher primes, 22edo approximates 17/16 as two steps and 32/29 as three steps, and one step of 22edo is extremely close to 32/31. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as 29/24, which is only about 0.35¢ off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
| 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) | |
Intervals of 22edo
A list of intervals is available at 22edo/Intervals, which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.
Edostep interpretations
22edo's edostep has the following interpretations in the 7-limit:
- 25/24 (the difference between 5/4 and 6/5)
- 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
- 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
- 49/48 (the difference between 8/7 and 7/6)
- 81/80 (the difference between 10/9 and 9/8)
Including prime 11, it additionally serves as:
- 22/21 (the difference between 7/6 and 11/9, or 14/11 and 4/3)
- 33/32 (the difference between 4/3 and 11/8, or 12/11 and 9/8)
- 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
- 56/55 (the difference between 5/4 and 14/11, or 11/8 and 7/5).
- 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]
Notation systems and a table of intervals
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).
The native-fifths or diatonic notation system is the most commonly used system, and the one that most microtonal notation systems support by default. A sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case). Ups and downs raise and lower by one edostep respectively.
22edo also supports any notation system for Pajara and Porcupine.
ADIN will be used for interval names in 22edo. This is also consistent with the interval names used on the Porcupine page.
JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
| Edostep | Cents | 11-limit add-17 JI approximation |
Notation | Interval category (ADIN) | ||
|---|---|---|---|---|---|---|
| Native-fifths (ups & downs) |
Blackdye/Zarlino (Vector) |
Pajara decatonic | ||||
| 0 | 0 | 1/1 | C | C | 0 | Perfect unison |
| 1 | 54.5 | 25/24, 28/27, [33/32], 36/35 | ^C, Db | C# | 1b | (Sub)minor second |
| 2 | 109.1 | [16/15], 15/14, 18/17, [17/16] | vC#, ^Db | Db | 1 | Nearminor second |
| 3 | 163.6 | 10/9, [11/10], 12/11 | C#, vD | D | 1# | Nearmajor second |
| 4 | 218.2 | 8/7, 9/8, [17/15] | D | D# | 2 | (Super)major second |
| 5 | 272.7 | 7/6 | ^D, Eb | Ebb / Dx | 2# | (Sub)minor third |
| 6 | 327.3 | 6/5, 11/9, 17/14 | vD#, ^Eb | Eb | 3b | Nearminor third |
| 7 | 381.8 | [5/4] | D#, vE | E | 3 | Nearmajor third |
| 8 | 436.4 | [9/7], 14/11, 32/25 | E | E# | 4b | (Super)major third |
| 9 | 490.9 | 4/3 | F | F | 4 | Perfect fourth |
| 10 | 545.5 | 11/8, 15/11 | ^F, Gb | F# | 4# | Near fourth |
| 11 | 600 | 7/5, 10/7, [17/12] | vF#, ^Gb | Gbb / Fx | 5 | Tritone |
| 12 | 654.5 | 16/11, 22/15 | F#, vG | Gb | 6b | Near fifth |
| 13 | 709.1 | 3/2 | G | G | 6 | Perfect fifth |
| 14 | 763.6 | [14/9], 11/7, 25/16 | ^G, Ab | G# | 6# | (Sub)minor sixth |
| 15 | 818.2 | [8/5] | vG#, ^Ab | Ab | 7 | Nearminor sixth |
| 16 | 872.7 | 5/3, 18/11, 28/17 | G#, vA | A | 7# | Nearmajor sixth |
| 17 | 927.3 | 12/7 | A | A# | 8b | (Super)major sixth |
| 18 | 981.8 | 7/4, 16/9, [30/17] | ^A, Bb | Bbb / Ax | 8 | (Sub)minor seventh |
| 19 | 1036.4 | 9/5, [20/11], 11/6 | vA#, ^Bb | Bb | 9b | Nearminor seventh |
| 20 | 1090.9 | [15/8], 28/15, 17/9, [32/17] | A#, vB | B | 9 | Nearmajor seventh |
| 21 | 1145.5 | 48/25, 27/14, [64/33], 35/18 | B | Cb | 9# | (Super)major seventh |
| 22 | 1200 | 2/1 | C | C | 0 | Octave |
Solfege
Solfege may use the porcupine solfege systems.
