22edo

Porcupine and pajara are the defining temperaments of 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. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor).

General theory

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.

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.

Approximation of prime harmonics 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	-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)

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).

Intervals and notation

Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef).

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).

The "native-fifths" system mentioned here is the most commonly used system, and the one that most microtonal notation systems support by default. It is derived through stacking 22edo's tempered version of 3/2 and assigning names accordingly. As a result, the nominals (C, D, E, F, G, ...) follow the diatonic MOS, where the distances between the large steps (C-D, D-E, F-G, G-A, A-B) are 4 EDO steps, and the distances between the small steps (E-F, B-C) are 1 EDO step. Therefore, a sharp corresponds to +3 EDO steps while a flat corresponds to -3 (representing the diatonic chroma in each case). Each sharp or flat can be split into three distinct notes, so we use the accidental ^ to raise by one EDO step and v to lower by one EDO step.

In the other proposed notation systems aside from native fifths, a sharp corresponds to +1 EDO step, while a flat corresponds to -1. To avoid ambiguity, these systems may use exclusively ups and downs, as those have the same meaning in 22edo. The Zarlino notation here uses the ternary Zarlino scale (see #Zarlino diatonic), or Ptolemy's intense diatonic, as its basic scale (which prioritizes the 5-limit, whereas native fifths prioritize 2.3.7), and uses to its advantage the fact that one step of 22edo maps to both 25/24 and 81/80 (a property of Porcupine temperament). Pajara uses the 10-note Pajara scale (see #Pajara) as its basis, which is generated by a perfect fifth but splits the octave in two to reach intervals of the full 7-limit relatively easily; the scale has four 2-step and one 3-step intervals per half-octave, which differ in size by a diatonic minor second, which is 1 step in 22edo.

The ADIN system uses the labels "nearminor" and "nearmajor" for intervals that may otherwise be called "classic(al)", "pental", or "ptolemaic" minor/major, which are terms used to describe the simple 5-limit intervals to which they correspond. Note also that as "subminor" and "supermajor" intervals are the minor and major intervals of the diatonic MOS in 22edo, unqualified "major" and "minor" should by default refer to these.

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 is notoriously hard to generalize. The solfege provided here will agree with 12edo solfege at the standard set of 12-13 MOS diatonic intervals. The nearmajor and nearminor intervals are solfeged the same way, but with a -n coda, emphasizing the two variants of each 12edo quality.

Note	Solfege (on Do)	Note	Solfege (on Do)
0	Do	11	Sen, Fin
1	Ra	12	Fi, Son
2	Ran	13	Sol
3	Ren	14	Le
4	Re	15	Len
5	Me	16	Lan
6	Men	17	La
7	Min	18	Te
8	Mi	19	Ten
9	Fa	20	Tin
10	Se, Fan	21	Ti

Derivation of 22edo

To construct the structure of 22edo, we 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.

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.

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. A stack of three quills represents the harmonic series segment 9:10:11:12 - if we stack this twice with a whole tone in the middle to close the octave, we get the 3-3-3-4-3-3-3 porcupine[7] scale (discussed further in 22edo#Scales). This means that with this particular scale, 22edo transforms an equal frequency division (the harmonic series segment) into an equal pitch division (of the perfect fourth).

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

Main article: 11edo

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, you 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.

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

PENDING REORGANIZATION FOLLOWING CONSENSUS OF XENGROVE

Tertian structure

22edo has four clear qualities of "thirds" that can serve as mediants in a chord bounded by a fifth. These are the subminor (273¢, 5\22), nearminor (327¢, 6\22), nearmajor (382¢, 7\22), and supermajor (436¢, 8\22) thirds, which reflect the intervals 7/6, 6/5, 5/4, and 9/7 respectively. As the gap between 6/5 and 5/4 is the same as that between 7/6 and 6/5 (or 5/4 and 9/7), 22edo's tertian structure is keemic.

Thirds in 22edo

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

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.

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 Nearmajor 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.)

It may be useful, perhaps for extraclassical tonality, to use the full 12-note form of superpyth's scale. This yields a softer (yet still hard) scale, alongside making 5/4 accessible as the major 3-step.

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 "Nearmajor second" along with being a chromatic semitone.

