22edo/V/Exposition

Just intonation

In music, the basic concept behind most tuning systems is the harmonic series, or the series of pitches whose frequencies are a multiple of some frequency called the fundamental. These are the frequencies that make up the sounds of notes on most instruments, and we hear concordance when two notes share frequencies. Sets of notes that do this are themselves tuned so that the intervals between them can be found somewhere in the harmonic series. These intervals are frequency ratios that can be expressed as rational numbers, and the set of rational intervals (and tuning systems that make exclusive use of them) is called just intonation.

Note that while frequency is linear, pitch is logarithmic, so a doubling in frequency sounds like linear steps in pitch. In specific, a doubling in frequency is the interval of an octave.

While concordance is limited to a small set of intervals such as 2/1, 3/2 (the perfect fifth), 4/3 (the perfect fourth), and often 5/4 (a kind of major third) and 5/3 (a kind of major sixth), it becomes much more prominent with multi-note chords, with the chord 4:5:6:7 often implying an octave above the 4 that isn't actually present (a technique which is used extensively in barbershop harmony). It is for this reason that just intonation presents a thorough structure around which to base tuning systems.

However, tuning systems that exclusively use just intervals encounter some problems. In Western music, it is important (for instrument design, for instance) for there to be a set of "all the notes", a concept which becomes rather arbitrary in just intonation. Additionally, we generally prefer to be able to transpose melodies and chords up and down and still be able to play them (that is, play in different keys). If we are to use only just intervals, then these two goals are in conflict with one another. For instance, if we construct a 12-note scale out of perfect fifths and 5/4 major thirds, we get duodene, which has approximately - but not exactly - equal intervals, and very complex dissonant intervals on some keys in the place of more consonant ones. If we instead only use perfect fifths, we get 12-note Pythagorean tuning (a chain of eleven fifths, closed with a "wolf" fifth), which is closer, but still not the best answer - at the same time, it leaves out the interval 5/4 entirely. We can extend to 53-note Pythagorean tuning, which gets us really close to a closed loop (and also gets us an interval that is essentially indistinguishable from 5/4) but it's still not exact and is a rather unwieldy number of notes to, say, include on an instrument.

Equal temperament

Therefore, we must simplify the structure of just intonation in some way. The way to do this is equal temperament, where we choose one interval to keep justly tuned (in this case, the octave) and equally divide it in pitch (so that each step is some fractional power of that interval in frequency space). The most common approach is 12-tone equal temperament, wherein the octave is 12 steps, and which finds approximations, albeit imperfect ones, to the perfect fifth at 7 steps and to the 5/4 major third at 4 steps. To measure the logarithmic size of any interval, we use a measure called cents, which is basically 1/100 of a 12edo semitone. In this system, the justly tuned perfect fifth is at about 702 cents, and 5/4 is at about 386 cents. We can see that 12edo very closely represents the perfect fifth and decently represents 5/4.

However, equal temperament is especially useful if we want to extend beyond the 5-limit and use intervals like 7/4 (a kind of minor seventh that shows up in the aforementioned barbershop chords). For this, we need to choose a different number of equal divisions of the octave.

Turning it up to 22

On "major" and "minor"

In the previous sections, if you're familiar with just intonation, you might notice that I've always been careful to specify "the 5/4 major third" as opposed to just "the major third" - this is because the quality represented by 5/4 is only one of several kinds of 'major'. In 12edo, intervals only have two qualities - major and minor. 22edo doubles that, giving us two kinds of major (supermajor and nearmajor), and two kinds of minor (nearminor and subminor). 22edo also represents 3/2 and 5/4 well, and 5/4 is the nearmajor third (its complement, 6/5, which the 12edo minor third approximates, is the nearminor third).

On the construction

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.

Steps of 22edo	Cents	Interval name
0	0	Unison
1	55	Subminor Second, Diatonic Semitone, Quarter Tone
2	109	Nearminor Second, Equal Semitone
3	164	Nearmajor Second, Minor Tone, Chromatic Semitone
4	218	Supermajor Second, Whole Tone
5	273	Subminor Third
6	327	Nearminor Third
7	382 (5/4)	Nearmajor Third
8	436	Supermajor Third
9	491	Perfect Fourth
10	545	Near Fourth, Wolf Fourth, Diminished Fifth
11	600	Semioctave, Tritone
12	655	Near Fifth, Wolf Fifth, Augmented Fourth
13	709 (3/2)	Perfect Fifth
14	764	Subminor Sixth
15	818	Nearminor Sixth
16	873	Nearmajor Sixth
17	927	Supermajor Sixth
18	982 (7/4)	Subminor Seventh
19	1036	Nearminor Seventh
20	1091	Nearmajor Seventh
21	1145	Supermajor Seventh
22	1200	Octave

You may notice that 22edo approximates 5/4 at the nearmajor third a lot closer than 12edo does, and its 3/2 is still pretty accurate as well at the perfect fifth.

Additionally, this expanded set of intervals allows for new chords, such as 4:5:6:7 (with the 7/4), and even some chords involving prime 11, such as 8:11:14 (made with the unison, the 10th step, and 18th step of the scale).

