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.

The interval qualities found in 22edo vs. those found in 24edo.

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 pajara scale and chords (0-3-6-8 major and minor)

22edo porcupine scale and chords (0-1-3 and 0-2-3)

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). 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.

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).
• 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

Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)

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" 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 (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).

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 (ups and downs notation). Other accidentals that are identified with this in 22edo include any accidental representing the syntonic comma (such as in Ben Johnston, sagittal, SRS, or FJS notation), any accidental representing 25/24 (the accidentals to be used for porcupine[7] or pajara[10] notation, also Ben Johnston), and any accidental representing 33/32 (such as the FJS or HEJI accidentals for 11).

That is, in the other notation systems proposed besides mosdiatonic, the difference between minor and major (as interval qualities in the scale being used for the notation) is 1 EDO step. To avoid ambiguity, these systems may use exclusively ups and downs, though it might be more natural to some to repurpose the diatonic # and b symbols (as is done here), especially if diatonic notation is not used simultaneously with these other schemes. 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 notation 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.

Both degrees and notes in 10-form pajara should be notated with 0-indexed numerals in text: the tonic is always 0, and absolute pitch should be specified in relation to standard diatonic notation. 0-indexing is used so that systems such as figured bass that depend on numerals being a single symbol each still work (if 1-indexing was used, the number 10 would indicate a degree). Additionally, Roman numeral analysis in this case would use N for zero.

As for notating the 10-form on the staff, there are a few different approaches. The first adds an extra line to each staff so that an octave can span 11 staff positions, but comes at the cost of losing intuition for people used to reading intervals from standard notation. The second uses the mosdiatonic notation for pentic, but uses an extra symbol to mark an alteration by a 109c semitone, allowing the full range of notes in pajara to be provided at the cost of potential overloading on symbols as opposed to visual distance to denote pitch. Meanwhile, the third option is simply to notate it starting from mosdiatonic as a base, with ups and downs notation.

22edo's qualities correspond neatly to the basic color qualities proposed by Kite: red (ru) = supermajor, yellow (yo) = nearmajor, green (gu) = nearminor, and blue (zo) = subminor. That these are equally spaced shows that 22edo is a keemic temperament, a quality shared with tunings such as 41edo, and their adjacency in the edo preserves the intended mnemonic framework of a "rainbow" of qualities.

On "major" vs. "supermajor"

A large number of resources, including Unque's theory page on xen.wiki, the Lumatone introductory video to 22edo, and the 31et.com page on 22edo, reserve "major/minor" for 5/4 and 6/5, distinguishing an unmarked "major" from "supermajor" (and "minor" from "subminor"). There is no obvious reason to do this other than 5-limit preferentialism, and it leads to ambiguity where "major" could either refer to specifically classical major intervals or more generally to any major interval. Diatonic notation has the opposite behavior, leaving 9/7 and 7/6 unmarked, while labelling 6/5 and 5/4 as "upminor" and "downmajor" respectively; while this is not a problem on its own, it leads to a high degree of ambiguity with the other naming scheme and obscures the idea of 22edo splitting each 12edo quality into two (therefore, 5/4 and 9/7 should both be equally major).

On this page, to resolve this and in accordance with the ADIN system of interval names, the two qualities of major are distinguished with "nearmajor" (5/4, 5/3) and "supermajor" (9/7, 12/7), and likewise the minor qualities are "nearminor" (6/5, 8/5) and "subminor" (7/6, 14/9, 7/4). Nearminor and nearmajor intervals 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.

On this page, unqualified "major" may refer to both nearmajor and supermajor collectively in cases like "either major key" or "the major thirds". However, when used to refer to a specific interval ("the major third"), it should refer to the supermajor intervals, as those are the major intervals of the diatonic MOS in 22edo. Likewise for minor.

Short forms of interval names should use ups and downs notation, however, so that "M3" is always the supermajor third. But also, see #Table of chords.

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

The intervals of 22edo

A table of intervals is available at 22edo/Intervals.

Solfege

Solfege is notoriously hard to generalize. A number of different approaches to solfege for microtonal tunings exist, including for 22edo and for more general systems that are applicable to 22edo. The core conflict at the heart of generalizing solfege is the lack of uniformity in standard solfege. While solfege uses alternative vowels for qualities, these alternate vowels are not fully uniform (due to historical conflicts with the original diatonic solfege, which took its syllables from a Latin hymn), and so generalizing them becomes rather complicated and idiosyncratic. It becomes especially so if you want to be cross-linguistically compatible, with five vowels and ~13 widespread consonants constituting a total of 60-70 combinations, if codas are ignored; while this is sufficient to solfege 22edo if you sing equivalent notes the same way, it fails for more general or higher-order systems.

A solfege used by Andrew Heathwaite (listed on the Xenharmonic Wiki) is as follows. It uses the standard labels for intervals of zarlino diatonic, rather than for mosdiatonic; the prevalence of zarlino is characteristic of a lot of early or idiosyncratic theories regarding non-meantone systems.

