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22edo chord and interval notation

The following ideas have been formed out as part of the first Tuning of the Year project (for year 2025), hereafter referred to as TOTY. The focus of the original TOTY was 22edo, which was voted on by the Xenharmonic Alliance discord.

Our objective was to flesh out 22edo music theory, arrive at a deeper understanding, and provide resources to help beginners learn how to think about and play in 22edo. The most rigorous work in this tuning had already been accomplished. Ups and downs notation sufficiently identifies the notes and chords, and the most important chords have already been identified and labelled.

Most of our work was in assessing our overall impressions of the tuning, discovering what language was available to discuss harmonies in this tuning, and to check existing nomenclature against our intuitive grasp of what things sound like. Ultimately we arrived at a nomenclature that explains 22edo in terms of how it sounds, as opposed to how it conforms to the logic of the pythagorean circle of fifths. Note names have been unaltered, but some chord and interval names have been altered and potentially refined.

When TOTY began, there was one existing system of chord nomenclature that covers 22edo, Kite Giedraitis' ups and down notation. Most of us were unfamiliar with the naming convention of this system. In developing our own systems of nomenclature, Kite reached out to us with information on his existing system.

Including ups and downs notation, we have at least four distinct methods of identifying chords in 22edo. These methods are all comprehensive, meaning one can reasonably identify any chord just as well as one can identify chords in 12edo using standard chord labels. In sum, we have ups and downs notation, classic notation, double-qualifier notation, and temperament notation.

Intervals

Using a "sounds-like" system of interval naming, we can identify 22edo as having the following intervals:

edosteps	symbol	spoken name	size (in cents)
1\22	s2	subminor second	54.5
2\22	m2	minor second	109.1
3\22	n2	neutral second	163.6
4\22	M2	major second	218.2
5\22	s3	subminor third	272.7
6\22	m3	minor third	327.3
7\22	M3	major third	381.8
8\22	S3	supermajor third	436.4
9\22	p4	perfect fourth	490.9
10\22	M4	major fourth	545.5
11\22	S4/s5	supermajor fourth subminor fifth	600
12\22	m5	minor fifth	654.5
13\22	P5	perfect fifth	709.1
14\22	s6	subminor sixth	763.6
15\22	m6	minor sixth	818.2
16\22	M6	major sixth	872.7
17\22	S6	supermajor sixth	927.3
18\22	m7	minor seventh	981.8
19\22	n7	neutral seventh	1036.4
20\22	M7	major seventh	1090.9
21\22	S7	supermajor seventh	1145.5
22\22	p8	octave	1200

Note that this is contrary to the established convention of referring to the 6\22 interval as the "upminor" and the 7\22 interval as the "downmajor". Generally speaking, we felt that these intervals simply sound like minor and major intervals, respectively. Also, their lesser and greater counterparts, 5\22 and 8\22, which are labelled on the 22edo page as "minor" and "major," sound more like subminor and supermajor intervals. The nearest LCJI approximations to these two intervals, 7/6 (266.8c) and 9/7 (435c) are typically defined as subminor and supermajor.

One can see that there is some inconsistency in qualities across the intervals. Seconds and sevenths are defined as minor, neutral, major, and supermajor, while thirds and sixths are defined as subminor, minor, major, and supermajor. This simply seems more accurate to the sound. Note that symmetry is still preserved across the tritone.

While obviously, there is some subjectivity here, and the possibility of an even more refined perspective. But this seems good enough to talk about 22edo in a way that is intuitive to what we are hearing.

Triads

In order to communicate the chords of 22edo, it is necessary to identify the names of the tertian triads. Adding sevenths to our triads gives us many practical chords that will be used by most composers and musicians. Adding our standard modifications to these, adding the sevenths, and alterations, gives us a complete system with which to name any chord. Here a triad is defined explicitly as a tertian triad, being a chord made out of two stacked thirds. Only the subminor, minor, major, and supermajor thirds (5\22, 6\22, 7\22, 8\22) are considered.

There are (at least) four comprehensive systems for naming chords in 22edo: ups and downs notation, classic notation, double-qualifier notation, and temperament notation. Ups and downs notation adheres to pythagorean logic in naming all chords and intervals, and is oldest and most established of the four. TOTY was not involved in its inception. The other three were created in tandem, and are quite closely related.

For consistency and convenience, all chords are defined on the root of C.

Classic Notation

The classic notation system is intended to be intuitive and clear. Chord labels are very similar to 12edo chord symbols. With the exception of ups and downs, which are standard accidentals in many microtonal systems, no new chord symbols are introduced.

Many tertian triads in 22edo have altered fifths. In classic notation, the alteration is spelled out explicitly. The chord C - E - vG would be spelled as a CS(v5).