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. 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. One way to use blackdye is to essentially treat it as multiple overlapping diatonics, which one can modulate between. Blackdye has the additional property of not having chirality, so that there is only a single form of the scale instead of a left- and right-handed version like with zarlino.

Equiheptatonic

The Greek equable diatonic, 1/(18:20:22:24:27:30:33:36) (or in this case, equivalently its otonal counterpart) is represented as the MOS scale sssLsss (3-3-3-4-3-3-3) in 22edo, as a result of porcupine temperament. It is reasonable to, for that structural reason, consider 3-3-3-4-3-3-3 the default mode, with a nearminor chord on the tonic - it is the unique mode which possesses both a perfect fifth and a perfect fourth. 3-3-3-4-3-3-3 is more generally the MOS porcupine[7]; altering several notes of this MOS yields the Zarlino diatonic, explaining 22edo Zarlino's heavy reliance on porcupine's equivalences. Porcupine also has an 8-note scale 3-3-3-3-1-3-3-3, and a chromatic scale 1-2-1-2-1-2-1-2-1-2-1-2-1-2-1, a form of the Roklotian scale that may also be derived by dividing the intervals of superpyth pentatonic: 4-5-4-5-4 -> [1-2-1] [2-1-2] [1-2-1] [2-1-2] [1-2-1].

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[10]" 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. The MODMOS [2-2-2-1-2-2-2-2-1-2-2-2] of Pajara[12] is the Delkian scale; it may be derived again by splitting 4-5-4-5-4 ([2-2] [2-1-2] [2-2] [2-1-2] [2-2]), or by splitting each whole tone of MOS diatonic ([2-2] [2-2] 1 [2-2] [2-2] [2-2] 1), which makes for a more xenharmonic way of translating 12edo music into 22edo than simply retuning the chain of fifths. ("When the 12edo goes chromatic, equally divide the whole tone!")

The following chart shows the modes of pajara[10]:

	Chart	2	3	4	6	8	9
Dynamic minor	├─┴─┴─┴─┴──┴─┴─┴─┴─┴──┤ 2 2 2 2 3 2 2 2 2 3	minor	minor	dim	perfect	minor	minor
Static minor	├─┴─┴─┴──┴─┴─┴─┴─┴──┴─┤ 2 2 2 3 2 2 2 2 3 2	minor	minor	perfect	perfect	minor	major
Static major	├─┴─┴──┴─┴─┴─┴─┴──┴─┴─┤ 2 2 3 2 2 2 2 3 2 2	minor	major	perfect	perfect	major	major
Dynamic major	├─┴──┴─┴─┴─┴─┴──┴─┴─┴─┤ 2 3 2 2 2 2 3 2 2 2	major	major	perfect	perfect	major	major
Augmented	├──┴─┴─┴─┴─┴──┴─┴─┴─┴─┤ 3 2 2 2 2 3 2 2 2 2	major	major	perfect	aug	major	major

And of a MODMOS of pajara[10], ssLsssssLs, the "pentachordal" pajara scale:

	Chart	2	3	4	6	8	9
(Minor)	├─┴─┴─┴─┴─┴──┴─┴─┴─┴──┤ 2 2 2 2 2 3 2 2 2 3	minor	minor	dim	perfect	minor	minor
Alternate minor	├─┴─┴─┴─┴──┴─┴─┴─┴──┴─┤ 2 2 2 2 3 2 2 2 3 2	minor	minor	dim	perfect	minor	major
(Minor)	├─┴─┴─┴──┴─┴─┴─┴──┴─┴─┤ 2 2 2 3 2 2 2 3 2 2	minor	minor	perfect	perfect	major	major
Standard major	├─┴─┴──┴─┴─┴─┴──┴─┴─┴─┤ 2 2 3 2 2 2 3 2 2 2	minor	major	perfect	perfect	major	major
(Major)	├─┴──┴─┴─┴─┴──┴─┴─┴─┴─┤ 2 3 2 2 2 3 2 2 2 2	major	major	perfect	aug	major	major
(Major)	├──┴─┴─┴─┴──┴─┴─┴─┴─┴─┤ 3 2 2 2 3 2 2 2 2 2	major	major	perfect	aug	major	major
Standard minor	├─┴─┴─┴──┴─┴─┴─┴─┴─┴──┤ 2 2 2 3 2 2 2 2 2 3	minor	minor	perfect	perfect	minor	minor
(Major)	├─┴─┴──┴─┴─┴─┴─┴─┴──┴─┤ 2 2 3 2 2 2 2 2 3 2	minor	major	perfect	perfect	minor	major
Alternate major	├─┴──┴─┴─┴─┴─┴─┴──┴─┴─┤ 2 3 2 2 2 2 2 3 2 2	major	major	perfect	perfect	major	major
(Major)	├──┴─┴─┴─┴─┴─┴──┴─┴─┴─┤ 3 2 2 2 2 2 3 2 2 2	major	major	perfect	perfect	major	major

