User:Stalefleas

[this page is still in progress]

todo: add a link to kite's notation

22edo chord 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.

Temperament Notation

The least conventional of these notation systems is the temperament notation. In this system, unique symbols are prescribed for every triad. The basic triads are given the same symbols as classic notation. But altered fifths are given specific names. There are also plusses and minuses added as qualifiers for various triads.

While temperament notation has the steepest learning curve, it has the advantage of highlighting relationships between chords that might otherwise be less obvious. It also is, at least in my opinion, actually quite intuitive, at least for 22edo.

Note names	Edosteps	Interval sizes	Temp label	Temp spoken name	Classic label
C Eb Gb	0 5 10	s3 s3	Cw	C orwell / subdiminished	Csb5
C Eb ^Gb	0 5 11	s3 m3	Cd-	C utonal diminished	Cs^b5
C ^Eb ^Gb	0 6 11	m3 s3	Cd+	C otonal diminished	Cdim
C Eb vG	0 5 12	s3 M3	Cw-	C orwell minor	Csv5
C ^Eb vG	0 6 12	m3 m3	Ck	C keemic	Cmv5
C vE vG	0 7 12	M3 s3	Cw+	C orwell major	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	CZ-	C sensaminor	Cm^5
C vE ^G	0 7 14	M3 M3	CJ	C magic	Caug
C E ^G	0 8 14	S3 m3	CZ+	C sensamajor	CS^5
C vE vG#	0 7 15	M3 S3	CJ-	C magic minor	Cv#5
C E vG#	0 8 15	S3 M3	CJ+	C magic minor	CSv#5
C E G#	0 8 16	S3 S3	CZ	C sensamagic	CS#5

The three orwell triads: subdiminished, orwell minor, and orwell major, are the only existing triads in the orwell [5] MOS scale in 22edo. The keemic triad is a reduced form of the keemic seventh chord. The magic MOS also includes the magic triads. CJ inverts to CJ- which inverts to CJ+. Similarly, CZ inverts to CZ+ and to CZ-.

This naming convention highlights the variety of diminished and augmented chords in 22edo. Instead of calling any specific triad the augmented triad, we have opted to consider the magic triads and the sensamagic triad their own distinct identities, though they are all functionally augmented triads in the right context.

An advantage to using temperament notation is that it is fairly clean and elegant. It is actually consistent with 12edo chord labels and logic, since 12edo uses unique symbols for every tertian triad. The disadvantage is that the names are not immediately obvious, especially to beginners who are unaware of temperaments.

Some of the chord symbols are also more or less arbitrary. It seemed necessary to use symbols that would not be confused for existing accidentals or existing note names. I chose letters that sound like they belong to the word they represent. K for keemic is quite obvious, but J for magic is a little less obvious--really it is just because the G in "magic" sounds like a J. As for Z, this seems perhaps the most arbitrary symbol, but I felt like the sensamagic triad is quite a big chord, and Z is quite a big letter, and Z looks like a backwards S.

Chord Families

In examining the 16 tertian triads of 22edo, we can see identify four categories: diminished, basic, augmented, and hybrid.

Diminished

The orwell subdiminished triad, the utonal and otonal diminished triads, and the keemic triad can all function as a diminished triad in the proper context. The orwell triad is the natural diminished triad of the super-pythagorean diatonic scale (the 5L2s mos-diatonic). The otonal diminished triad is the natural diminished triad of nicetone, as well as the upper triad of the harmonic seventh chord. The utonal diminished is the natural diminished ii chord of nicetone. Keemic does not occur in either of these diatonic scales of 22edo, but being a stack of two minor thirds, can sound quite like a diminished chord.

Basic

Triads that form a perfect fifth can be considered basic. They are in all probability the most commonly used and explored chords. These include the subminor, minor, major, and supermajor triads. Of these, the supermajor triad in 22edo is probably the most difficult to use.

Augmented

The magic triads, including magic major and magic minor, can all function as augmented triads in the proper context. The sensamagic triad can also function as an augmented triad. Augmented triads are not naturally occurring in any diatonic scale, but the sensamagic triad would be the logical augmented triad of a super-pythagorean diatonic, and the magic minor would be the natural extension of the nicetone diatonic.

Hybrid

Four chords of less obvious function remain. These might sound like wolf triads, or less conventional versions of basic triads. Orwell major and orwell minor sound somewhat like major and minor chords, as do sensamajor and sensaminor. These chords are not quite diminished or augmented, and not quite basic. Thus, they can be conceptualized as hybrid triads.

Triads Overview

The following table displays the tertian triads, their respective labels, and the families to which they belong.