The tertian triads in classic notation would be spelled thusly:

Note names	Edosteps	Interval sizes	Chord label	Spoken name
C Eb Gb	0 5 10	s3 s3	Csb5	C subdiminished
C Eb ^Gb	0 5 11	s3 m3	Cs^b5	C subminor up-flat five
C ^Eb ^Gb	0 6 11	m3 s3	Cdim	C diminished
C Eb vG	0 5 12	s3 M3	Csv5	C subminor down five
C ^Eb vG	0 6 12	m3 m3	Cmv5	C minor down five
C vE vG	0 7 12	M3 s3	Cv5	C down-five
C Eb G	0 5 13	s3 S3	Cs	C subminor
C vEb G	0 6 13	m3 M3	Cm	C minor
C vE G	0 7 13	M3 m3	C	C (major)
C E G	0 8 13	S3 s3	CS	C supermajor
C vEb ^G	0 6 14	m3 S3	Cm^5	C minor up five
C vE ^G	0 7 14	M3 M3	Caug	C augmented
C E ^G	0 8 14	S3 m3	CS^5	C supermajor up five
C vE vG#	0 7 15	M3 S3	Cv#5	C down-sharp five
C E vG#	0 8 15	S3 M3	CSv#5	C supermajor down-sharp five
C E G#	0 8 16	S3 S3	CS#5	C supermajor sharp five

In this system, chords are quite clear. For maximum clarity, one could also opt to call the diminished chord "C minor up-flat five" and call the augmented chord "C major up five".

While the chord labels can be completely unambiguous written in this way, many people do not write ups and downs before the note they modify. In this case, it would be necessary to include the alterations of the fifth either in parenthesis or otherwise demarcated. However, it's quite clean if ups and downs are written before the note they modify. For instance, written correctly, ^D - F# ^A would be written as:

^D5

or written without respect to this convention:

D^(5)

since if this chord were written as D^5, it would be unclear if the root was being modified or the fifth.

Double-Qualifier Notation

In order to avoid using many alterations, we can actually describe every triad based on the quality of its third and the quality of its fifth. Using the interval system above, and the necessary triads, we have seven qualities of fifth: diminished, subminor, minor, perfect, major, supermajor, and augmented. These intervals are 10\22, 11\22, 12\22, 13\22, 14\22, 15\22, and 16\22.

While double-qualifier notation is not immediately obvious, it is quite easy to understand, and is basically interchangeable with classic notation. The advantage to using double-qualifier notation it does not require writing alterations for every type of fifth. In actuality, this is somewhat consistent with 12-edo nomenclature, which usually doesn't explicitly notate altered fifths in chord symbols (except in modern styles like jazz). For instance, the diminished fifth is communicated by the symbol for diminished triad, and the augmented fifth is communicated by the symbol for the augmented triad.

Note names	Edosteps	Interval sizes	DQ Label	DQ spoken name	Classic label
C Eb Gb	0 5 10	s3 s3	Csd	C subminor diminished	Csb5
C Eb ^Gb	0 5 11	s3 m3	Css	C double subminor	Cs^b5
C ^Eb ^Gb	0 6 11	m3 s3	Cms	C minor subminor	Cdim
C Eb vG	0 5 12	s3 M3	Csm	C subminor minor	Csv5
C ^Eb vG	0 6 12	m3 m3	Cmm	C doubleminor	Cmv5
C vE vG	0 7 12	M3 s3	CMm	C major minor	Cv5
C Eb G	0 5 13	s3 S3	Cs	C subminor	Cs
C vEb G	0 6 13	m3 M3	Cm	C minor	Cm
C vE G	0 7 13	M3 m3	C	C (major)	C
C E G	0 8 13	S3 s3	CS	C supermajor	CS
C vEb ^G	0 6 14	m3 S3	CmM	C minor major	Cm^5
C vE ^G	0 7 14	M3 M3	CMM	C double major	Caug
C E ^G	0 8 14	S3 m3	CSM	C supermajor major	CS^5
C vE vG#	0 7 15	M3 S3	CMS	C major supermajor	Cv#5
C E vG#	0 8 15	S3 M3	CSS	C double supermajor	CSv#5
C E G#	0 8 16	S3 S3	CSA	C supermajor augmented	CS#5

If writing triads, including the 5 is optional. One could opt to include the 5, especially to maintain clarity when writing seventh chords or extended chords (explored in more detail below). For instance, Css could be written as Css5 or Cs(s5).

An advantage to this system is that ups and downs can be used to define alterations beyond fifths and sevenths. A disadvantage is that one must learn the various qualities of the fifth as defined by this system, and the chord names sound very similar. It is also contrary to the usual method of chord qualification, where two qualifiers typically define the quality of the triad and the seventh. For instance, C minor-major is shorthand in 12edo for a minor triad with a major seventh. In this double qualifier system, the same chord should be explicitly called a C minor major-seventh, as opposed to C minor major-fifth.