Tempering properties
Tempered commas
Important commas tempered out by the 11-limit of 22et include:
- 50/49 (jubilismic), equating 7/5 and 10/7 to exactly half an octave.
- 55/54 (telepath), equating 6/5 with 11/9
- 64/63 (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to 7/4
- 99/98 (mothwellsmic), equating 14/11 with 9/7
- 100/99 (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to 16/11
- 121/120 (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with 15/11
- 176/175 (valinorsmic), equating a stack of two 5/4s to 11/7
- 225/224 (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to 14/9
- 245/243 (sensamagic), equating a stack of two 9/7s to 5/3
- 250/243 (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
- 385/384 (keenanismic), equating the product of 7/6 and 5/4 to 16/11
Regular temperaments associated with these are discussed in #Notable structural chains. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form 8/5 (orwellismic). In terms of S-expressions, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.
Arithmetic progressions
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.
This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh val (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming Ringer 22fh: 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.
Notable structural chains
22edo has five distinct intervals that generate octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).
3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a quill defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called Porcupine, and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like 15 and 37, as well as 29edo aside from the mapping of 7.
5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in Orwell temperament, while if stacked further, four 7/6s are made to reach 15/8, so that 3/1 is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like 31 and 53edo, though note that the 11-limit is less accurate than the 7-limit component in general.
7\22 represents a flattened 5/4, five of which stack to 3/1, which is Magic temperament. The deficit between the octave and three 5/4s, 128/125, is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like 19 and 41edo.
Finally, 9\22 represents 4/3, two of which stack to 7/4 in Archy/Superpyth temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in 27 and 49edo; this is not shared with even sharper tunings of Archy, such as 37edo.
22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is Pajara, generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings 10edo and 12edo). This general structure without prime 7, known as Diaschismic, however, is supported by notable EDOs such as 34 and 46edo.
In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see #Consonance and dissonance properties) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.
11edo temperaments
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset.
Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the sensamagic category of temperaments, which in 11edo specifically becomes sentry. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "father" temperament, although that is not supported by 11edo patent.)
Another temperament that resides in 11edo is called orgone, and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.
Compositional theory
Tertian structure
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.
| Quality | Subminor | Nearminor | Nearmajor | Supermajor |
|---|---|---|---|---|
| Cents | 273 | 327 | 382 | 436 |
| Just interpretation | 7/6 (+5.9¢) | 6/5 (+11.6¢) | 5/4 (-4.5¢) | 9/7 (+1.3¢) |
Diatonic thirds are bolded.
Scales
Scale theory
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, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. 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. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).
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.
22edo supports the various heptatonic scales supported by porcupine (see Porcupine#Scales) - namely, superpyth diatonic, zarlino diatonic, and porcupine equiheptatonic.
It also supports the Pajara[10] scale, which evenly divides each step of the MOS pentatonic scale.
More scales may be found at 22edo/Scales.
Generator sequences
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce a similar 8-note scale to the original, but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].
Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.
Building scales from tetrachords
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). More info can be found at Tetrachord.
There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12). In 22edo, there are also four chromatic tetrachords (5-2-2, 5-3-1, 6-2-1, and 6-1-2), and one enharmonic tetrachord (7-1-1).
Other polychordal structures
It's also possible to use trichords to build scales in 22edo. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same. Pentachords may also be used; the most common pentachord is the pajara pentachord. The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.
Tables of scales
The following is a table of scales in 22edo.