The diatonic scale

In 22edo, because each quality is split into two new ones, we are left with a choice of how to represent the standard diatonic scale in the system. The best approach, at least in order to define a somewhat neutral standard for notation and interval categorization, is to utilize the supermajor and subminor intervals in diatonic. This sounds very different from a standard 12edo diatonic, but it has the shared advantage of being created from a chain of stacked fifths and of being a moment-of-symmetry scale (with only 2 possible sizes of each interval); it is also, aside from the augmented fourth and diminished fifth, a proper scale. It is the diatonic scale which the aforementioned diatonic and chromatic semitones apply to, with, for instance, the diatonic major third, as the supermajor third, being separated from the diatonic minor third by a 3-step chromatic semitone, and from the perfect fourth by a 1-step diatonic semitone. The step sizes for the diatonic scale are 4-4-1-4-4-4-1, compared to 12edo's 2-2-1-2-2-2-1.

A clarification on notation

In 22edo, the most common approach is for the sharp (#) sign to represent going from diatonic minor to diatonic major, or a chromatic semitone of 3 steps - the flat (b) sign reverses this. However, because of the utility of a single step for adjusting intervals (especially to reach intervals of 5, which are skipped over by the standard diatonic), 22edo also has the up (^) and down (v) signs for adjusting by a quarter-tone. Keep in mind that the quarter tone is also the diatonic semitone, so C-vF is the same size of interval as C-E.

This means that standard enharmonic equivalences no longer apply (enharmonic intervals are in fact specifically separated by the equal semitone). In fact, 22edo has its own set of equivalences. This chart shows the equivalent notes in 22edo from A to C.

A		vCb
Bb	^A	vvA#
Cb	^^A, vvB	vA#, ^Bb
A#	vB	^^Bb
B	vC	^A#
C	^B

Functional harmony

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.

Consonance, dissonance, and just intonation

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 (as previously mentioned) 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.

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.

For future reference, here is a table of 22edo intervals and just intervals represented. "Nm" and "NM" are nearminor and nearmajor; "sm" and "SM" are subminor and supermajor. TT indicates the tritone.

1 (sm2)	2 (Nm2)	3 (NM2)	4 (SM2)	5 (sm3)	6 (Nm3)	7 (NM3)	8 (SM3)	9 (P4)	10 (d5)	11 (TT)
25/24, 33/32, 49/48	15/14, 16/15, 17/16	10/9, 11/10, 12/11	9/8, 8/7	7/6	6/5	5/4	9/7	4/3	11/8	7/5, 10/7
12 (A4)	13 (P5)	14 (sm6)	15 (Nm6)	16 (NM6)	17 (SM6)	18 (sm7)	19 (Nm7)	20 (NM7)	21 (SM7)	22 (P8)
16/11	3/2	14/9	8/5	5/3	12/7	7/4	11/6, 9/5	15/8, 17/9	48/25	2/1

Structural properties of 22edo

The most significant property of 22edo coming from 12edo is that the 5/4 nearmajor third is not found within the MOS form of the diatonic scale. Therefore, to make use of it, one must use other diatonic scales or a completely different scale structure entirely. Instead, the MOS diatonic reaches the 9/7 supermajor third (and the 7/6 subminor third). This is called archy temperament.

However, a much more significant property with greater implications on the structure of chords is that the nearmajor third (5/4) and the subminor seventh (7/4) are separated by a perfect tritone, and that the nearminor third and supermajor sixth are separated by that very same interval, so that the interval between 5/4 and 6/5 becomes the same as the interval between 7/4 and 12/7 (exactly one edostep). Suddenly, 4:5:6:7 strikes not as a detuned dominant tetrad, but as the prototype of a new kind of chord built from these tritone-separated intervals. This fact, combined with the above, represents pajara temperament, one of the defining features of 22edo, and which will eventually allow us to build a new kind of scale to access intervals of both 5 and 7.

The counterpart to 4:5:6:7 (P1-NM3-P5-sm7) is obtained by flattening the third and seventh each by an edostep, creating a canonical "minor" version of the chord, P1-Nm3-P5-SM6. Notice that the seventh actually crossed categories into being a sixth - this is the key to unlocking the special new kind of scale that pajara offers. We can relabel our intervals to reflect this emergent categorical variation and define this as a new interval ordinal - call it the antilatus - and make its complement (found in the 6:7:8 chord and its minor counterpart) the unilatus. And because the tritone defines this structure, we will give it its own ordinal as well, distinct from the fourth and fifth. If we build a scale from these steps, we create the 2-2-3-2-2-2-2-3-2-2 pajara[10] scale. The way in 22edo to access the full potential of the 7-limit is not to cram it into the existing diatonic structure, but to add new notes to give the intervals of 7 and 5 both their own spots on the scale. As it turns out, we've actually already done this - if you take the 4-4-1-4-4-4-1 diatonic scale, and divide each of the 4-step whole tones into two equal semitones, you receive 2-2-2-1-2-2-2-2-1-2-2-2 - which is a 12-note superset of pajara[10]!