Note	Solfege (on Do)	Note	Solfege (on Do)
0	Do	11	Fi
1	Di	12	Su
2	Ra	13	Sol
3	Ru	14	Lo
4	Re	15	Le
5	Ma	16	La
6	Me	17	Li
7	Mi	18	Ta
8	Mo	19	Tu
9	Fa	20	Ti
10	Fu	21	Da

Another solfege proposed by Kite Giedraitis uniformizes the vowels corresponding to qualities, at the cost of losing compatibility with standard diatonic solfege, representing the central tradeoff when it comes to generalizing solfege:

Note	Solfege (on Da)	Note	Solfege (on Da)
0	Da	11	Fo/Su
1	Ri	12	So
2	Ru	13	Sa
3	Ro	14	Li
4	Ra	15	Lu
5	Mi	16	Lo
6	Mu	17	La
7	Mo	18	Ti
8	Ma	19	Tu
9	Fi	20	To
10	Fu	21	Ta

A solfege provided by Vector 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

An alternative approach is to discard the inclusion of qualities altogether, and just solfege the 7-form (or the 10-form, with the added syllables na for 2\10, zi for 5\10, and be for 8\10).

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.

22edo's primary structures.

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.

Sensamagic demonstration

11edo temperaments

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. (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 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.

In fact, blackdye or several characteristics of it are likely to naturally emerge in tonal harmony in the first place, given an overall desire to avoid wolf fifths in nearmajor and nearminor tonalities, resulting in certain intervals being doubled up. Most notably, it is reasonable to consider the two forms of major second equally part of a nearmajor tonal system, so that 5-2 and 2-6 can both be perfect fifths with different versions of the 2 degree (although unless 6 is additionally sharpened, some additional harmonic movements are needed to resolve the edostep offset that results from a pumped syntonic comma if you move from fifth-bounded triads on 5 to 2 in a single motion).

Equiheptatonic

The 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

The 12edo concept of consonance and dissonance as seen in modern harmony may be generalized via the membership of intervals 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. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.

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; similarly, the nearmajor second and nearminor seventh's proximity to the unison and octave have a similar effect. 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).

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.

Building scales from tetrachords

A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). They are useful for building scales because two perfect fourths and a whole tone may be stacked in order to complete the octave with a heptatonic scale; when singing a scale, heptatonic or pentatonic forms are naturally common due to the fact that 4/3 and especially 3/2 can be easily sung without a reference, and tetrachords formalize this natural occurrence into scale-building. Tetrachords may be classified based on the size of their largest interval; there are nine tetrachords in 22edo that fall under this classification scheme.

Diatonic tetrachords

In diatonic tetrachords, the largest interval is a major second (less than half of the perfect fourth). 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).

Chromatic tetrachords

Here, "chromatic" doesn't have to do with the modern sense of the chromatic scale, but instead indicates a tetrachord where the largest interval is some kind of minor third. In 22edo, there are also four chromatic tetrachords: 5-2-2, 5-3-1, 6-2-1, and 6-1-2. When built up into scales, we get scales with two adjacent semitones, which is a very different sound from a standard diatonic scale.

Enharmonic tetrachords

Similarly, "enharmonic" has little to do with enharmonic notes here, but instead with tetrachords where the largest interval is some kind of major third. In 22edo, one enharmonic tetrachord exists: 7-1-1, where the 7 is the nearmajor third. The scale ends up having two adjacent quarter tones, making for somewhat of a "shimmery" sound.

Tetrachords were historically used in largely monophonic music, so that additional structural constraints didn't become relevant until additional harmonic complexity entered the scene, and music switched primarily to using diatonic. Therefore, tetrachord-based scales, especially chromatic and enharmonic ones, remain useful in a monophonic or homophonic context, where the harmonic relations between notes do not matter as much. One can think of tetrachordal music as entirely "degree-based", whereas modal music is to an extent "interval-based".

However, tetrachords are often still theoretically relevant in discussing the construction of modern scales, for example in the examination of the conventional framing of the melodic minor scale, where the upper tetrachord varies when ascending vs. when descending; additionally the Ionian major scale can be thought of as a M2-M2-m2 tetrachord stacked into an octave-repeating scale.

Other polychordal structures

Trichord

It's also possible to use trichords (subscales spanning a perfect fourth and consisting of only 3 notes) to build scales in 22edo; these scales will be pentatonic and vary entirely on what middle interval is used in the trichord. 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.

A trichord is structurally equivalent to a fourth-bounded chthonic triad, and so a trichordal scale may be conceptualized as a chthonic generator chain, especially since the whole tone separating the trichords is itself a form of chthonic. Interestingly, this implies that trichordal scales place no notes in between the notes of their chords, having the three notes of a chthonic chord fall on consecutive scale steps.