Some names are from Paul Erlich.

Generator sequences

Now, let us return to sentry (as discussed in the 22edo#Tempering properties section). 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.

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.

Consonance and dissonance properties

In 12edo, consonance and dissonance is generally defined via membership to groups of intervals called odd-limits. The 3-odd-limit consists of intervals with 1, 2, 3, and 4 in their numerators or denominators, and is so called because the maximum value that either the numerator or denominator can have once all factors of two are removed is 3. The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third). The 5-odd-limit expands the range to include imperfect consonances, which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the major and minor thirds, and the minor and major sixths, found in 12edo. They are also present in 22edo as the nearmajor and nearminor intervals.

To expand the range of consonances further in 22edo, we may now consider the intervals of the 9-odd-limit. These include all the previous intervals, as well as intervals involving 7, 9, 10, 12, 14, and 16. In 22edo, this allows the whole tone, subminor third, supermajor third, and tritone to function as secondary consonances, although because the tritone is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance. The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths other than the subminor seventh) are the rest of the dissonances.

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).

However, at the same time, this means that when it comes to the supermajor and subminor thirds, the former is actually less stable in chords, due to sitting awkwardly between 5/4 and 4/3. We may loosely understand the stability of chords by examining their complexity when taken out of the harmonic series: the standard nearmajor triad is 4:5:6 (because its intervals are 5/4 and 3/2 (aka 6/4), but the standard nearminor triad is 10:12:15, which is somewhat more complex. Conversely, when considering supermajor and subminor, it is the subminor triad that is simpler at 6:7:9 (the subminor third is 7/6), meanwhile the supermajor triad is found all the way up at 14:18:21.

Functional harmony

Leading tones

When considering Secor's supposed optimal leading tone at 70 cents, one may notice that 22edo skips this category entirely. However, 22edo instead matches with Aura's theory of functional harmony, which places the 70-cent leading tone at the intersection of two other functional categories at around 110 cents and 50 cents respectively - the collocant and gradient functions. The collocant functions as a conventional leading tone, whereas the gradient functions as a passing tone to either jump past the tonic or resolve to the collocant. 22edo represents both of these separately, and in doing do presents a distinct approach to leading tones from systems like 17edo and 31edo that have Secor's leading tone instead.

Further functional harmony

Beyond the leading tones, we may adopt two distinct approaches to 22edo functional harmony. The first takes the diatonic scale and its functions as a base, and instead utilizes the multitude of new interval qualities to allow for new harmonic functions. Meanwhile, the second potential approach introduces new functions based on the scale degrees present in pajara[10].

First of all, we will explore a heptatonic tonal approach to 22edo. Using tonal harmony solves our diatonic conundrum from earlier, as instead of having to remain in a scale, we can simply play over whatever the current chord is.

The standard qualities of major and minor retain their "bright" and "dark" feels respectively, which can be broken down into a combination of their complexity in triads and the actual width of the intervals. This suggests that the four qualities in 22edo should have more granularity in their feels, and can be broken down into stable/unstable and bright/dark.

	Stable	Unstable
Bright	Nearmajor (warm, pleasant, comforting)	Supermajor (excited, animated, active)
Dark	Subminor (depressive, sad, bluesy)	Nearminor (angry, tense, stressful)

These proceed to define tonal hierarchies that are loosely independent of each other. The heptatonic interval functions remain as they are in 12edo, although with the caveat that the ideal leading tone ends up at the equal semitone rather than the semitone found in MOSdiatonic, which has implications for the subminor and supermajor keys.