Porcupine scales
MOS scales generated by a nearmajor second.
| Name | Chart | Notes |
|---|---|---|
| Onyx | ├──┴──┴──┴───┴──┴──┴──┤ 3 3 3 4 3 3 3 | The same as the "equable Dorian" discussed above. |
| Pine | ├──┴──┴──┴──┴┴──┴──┴──┤ 3 3 3 3 1 3 3 3 | |
| Roklotic | ├┴─┴┴─┴┴─┴┴─┴┴─┴┴─┴┴─┴┤ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 | The "Roklotian" scale mentioned in the #Equiheptatonic section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale. |
Orwell scales
MOS scales generated by a subminor third.
| Name | Chart | Notes |
|---|---|---|
| Manual | ├────┴────┴────┴────┴─┤ 5 5 5 5 2 | The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11. |
| Gramitonic | ├──┴─┴──┴─┴──┴─┴──┴─┴─┤ 3 2 3 2 3 2 3 2 2 | The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale. |
| Antiparagonic | ├┴─┴─┴┴─┴─┴┴─┴─┴┴─┴─┴─┤ 1 2 2 1 2 2 1 2 2 1 2 2 2 | A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat. |
Magic scales
MOS scales generated by a nearmajor third.
| Name | Chart | Notes |
|---|---|---|
| Mosh | ├─────┴┴─────┴┴─────┴┴┤ 6 1 6 1 6 1 1 | Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualizes as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree. |
| Sephiroid | ├────┴┴┴────┴┴┴────┴┴┴┤ 5 1 1 5 1 1 5 1 1 1 | |
| Antiluachoid | ├───┴┴┴┴───┴┴┴┴───┴┴┴┴┤ 4 1 1 1 4 1 1 1 4 1 1 1 1 |
Superpyth scales
MOS scales generated by a perfect fifth.
| Name | Chart | Notes |
|---|---|---|
| Pentic | ├───┴────┴───┴────┴───┤ 4 5 4 5 4 | One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic". |
| Mosdiatonic | ├───┴┴───┴───┴───┴┴───┤ 4 1 4 4 4 1 4 | A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo. |
| P-chromatic | ├──┴┴┴──┴┴──┴┴──┴┴┴──┴┤ 3 1 1 3 1 3 1 3 1 1 3 1 |
Half-octave scales
MOS scales generated against the half-octave.
| Temperament | Name | Chart | Notes |
|---|---|---|---|
| Pajara | jaric | ├─┴─┴──┴─┴─┴─┴─┴──┴─┴─┤ 2 2 3 2 2 2 2 3 2 2 | |
| telluric | ├─┴─┴─┴┴─┴─┴─┴─┴─┴┴─┴─┤ 2 2 2 1 2 2 2 2 2 1 2 2 | Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space. | |
| Hedgehog | malic | ├──┴──┴────┴──┴──┴────┤ 3 3 5 3 3 5 | One of three tunings of malic available in 22edo. |
| ekic | ├──┴──┴──┴─┴──┴──┴──┴─┤ 3 3 3 2 3 3 3 2 | ||
| - | ├┴─┴┴─┴┴─┴─┴┴─┴┴─┴┴─┴─┤ 1 2 1 2 1 2 2 1 2 1 2 1 2 2 | ||
| Astrology | citric | ├──┴───┴───┴──┴───┴───┤ 3 4 4 3 4 4 | One of two tunings of citric available in 22edo. |
| lemon | ├──┴──┴┴──┴┴──┴──┴┴──┴┤ 3 3 1 3 1 3 3 1 3 1 | ||
| Doublewide | citric | ├┴────┴────┴┴────┴────┤ 1 5 5 1 5 5 | One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case. |
| lime | ├┴┴───┴┴───┴┴┴───┴┴───┤ 1 1 4 1 4 1 1 4 1 4 |
Additional scales
| Name | Chart | Notes |
|---|---|---|
| Zarlino pentatonic | ├─────┴──┴───┴─────┴──┤ 6 3 4 6 3 | One possible pentatonic analog to the Zarlino diatonic. |
| Zarlino | ├─┴───┴──┴───┴─┴───┴──┤ 2 4 3 4 2 4 3 | The 5-limit diatonic in 22edo. |
| Pentachordal pajara | ├─┴─┴──┴─┴─┴─┴──┴─┴─┴─┤ 2 2 3 2 2 2 3 2 2 2 | |
| Tellurian | ├─┴─┴─┴┴─┴─┴─┴─┴┴─┴─┴─┤ 2 2 2 1 2 2 2 2 1 2 2 2 |
Harmony
Consonance and dissonance properties
Generally, the set of consonances in 22edo is considered to be the 9-odd-limit, with some exceptions: because the tritone (7/5 or 10/7) is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second (10/9) and nearminor seventh (9/5)'s proximity to the unison and octave have a similar effect, along with being closer to 11/10 and 20/11 (which are in the 11-odd-limit). The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.