Pentachord

A pentachord consists of five notes spanning a perfect fourth. A pentachord may be constructed by dividing the steps of a trichord; therefore, a trichord or a chthonic triad consists of alternating notes of a pentachord. Because of this, it is also structurally useful to insert the perfect tritone (or augmented fourth/diminished fifth) between the fourth and the fifth to maintain a relatively even spacing of intervals. The most common pentachord is the pajara pentachord, consisting of 2-2-2-3, which extends out into the pentachordal scale 2-2-2-3-2-2-2-2-2-3.

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.

Pentachordal extensions may be used to improve melodic cohesion in trichordal scales.

A demonstration of 22edo diatonic functional harmony.

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 and new interpretations of conventional chord structures, mostly focusing on the dichotomy between near- and super/sub versions of the conventional qualities in 22edo. Meanwhile, the second potential approach introduces new functions based on the scale degrees present in pajara[10], keeping two qualities but filling the gaps with new ordinals and different varieties of chord structures.

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, or otherwise alter the scale to fit the harmony we're using. This generally requires, even more than in cases like 12edo minor, that a key not necessarily be considered as having a base scale.

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)

Therefore, there are four distinct "keys" in 22edo, as compared to two in 12edo, where a key is defined as a system of tonal hierarchy based around a certain interval quality or tonic chord (independent of absolute pitch), which will be elaborated on below. When counting absolute pitch, there are 88 (4 x 22) keys. Note that relative major or minor depends on whether the key is near- or super/sub, and that, for instance, nearmajor and supermajor use different scales that are not rotations of one another. In specific, using ups and downs notation, C Nearmajor corresponds to vA Nearminor, meanwhile C Supermajor corresponds to A Subminor, and in general nearmajor-nearminor relative correspondences acquire an additional down accidental compared to standard MOSdiatonic correspondences.

The chirality of the nearmajor or nearminor scale in question is ultimately of little relevance (see #Blackdye; the major second in nearmajor (and the fourth in nearminor) may be either note depending on context), but in general the right-handed version of nearmajor is assumed due to having a non-wolf V chord, and the left-handed version of nearminor is assumed due to having a non-wolf fourth over the tonic.

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 and turns the use of the diatonic scale into a balancing act between the functional utility of MOSdiatonic and the tension of the leading tones in zarlino diatonic. (In particular, it suggests the use of a "harmonic supermajor" by flattening the seventh of supermajor by an edostep.)

Nearmajor key

Natural nearmajor scale and tonic chord

In nearmajor (the key with the nearmajor tonic chord), the fourth acts as it usually does in 12edo major, serving as a tendency tone towards the third. The basic tonal identity for nearmajor is 4:5:6, which extends generally to a nearmajor seventh chord, although a dominant (harmonic) seventh is also possible, and more justified in 22edo due to naturally extending the harmonic series segment corresponding to 4:5:6.

Nearminor key

Natural nearminor scale and tonic chord

Nearminor harmony functions somewhat similarly to how you expect, with the nearminor sixth functioning as a leading tone down to the fifth and the seventh being able to be raised to a nearmajor seventh in order to give a more directed dominant resolution. The whole tone also provides a lead up to the minor third, like in standard diatonic.

Melodic minor scales are somewhat interesting here as well, as there are a couple different reasonable ways to construct them, which would likely depend on the chords being used and the desired melodic contour.

Supermajor key

Natural supermajor scale and tonic chord

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. Therefore, an alternate version of the supermajor scale usable in certain contexts makes the fourth wolf and the seventh nearmajor.

This also means that the regular perfect fourth isn't as unstable an interval or as functionally dissonant in supermajor - in fact, the third is actually somewhat of a tension compared to it (though the step between them is half the size of a conventional leading tone).

Subminor key

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).

Natural subminor scale and tonic chord

In general, 22edo's functional harmony ends up a lot more context-bound and much less scale-bound than 12edo's, due to the multiple different qualities of intervals and notes doing different things, and the ideal leading tone not matching the standard diatonic structure.

Alternative leading tones

An alternative approach to simplify things is instead to discard Aura's theory of leading in favor of treating the quartertone as the optimal leading tone (as it is the diatonic major seventh), an entirely different paradigm emerges. Supermajor and subminor become definitive, stable diatonic tonality systems, with no awkwardness around leading tones and similar mechanics to their 12edo counterparts (albeit with the different, somewhat inverted "moods" presented by the supermajor and subminor intervals). Meanwhile, the nearmajor and nearminor scales acquire new "harmonic" variations, with the final note raised up to a quartertone below the tonic. In effect, supermajor/subminor and nearmajor/nearminor "switch" in regards to some functions. Instead of raising the fourth in supermajor, it is in this system viable to lower it in nearmajor. The best dom7 to resolve to a nearmajor triad on the tonic is not actually a seventh chord at all, but instead features the supermajor sixth and third, and can consequently be reanalyzed as a subminor seventh chord on the 9/7 over the tonic. The MOSdiatonic dominant seventh serves to resolve to a MOSdiatonic major triad, as in 12edo.