In nearmajor, the fourth acts as it usually does in 12edo major, serving as a tendency tone towards the third. The same, however, is not quite the case for supermajor. In supermajor, a lead to the third would be a diminished fifth (11/8), perhaps justifying its inclusion in the scale over the fourth proper, or the functional alternation between the two in different contexts. The functionality of the seventh grows increasingly complicated in supermajor - while in 12edo, one may only see, for instance, a dominant chord replacing the I chord, in 22edo there are four different potential types of seventh, all with justifications. A fifth over the third would be a supermajor seventh (notably serving as the diatonic maj7, and distinguishing itself from the 12edo maj7 by not leading up to its own root), a tritone over the third would be a nearminor seventh, a lead up to the tonic would be a nearmajor seventh, and finally the MOS diatonic dominant chord utilizes a subminor seventh.

The same kind of justification emerges for harmonic subminor, except that there is little reason to alter the seventh all the way up to a supermajor seventh if the objective is for it to function as a leading tone. In fact, the same logic can be used against the conventional dominant chord in nearmajor - leading inwards to a nearmajor third by equal semitones on either side requires that the initial interval be a perfect tritone, and that the chord to be used as a dominant is actually a harmonic 4:5:6:7 on the fifth. (Resolving to a supermajor chord actually wants a dom7 with a nearmajor third and nearminor seventh, if quartertones are not to be used).

The remainder of the discussion of functional harmony is simply the assignment of placements in the tonal hierarchy to the new degrees added by the 10-tone system. To put it simply, the antilatus and unilatus become the varicant and subvaricant, which sit between the mediant/submediant and supertonic/subtonic in terms of stability (and feature as elements of chthonic chords like 6:7:8, an alternative to standard diatonic chords available in the 10-form). Additionally, the tritone acquires the antitonic function. While the dominant serves as a stable "structural anchor" in diatonic, here the antitonic serves as an unstable structural anchor - the opposite of the tonic both in placement and stability.

Tables

Table of chords

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)	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 (X+)	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 (X°)	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 (X7)	supermajor	perfect	subminor	10	[0 8 13 18]
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]	4:5:6:7. Standard nearmajor dominant.
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.

Diminished 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. Uses the aforementioned nearmajor up 4th.
suspended up4th (sus^4)	up 4th	perfect fifth	[0 10 13]	Suspension resolves to supermajor
suspended 2nd (sus2)	supermajor 2nd	perfect fifth	[0 4 13]	Suspension resolves to nearminor
suspended down2nd (susv2)	nearmajor 2nd	perfect fifth	[0 3 13]	Suspension resolves to subminor
naiadic minor (X+m)	nearmajor third	nearmajor sixth	[0 7 16]
naiadic major (X+M)	perfect fourth	nearmajor sixth	[0 9 16]	3:4:5 chord.

Table of MOS scales

Porcupine scales

MOS scales generated by a nearmajor second.

Name	Chart	Notes
Onyx	├──┴──┴──┴───┴──┴──┴──┤ 3 3 3 4 3 3 3
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

Orwell scales

MOS scales generated by a subminor third.

Name	Chart	Notes
Manual	├────┴────┴────┴────┴─┤ 5 5 5 5 2
Gramitonic	├──┴─┴──┴─┴──┴─┴──┴─┴─┤ 3 2 3 2 3 2 3 2 2
Antiparagonic	├┴─┴─┴┴─┴─┴┴─┴─┴┴─┴─┴─┤ 1 2 2 1 2 2 1 2 2 1 2 2 2

Magic scales

MOS scales generated by a nearmajor third.

Name	Chart	Notes
Mosh	├─────┴┴─────┴┴─────┴┴┤ 6 1 6 1 6 1 1
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
Mosdiatonic	├───┴┴───┴───┴───┴┴───┤ 4 1 4 4 4 1 4
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
Hedgehog	malic	├──┴──┴────┴──┴──┴────┤ 3 3 5 3 3 5
	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
	lemon	├──┴──┴┴──┴┴──┴──┴┴──┴┤ 3 3 1 3 1 3 3 1 3 1
Doublewide	citric	├┴────┴────┴┴────┴────┤ 1 5 5 1 5 5
	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
Delkian	├─┴─┴─┴┴─┴─┴─┴─┴┴─┴─┴─┤ 2 2 2 1 2 2 2 2 1 2 2 2

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.