An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.
An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).
Modal and functional harmony.
There are two distinct approaches to harmony in 22edo: pajara and diatonic (porcupine) (more info found on their respective pages), each with tonal and modal approaches.
In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.
However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.
Tables of chords
The following is a table of chords in 22edo.
The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see Vector's chord names. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.
Fifth-bounded tertian triads
Three-note chords built out of thirds, bounded by a perfect fifth.
| Name | Third | Fifth | Edostep |
|---|---|---|---|
| supermajor (M) | supermajor | perfect | [0 8 13] |
| nearmajor (P, unmarked) | nearmajor | perfect | [0 7 13] |
| nearminor (p) | nearminor | perfect | [0 6 13] |
| subminor (m) | subminor | perfect | [0 5 13] |
Other tertian triads
Additional three-note chords built out of thirds.
Augmented triads
| Name | Third | Fifth | Edostep | Notes |
|---|---|---|---|---|
| near augmented (z+) | nearmajor | up | [0 7 14] | Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad. |
| exo augmented (S+) | supermajor | augmented | [0 8 16] | "Neutral" counterpart of 5/3-bounded chords. |
Diminished triads
| Name | Third | Fifth | Edostep | Notes |
|---|---|---|---|---|
| near diminished (z°) | nearminor | down | [0 6 12] | Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord. |
| major diminished (°) | nearminor | updiminished (tritone) | [0 6 11] | 5:6:7. Found in harmonic 4:5:6:7. |
| minor diminished (m°) | subminor | updiminished (tritone) | [0 5 11] | |
| exo diminished (S°) | subminor | diminished | [0 5 10] | Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord. |
Tetrads
Supermajor tetrads
| Name | Third | Fifth | Seventh | Steps between 3 and 7 | Edostep | Notes |
|---|---|---|---|---|---|---|
| exodominant seventh (S7) | supermajor | perfect | subminor | 10 | [0 8 13 18] | As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals. |
| supermajor seventh (M7, Δ7) | supermajor | perfect | supermajor | 13 (P5) | [0 8 13 21] | Seventh chord of supermajor. |
| supermajor nearmajor seventh (MP7) | supermajor | perfect | nearmajor | 12 | [0 8 13 20] | Acts as a more directed version of a M7 chord. |
Nearmajor tetrads
| Name | Third | Fifth | Seventh | Steps between 3 and 7 | Edostep | Notes |
|---|---|---|---|---|---|---|
| harmonic seventh (7), major harmonic (H) | nearmajor | perfect | subminor | 11 (tritone) | [0 7 13 18] | There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm). |
| neardominant seventh (z7) | nearmajor | perfect | nearminor | 12 | [0 7 13 19] | |
| nearmajor seventh (P7) | nearmajor | perfect | nearmajor | 13 (P5) | [0 7 13 20] | Seventh chord of nearmajor. |
| nearmajor supermajor seventh (PM7) | nearmajor | perfect | supermajor | 14 | [0 7 13 21]1] | Acts as a less directed version of a P7 chord. |
Nearminor tetrads
| Name | Third | Fifth | Seventh | Steps between 3 and 7 | Edostep | Notes |
|---|---|---|---|---|---|---|
| minor harmonic (Hm) | nearminor | perfect | supermajor 6th | 11 (tritone) | [0 6 13 17] | |