10-tone functional harmony

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. Note that in the 10-tone system, we return to having two distinct interval qualities down from four, so we go back to having two different keys. 10-tone harmony is also useful in modal music. Also, note that 109-cent leading tones comprise the majority of the intervals in pajara[10], so their impact may be reduced and in fact one might depend more on the few larger nearmajor seconds that exist in the scale, or skip steps entirely and use subsets.

10-form harmony demonstration in 22edo

Another thing to note about the 10-tone system is that it is possible to constrain oneself entirely to chthonic harmony, in which case a lot of the familiar functional harmony language somewhat breaks. The role of the traditional dominant with respect to the tonic disappears completely (even if, for instance, the root position of a chord is assumed to be 4:6:7), with instead the chords on the tritone and the sixth including a leading tone up to the tonic (in fact, the dominant in this system becomes a stable chord rather than a tense one, assuming a 6:7:8 root position).

We may contextualize these differences by examining the 3-function analysis of functional harmony, which in the 7-form (as in 22edo) places the tonic function on the degrees (1-indexed) 1, 3, and 6, the subdominant function on 2 and 4, and the dominant function on 5 and 7. In the chthonic 10-form, however, it requires some amount of reorganization. A theory for Vector's abandoned Earth#Pajara project utilizes four functional categories for chords, rather than three (0-indexed): tonic (0, 2, 8), dominant (4, 6), antitonic (3, 5, 7) and antidominant (1, 9) - the "dominant" function here acts similarly to the heptatonic subdominant (stable, dynamic), with the antitonic and antidominant serving as tense "static" and "dynamic" functions respectively. This essentially splits the 10-form into two pentatonic subscales, one built on the tonic and one built on the antitonic (which is actually how pajara[10] is constructed, but in this tonal system you can pretty strongly feel that construction determining how harmony is structured).

If this theory is also altered to work with tertian harmony instead, the functions follow a chain of thirds rather than a chain of chthonics, so that tonic is (0, 3, 7), a more traditional subdominant is (4, 1), dominant is thus (6, 9), and the remaining degrees (2, 5, 8) constitute antitonic. In this case, dominant is the "tense dynamic" function and antitonic is the "tense static" function.

Modal harmony

The modes presented here, arranged in a Tetrahedron.

Modal harmony further emphasizes the qualities of the various intervals and chords found in the different scales used in music, as opposed to things like leading tendencies. It is within modal harmony that clear "supermajor", "nearmajor", "nearminor", and "subminor" diatonic scales can be defined, rather than used as context-dependent tonal systems. These mostly follow the interval qualities suggested above, except this time it becomes applicable to an entire scale rather than just to specific chords. (And of course, additional modes of zarlino or mosdiatonic may be used.)

Given this, it's also useful to enumerate various modal scales, as a counterpart to the various non-Ionian/Aeolian modes used throughout standard modal harmony. These will not be exclusively "real" diatonic modes, but rather combinations of qualities loosely analogous to standard modes (and sharing the quality of having notes constrained to range in certain qualities), in six "series" comprising 22 unique modes, visible on the right. This setup overall aims to generalize the idea that diatonic modes exist on a gradation of "brightness" in 12edo, where successive alterations make a mode brighter or darker. Here, bright vs. dark isn't the only axis, however - there's near vs. super/sub and stable vs. unstable as well, so series of alterations along those allow for a much more complex selection of modes to choose from. Additionally, each series of modes has one quality in common, which I've labelled here, so all the "stable" modes have only stable intervals, even if they might contain both nearmajor and subminor ones, and all the "bright" modes contain only major intervals, even if they might be both nearmajor and supermajor. Holding the fourth and fifth constant (as Lydian and Locrian are rarely used in standard diatonic modal harmony) means that there are four "vertex" modes, corresponding to the pure nearmajor, nearminor, supermajor, and subminor qualities, as well as the Ionian and Phrygian modes of Zarlino and mosdiatonic.

Choosing a mode

Much as the choice of mode in 12edo largely depends on its position on the scale from bright to dark, you might choose a mode here by selecting a series based on the common sound you want your song or section to have, and then choosing a position on that series between its two extremes. For example, for something intense and somewhat uncanny, you might start by choosing the Unstable series, and then proceed to select a mode along that series between bright/super and near/dark that embodies the feel you want, such as Unstable Dorian. Alternatively, for an excited, cheerful sound, you might choose the Bright series and a mode between the near/stable and super/unstable extremes of it, such as Didymic Major.

The "Equable" mode serves as a somewhat 'neutral' sound - despite the lack of neutral intervals in 22edo, it still occupies that somewhat soft position in between major and minor qualities, while at the same time being more equally distributed than any other version of Dorian available. Quality-wise, it has a mix of nearmajor (bright, stable) and nearminor (dark, unstable) intervals, serving as the opposite polarity to MOS Dorian, and its equidistant nature somewhat overrides other quality-based properties from a melodic perspective. On the opposite side of things, MOS Dorian can be seen as somewhat aggressively defined by its qualities, being a mix of subminor (stable, dark) and supermajor (unstable, bright), with a subminor third on the tonic.

Other pairs of "opposing" modes include unstable Dorian vs. stable Dorian, and didymic major vs. didymic minor, both of which unlike equable vs. MOS Dorian form complementary pairs similar to Ionian and Phrygian in 12edo.

Here they have been organized into two "loops"; bolded entries represent modes that differ along the loops, and italicized entries have had their positions flipped.

Class A

Loop A				Loop B
Series	Mode	Type	Name	Series	Mode	Type	Name
Super/Sub	├───┴┴───┴───┴┴───┴───┤ 4 1 4 4 1 4 4	Aeolian	Aeolian	Super/Sub	├───┴┴───┴───┴┴───┴───┤ 4 1 4 4 1 4 4	Aeolian	Aeolian
Super/Sub	├───┴┴───┴───┴───┴┴───┤ 4 1 4 4 4 1 4	Dorian	Dorian	Super/Sub	├───┴┴───┴───┴───┴┴───┤ 4 1 4 4 4 1 4	Dorian	Dorian
Super/Sub	├───┴───┴┴───┴───┴┴───┤ 4 4 1 4 4 1 4	Mixolydian	Mixolydian	Super/Sub	├───┴───┴┴───┴───┴┴───┤ 4 4 1 4 4 1 4	Mixolydian	Mixolydian
Bright, Super/Sub, Unstable	├───┴───┴┴───┴───┴───┴┤ 4 4 1 4 4 4 1	Ionian	Ionian ("Supermajor")	Bright, Super/Sub, Unstable	├───┴───┴┴───┴───┴───┴┤ 4 4 1 4 4 4 1	Ionian	Ionian ("Supermajor")
Bright	├───┴───┴┴───┴───┴──┴─┤ 4 4 1 4 4 3 2	Ionian	Harmonic major	Unstable	├───┴───┴┴───┴───┴─┴──┤ 4 4 1 4 4 2 3	Mixolydian	Unstable Mixolydian
Bright	├───┴──┴─┴───┴───┴──┴─┤ 4 3 2 4 4 3 2	Ionian	Didymic major	Unstable	├───┴─┴──┴───┴───┴─┴──┤ 4 2 3 4 4 2 3	Dorian	Unstable Dorian
Bright	├───┴──┴─┴───┴──┴───┴─┤ 4 3 2 4 3 4 2	Ionian	RH-Ionian	Unstable	├───┴─┴──┴───┴─┴───┴──┤ 4 2 3 4 2 4 3	Aeolian	LH-Aeolian
Near, Bright, Stable	├──┴───┴─┴───┴──┴───┴─┤ 3 4 2 4 3 4 2	Ionian	LH-Ionian ("Nearmajor")	Near, Dark, Unstable	├─┴───┴──┴───┴─┴───┴──┤ 2 4 3 4 2 4 3	Phrygian	RH-Phrygian ("Nearminor")
Near	├──┴───┴─┴───┴──┴──┴──┤ 3 4 2 4 3 3 3	Mixolydian	Major equable	Near	├──┴──┴──┴───┴─┴───┴──┤ 3 3 3 4 2 4 3	Aeolian	Minor equable
Near	├──┴──┴──┴───┴──┴──┴──┤ 3 3 3 4 3 3 3	Dorian	Equable	Near	├──┴──┴──┴───┴──┴──┴──┤ 3 3 3 4 3 3 3	Dorian	Equable
Near	├──┴──┴──┴───┴─┴───┴──┤ 3 3 3 4 2 4 3	Aeolian	Minor equable	Near	├──┴───┴─┴───┴──┴──┴──┤ 3 4 2 4 3 3 3	Mixolydian	Major equable
Near, Dark, Unstable	├─┴───┴──┴───┴─┴───┴──┤ 2 4 3 4 2 4 3	Phrygian	RH-Phrygian ("Nearminor")	Near, Bright, Stable	├──┴───┴─┴───┴──┴───┴─┤ 3 4 2 4 3 4 2	Ionian	LH-Ionian ("Nearmajor")
Dark	├─┴───┴──┴───┴─┴──┴───┤ 2 4 3 4 2 3 4	Phrygian	LH-Phrygian	Stable	├──┴───┴─┴───┴──┴─┴───┤ 3 4 2 4 3 2 4	Mixolydian	RH-Mixolydian
Dark	├─┴──┴───┴───┴─┴──┴───┤ 2 3 4 4 2 3 4	Phrygian	Didymic minor	Stable	├──┴─┴───┴───┴──┴─┴───┤ 3 2 4 4 3 2 4	Dorian	Stable Dorian
Dark	├─┴──┴───┴───┴┴───┴───┤ 2 3 4 4 1 4 4	Phrygian	Subharmonic minor	Stable	├──┴─┴───┴───┴┴───┴───┤ 3 2 4 4 1 4 4	Aeolian	Stable Aeolian
Dark, Super/Sub, Stable	├┴───┴───┴───┴┴───┴───┤ 1 4 4 4 1 4 4	Phrygian	Phrygian ("Subminor")	Dark, Super/Sub, Stable	├┴───┴───┴───┴┴───┴───┤ 1 4 4 4 1 4 4	Phrygian	Phrygian ("Subminor")