| nearminor seventh (p7) | nearminor | perfect | nearminor | 13 (P5) | [0 6 13 19] | Seventh chord of nearminor. |
| nearminor nearmajor seventh (pP7) | nearminor | perfect | nearmajor | 14 | [0 6 13 20] | Seventh chord of harmonic nearminor. |
| nearminor subminor seventh (pm7) | nearminor | perfect | subminor | 12 | [0 6 13 18] |
Subminor tetrads
| Name | Third | Fifth | Seventh | Steps between 3 and 7 | Edostep | Notes |
|---|---|---|---|---|---|---|
| subminor seventh (m7) | subminor | perfect | subminor | 13 (P5) | [0 5 13 18] | Seventh chord of subminor. |
| subminor nearminor seventh (mp7) | subminor | perfect | nearminor | 14 | [0 5 13 19] | |
| subminor nearmajor seventh (mP7) | subminor | perfect | nearmajor | 15 | [0 5 13 20] | Seventh chord of harmonic subminor. |
Non-tertian functional chords
| Name | Mediant | Bounding interval | Edostep | Notes |
|---|---|---|---|---|
| chthonic minor (Lm) | minor unilatus (whole tone) | perfect fourth | [0 4 9] | |
| chthonic major (LM) | major unilatus (subminor third) | perfect fourth | [0 5 9] | 6:7:8 chord. |
| suspended 4th (sus4) | perfect 4th | perfect fifth | [0 9 13] | Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord. |
| suspended up4th (sus^4) | up 4th | perfect fifth | [0 10 13] | Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th. |
| suspended 2nd (sus2) | supermajor 2nd | perfect fifth | [0 4 13] | Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord. |
| suspended down2nd (susv2) | nearmajor 2nd | perfect fifth | [0 3 13] | Suspension resolves to subminor |
| naiadic minor (S+m) | nearmajor third | nearmajor sixth | [0 7 16] | |
| naiadic major (S+M) | perfect fourth | nearmajor sixth | [0 9 16] | 3:4:5 chord. |
Isomorphic layouts and other instrument designs
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the standard diatonic layout places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a pajara-based layout. The harmonic table is also supported, though it is not as structurally critical as in 15edo.
The standard diatonic layout follows:
Supersets and subsets
44edo
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself inconsistent) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +7.1 | -4.5 | +13.0 | -5.9 | +4.9 | +4.1 | +2.5 | -1.0 | +6.8 | +0.4 |
| Relative (%) | 0.0 | +26.2 | -16.5 | +47.6 | -21.5 | +18.1 | +15.2 | +9.1 | -3.7 | +24.9 | +1.5 | |
| Steps
(reduced) |
44
(0) |
70
(26) |
102
(14) |
124
(36) |
152
(20) |
163
(31) |
180
(4) |
187
(11) |
199
(23) |
214
(38) |
218
(42) | |
Comparisons to other tuning systems
- 15edo shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.
- 24edo essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.
- 26edo can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks (e.g. compressed 5-limit thirds and 7/5~10/7 being mapped to the semioctave).
- 27edo shares Superpyth, and 32edo, also an Archy tuning, shares Pajara with a particularly sharp tuning.
- 31edo shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.
- 41edo shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.
Music in 22edo
Vector - What Happens After
See also
- User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)
- 22edo/Chords
- 22edo/Scales
- 22edo/Intervals
- 22edo/V/Exposition - an introduction to 22edo written by Vector