Note that Aeolian is not a vertex. Because of this, it might be prudent to construct a secondary, smaller tetrahedron that holds the major second constant alongside the fourth and fifth. Doing so yields six additional modes:

• A set of two additional modes between RH-Ionian and LH-Aeolian, acting as alternative near forms of Mixolydian/major equable (├───┴──┴─┴───┴──┴──┴──┤ 4 3 2 4 3 3 3) and Dorian/equable (├───┴─┴──┴───┴──┴──┴──┤ 4 2 3 4 3 3 3 )

• A set of two additional modes between LH-Aeolian and mosdiatonic Aeolian, acting as alternative dark/minor scales (├───┴─┴──┴───┴─┴──┴───┤ 4 2 3 4 2 3 4, ├───┴┴───┴───┴─┴──┴───┤ 4 1 4 4 2 3 4 ).

• Alternative stable forms of Mixolydian (├───┴──┴─┴───┴──┴─┴───┤ 4 3 2 4 3 2 4) and Dorian (├───┴┴───┴───┴──┴─┴───┤ 4 1 4 4 3 2 4 ). These differ by one note varying by two steps; between them is in fact the simplest possible 5-limit Dorian, at (├───┴─┴──┴───┴──┴─┴───┤ 4 2 3 4 3 2 4 ), which is not a mode of zarlino due to distributing the large and medium steps differently. This appears to suggest that the sum total of all theoretically possible modes existing under this system is the complete volume of a tetrahedron with endpoints at near- Locrian and Lydian and at sub-Locrian and super-Lydian. There are 84 total modes in the scheme, which are the rotations of the following 8 base scales, including chirality. These are the set of scales that have the property that all instances of any diatonic interval between any two notes in the scale are either supermajor, nearmajor, nearminor, or subminor, which is the property that constrains the tetrahedron:

Name	Scale	Note	Symmetrical?	Exists in the set of 22 modes?
mosdiatonic	├───┴───┴┴───┴───┴───┴┤ 4 4 1 4 4 4 1		No	Yes
harmonic major	├───┴───┴┴───┴───┴──┴─┤ 4 4 1 4 4 3 2		Yes	Yes
didymic	├───┴──┴─┴───┴───┴──┴─┤ 4 3 2 4 4 3 2		Yes	Yes
zarlino	├──┴───┴─┴───┴──┴───┴─┤ 3 4 2 4 3 4 2		Yes	Yes
diatonyx-A	├───┴──┴─┴───┴──┴──┴──┤ 4 3 2 4 3 3 3	The upper tetrachord is a porcupine tetrachord.	No	No
diatonyx-B	├──┴──┴──┴───┴─┴───┴──┤ 3 3 3 4 2 4 3	The lower tetrachord is a porcupine tetrachord.	Yes	Yes
equable / onyx	├──┴──┴──┴───┴──┴──┴──┤ 3 3 3 4 3 3 3		No	Yes
symmetrical dorian	├───┴─┴──┴───┴──┴─┴───┤ 4 2 3 4 3 2 4		No	No

This reduces to a set of 35 if the fourth and fifth are held fixed, and 55 if only the fifth is.

Every mode of one of these scales has a pattern of broadly major and minor intervals corresponding to one of the standard diatonic modes. For example, the equable scale in its primary mode is a form of Dorian, as its pattern is major-minor-perfect-perfect-major-minor (in this case, nearmajor and nearminor). However, there are not in fact 12 instances of each mode! The equable scale only has Mixolydian, Dorian, and Aeolian modes, and the symmetrical Dorian scale lacks a Locrian or Lydian mode.

Mode type	84-set	35-set	22-set
Locrian	7	-	-
Phrygian	12	5	5
Aeolian	15	8	4
Dorian	16	9	4
Mixolydian	15	8	4
Ionian	12	5	5
Lydian	7	-	-

Regardless, this is simply a mathematically complete enumeration - for actual modal music, it is best to stick to the list of 22 modes provided above, as those are the ones that have clear common qualities alongside the functionally important perfect fifth and fourth.

Additional work needs to be done to determine if this can be generalized.

Tetrachords in modal analysis

Tetrachords, mentioned previously in an adaptation of their original Greek form, can be used in a different way, more in accordance with their use in modern 12edo theory. In modes where the fourth and fifth are perfect, the mode can always be thought of as being comprised of two tetrachords separated by a whole tone, although the constraints on these tetrachords are entirely different from the Greek versions. In short, a modal tetrachord must comprise the unison, the fourth, a second of one of the four qualities, and a third of one of the four qualities, such that the interval between two adjacent tones is never more than four steps. This is a generalization of the constraints on modal tetrachord patterns in 12edo, which must always contain either whole tones or semitones. By this constraint there are ten distinct tetrachords in 22edo. Considering all the possible scales constructed from these, there are 10x10 = 100 distinct possibilities, compared to the 3x3 = 9 options found in 12edo. This provides an extended set, including not only the 35 modes corresponding to diatonic but 65 additional scales corresponding in some regard to melodic minor or neapolitan major. Not all intervals are necessarily within their expected quality ranges.

Loosening the constraint further to only necessitate that the two movable tones remain within their respective halves of the tetrachord allows for the generalization to a set of scales analogous to harmonic minor or double harmonic major, with 156 additional possibilities.

The fact that while in 12edo there are 2^4=16 distinct tetrachordal scales under the extended definition, in 22edo there are 4^4=256, is not a coincidence: it is a direct result of the doubling of interval qualities within a tetrachord's span of the perfect fourth that leads to 22edo's construction in the first place. This extreme number of scales is overwhelming, but it simply shows the sheer inexhaustibility of 22edo modal harmony.

In the 10-tone system

As pajara[10] is a MOS, its modal harmony doesn't have as much of the same complexity as diatonic modal harmony does. However, it is useful to consider the previously mentioned MODMOSes as roughly on the same level as the MOS form of the scale (as the only additional variety they introduce is in the tritone), giving 15 distinct modes available to choose from, each with a degree of brightness or darkness to them, much more like conventional 12edo modal harmony. It is for this reason and the reasons given above with 10-tone functional harmony that pajara[10] can be considered a much more "familiar" approach to 22edo, despite being such a completely different scale.

A pajara[10] pentachord may be considered to consist of five tones. To continue the theme of pajara mirroring conventional harmony with two qualities from 12edo, we may constrain this specific set of pentachords such that they must be comprised entirely of semitones and nearmajor seconds, which is an analogous constraint to the one stating that a 12edo tetrachord must be comprised entirely of tones and semitones, as it leads to four distinct pentachords.

Alternatively in the more general interpretation, there are four additional "harmonic" pentachords. Note that in either case, no note in a pentachord may occupy the first or last step of the perfect fourth.

Chromatic subsets

In 22edo, multiple qualities may be combined together into a compound system. In 12edo, there is little reason to do this, because there are only two qualities available, so the scale combining them (the 12edo chromatic scale, or some other large scale like ├─┴┴┴┴─┴┴┴─┴┤ 2 1 1 1 2 1 1 2 1 ) is not particularly engaging from either a tonal or modal perspective. However, in 22edo, there are four different qualities, from which two may be selected to share characteristics.

The standard chromatic scales combine nearmajor+nearminor and supermajor+subminor, which lead to somewhat of the same problem as 12edo chromatic; they are opposing pairs of qualities. However, if we make an asymmetric chromatic scale, with (for instance) supermajor and nearminor, we get a scale with the trait they have in common: being "unstable". Alternatively, you could get a generally "dark" system by combining subminor and nearminor qualities.

The following are a few examples of these kinds of scales, including diatonic and chromatic variations. (Note that in tonal music, these become less distinct from standard counterparts, as degrees are already expected to be altered between different qualities depending on context.)

Nearminor (harmonic): P1 - SM2 - nm3 - P4 - P5 - nm6 - NM7 - P8 ( ├───┴─┴──┴───┴─┴────┴─┤ 4 2 3 4 2 5 2 )

Subminor (harmonic): P1 - SM2 - sm3 - P4 - P5 - sm6 - NM7 - P8 ( ├───┴┴───┴───┴┴─────┴─┤ 4 1 4 4 1 6 2 )

Compound minor (harmonic + natural): P1 - SM2 - sm3 - nm3 - P4 - P5 - sm6 - nm6 - NM7 - P8 ( ├───┴┴┴──┴───┴┴┴────┴─┤ 4 1 1 3 4 1 1 5 2 )

Supermajor (harmonic): P1 - SM2 - SM3 - P4 - P5 - SM6 - NM7 - P8 ( ├───┴───┴┴───┴───┴──┴─┤ 4 4 1 4 4 3 2 )

Compound unstable (harmonic + natural): P1 - SM2 - nm3 - SM3 - P4 - P5 - nm6 - SM6 - nm7 - NM7 - P8 ( ├───┴─┴─┴┴───┴─┴─┴─┴┴─┤ 4 2 2 1 4 2 2 2 1 2 )

Aberrismic scales may also be leveraged for this purpose.

Consonant vs. tense suspended chords

The wider supermajor second and contrast with the supermajor third actually makes suspended chords somewhat of a point of resolution, rather than a point of tension like in 12edo. It's reasonable to have a suspended chord that doesn't resolve, perhaps making the term "suspended" inaccurate. These suspended chords can function like arto and tendo chords, with a 1-2-4-5 chord structure being plausible, or can be used in modal harmony as a form of "mode-agnostic" anchor point. The sus4 chord in particular is composed of the three octave-reduced perfect consonances, and thus can also be considered the most basic polychordal scale (perhaps a/the "dichordal" scale). The consonance of the supermajor second is additionally what allows chthonic harmony to function. However, suspensions that function more like 12edo ones in leading into the MOS diatonic intervals and being more tense can still be found with the nearmajor sus2 and wolf sus4, which lose some of the structural elegance of standard Pythagorean suspensions in favor of a more tense, crowded sound that can easily resolve to even the rather tense supermajor triad.

Pajara vs. diatonic: a summary

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

Table of chords

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.

Table of MOS scales

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	As elaborated on previously, this acts as the 10-form analog to diatonic, retaining a simple dichotomy of interval qualities and familiar styles of leading harmony, but also adds in new chthonic harmony.
	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

11edo scales

MOS scales generated from 11edo temperaments.

Temperament	Name	Chart	Notes
Sensamagic	antipentic	├─────┴─┴─────┴─┴─────┤ 6 2 6 2 6	Has swapped interval sizes relative to the normal pentatonic scale.
	checkertonic	├─┴───┴─┴─┴───┴─┴─┴───┤ 2 4 2 2 4 2 2 4	Can be considered the "basic" 8-form scale of sensamagic harmony. The basic 1/1-9/8-5/3 chord exists on almost all keys, and inversions thereof can be considered as the basic chords of the system. Has similar step sizes to 12edo diatonic, allowing for familiar kinds of melodic shapes to be used against entirely novel harmony
Machine	pedal	├───┴───┴───┴───┴─────┤ 4 4 4 4 6
	machinoid	├───┴───┴───┴───┴───┴─┤ 4 4 4 4 4 2	Acts as one possible 22edo counterpart to the "whole tone scale". Highlights the chords used by antidiatonic, but now features both simultaneously on 3 of its degrees.
Orgone	smitonic	├─┴───┴─┴───┴─┴───┴───┤ 2 4 2 4 2 4 4	Acts as an altered version of 12edo diatonic, and can be seen as an 11edo analog to diatonic, being generated by its primary consonance (5/3 in this case). It reaches 9/7 after five scale steps and places it on the same degree as 11/8.
Joan	pentic	├─┴───────┴─┴─┴───────┤ 2 8 2 2 8	One of two tunings of pentic available in 22edo.
	antidiatonic	├─┴─┴─────┴─┴─┴─┴─────┤ 2 2 6 2 2 2 6	Has swapped interval sizes relative to diatonic, and therefore diatonic chord/harmony logic can be loosely "translated" into antidiatonic. This can be seen as a system generated by wolf fifths. A fifth tuned flat enough to generate a hard antidiatonic scale is called a "zavala" fifth. The "third" in this system is either 8/7 or 9/7. The "fifth" is 16/11, meaning that the counterparts of diatonic major and minor chords are "essentially tempered", because they are bounded by 16/11 yet composed of only 7-limit intervals. Additionally, this system allows /7 harmony, with 7:9:11 being an available chord on some degrees.
	balzano	├─┴─┴─┴───┴─┴─┴─┴─┴───┤ 2 2 2 4 2 2 2 2 4	Acts as an expanded version of antidiatonic with additional notes. Many of the same harmonic rules apply, except that in this case 8/7 and 9/7 fall on different degrees (both shared with 6/5), so two possible "tertian" chords exist on a single degree.

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	The 10-note counterpart of the Delkian scale, which contributes 10 of the total 15 pajara modes, and can be considered somewhat of a pajara analog of melodic minor. Constructed from two sssL "pentachords" joined by an ss whole tone.
Delkian	├─┴─┴─┴┴─┴─┴─┴─┴┴─┴─┴─┤ 2 2 2 1 2 2 2 2 1 2 2 2	A scale which represents the dual correspondence of each 12edo interval, having 12 different modes as opposed to the 6 of the normal pajara[12]. This scale additionally is significant for being used in Famana's music theory system (albeit in a different tuning). It may be constructed by splitting pentic scale steps: 4-5-4-5-4 becomes [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.

Supersets and subsets

11edo

Main article: 11edo

22edo is, as mentioned previously, the double of 11edo, which serves as a strong system in certain composite subgroups.

44edo

22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure. 22edo's 7/4 becomes particularly inaccurate 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. It's something like 12edo's 5/4 in a system like 24edo, where it's structurally justified by the subset edo while losing accuracy. 44edo also contains accurate approximations of the 13th and 19th harmonics as well as the 23rd harmonic.

Comparisons to other related 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.
• 31edo shares orwell.
• 41edo shares the keemic tertian structure, and more specifically magic.
• 27edo shares superpyth, and 32edo, also an archy tuning, shares pajara with a particularly sharp tuning.
• 24edo essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.

Music in 22edo

Vector - What Happens After

See also

• User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)