Note names	Edosteps	Interval sizes	Classic	DQ	Kite	Temp	Chord family
C Eb Gb	0 5 10	s3 s3	Csb5	Csd	Cd	Cw	diminished
C Eb ^Gb	0 5 11	s3 m3	Cs^b5	Css	Cd(^5)	Cd-	diminished
C ^Eb ^Gb	0 6 11	m3 s3	Cdim	Cms	C^d(^5)	Cd+	diminished
C Eb vG	0 5 12	s3 M3	Csv5	Csm	Cm(v5)	Cw-	hybrid
C ^Eb vG	0 6 12	m3 m3	Cmv5	Cmm	C^m(v5)	Ck	diminished
C vE vG	0 7 12	M3 s3	Cv5	CMm	Cv(v5)	Cw+	hybrid
C Eb G	0 5 13	s3 S3	Cs	Cs	Cm	Cs	basic
C vEb G	0 6 13	m3 M3	Cm	Cm	C^m	Cm	basic
C vE G	0 7 13	M3 m3	C	C	Cv	C	basic
C E G	0 8 13	S3 s3	CS	CS	C	CS	basic
C vEb ^G	0 6 14	m3 S3	Cm^5	CmM	C^m(^5)	CZ-	hybrid
C vE ^G	0 7 14	M3 M3	Caug	CMM	Cv(^5)	CJ	augmented
C E ^G	0 8 14	S3 m3	CS^5	CSM	C(^5)	CZ+	hybrid
C vE vG#	0 7 15	M3 S3	Cv#5	CMS	Cv(v#5)	CJ-	augmented
C E vG#	0 8 15	S3 M3	CSv#5	CSS	C(v#5)	CJ+	augmented
C E G#	0 8 16	S3 S3	CS#5	CSA	Ca	CZ	augmented

Seventh Chords

Notating seventh chords is fairly straightforward. One simply identifies the type of triad in their preferred system, and appends the quality of the seventh.

If we follow the convention of 12edo, neither major triads nor minor sevenths require clarification. So, an unqualified triad is assumed to be major, and an unqualified seventh is assumed to be minor.

If we define the sevenths as being either minor, neutral, major, and supermajor, some of our common seventh chords might include:

Notes	Chord label	Spoken name
C vE G vB	CM7	C major seventh
C ^Eb G ^Bb	Cmn7	C minor neutral seventh
C vE G Bb	C7	C dominant / C seven
C E G B	CSS7	C supermajor supermajor seventh
C Eb G Bb	Css7	C subminor subminor seventh

Some less common chords would be expressed differently in different notation systems:

Notes	Classic	Classic spoken	DQ	DQ spoken	Temp	Temp spoken
C vE vG B	Cv5(S7)	C down five supermajor seventh	CMm5(S7)	C major minor fifth super seventh	Cw+S7	C orwell major super seventh
C E ^G ^Bb	C^5(n7)	C up five neutral seventh	CMM5(n7)	C major fifth neutral seventh	CJn7	C magic neutral seventh
C Eb ^G Bb	Cs^5(m7)	C subminor up five minor seventh	CsM5(m7)	C subminor major fifth minor seventh	CZ-7	C sensaminor seven

Disambiguation

In 22edo, the chord that would typically be expressed as the basic "minor seventh" chord actually has a neutral seventh, and the chord that would be expressed as the basic dominant chord has the subminor seventh, here notated simply as the minor seventh. One would expect that many will simply call the minor neutral seventh chord a minor seventh chord for short, and even call the chord C ^Eb G Bb the minor subminor seventh chord. This is fine for conversation, conceptualization, and disambiguation. However, here we are defining the lesser seventh as the minor seventh and the larger minor seventh as the neutral seventh, in part because this preserves tritone symmetry with the seconds, and in part because this seems like an adequate description of the sound of these chords.

It is common to call a chord with a supermajor third and a supermajor seventh simply a supermajor seventh chord, and even to notate it as CS7. However, this is not a clearly defined chord. Here the S could be modifying the triad or the seventh.

So we might have either

C - vE - G - B

or

C - E - G - Bb

By explicitly defining the quality of the triad and the seventh as CSS7, we make it clear that we want the chord

C - E - G - B

What if we did want the chord C - vE - G - B? CS7 would still not suffice. In this chord, it is necessary to separate the seventh from the triad. So here we could write C(S7) or even C.S7 to clarify the seventh is in fact supermajor, and not the triad. And if we wanted the other chord C - E - G - Bb, we could write CS(7) or CS.7 or even CSm7

Disambiguation is not necessary for all chords, but defaulting to using it could lead to greater consistency in notation.

Extended chords

Following the convention of 12edo notation, we have qualifiers for both triads and sevenths. In 22edo these have been described above. To add additional extensions, we can also follow the example provided by standard notation. All extensions will be assumed to be mos-diatonic (super-pythagorean) unless otherwise qualified. So, seconds are supermajor by default, fourths are perfect, and sixths are supermajor.

Thus the chord C11 could include the following notes:

C - vE - G - Bb - D - F

However, in 22edo, it is quite likely that we might want the ^11, as it approximates the eleventh harmonic and want the chord that in total approximates the 4:5:6:7:9:11 chord:

C - vE - G - Bb - D - ^F

This could be notated as C9(^11)

Various extended diminished chords can be specified by appending a sixth. If we alternate minor and subminor thirds, we have

C - ^Eb - ^Gb - vBb

since vBb is enharmonically A, we can consider this chord Cd+6, Cdim6. Or, if one prefers, they might opt for the unwieldy Cm^b5(vb7)

We can also add extensions to triads using "add" or by using the comma from Kite notation.

C - vE - G - D

could be written as Cadd9 or C,9

Non-tertian chords

Of course, any chord that exists in 12edo can still be expressed in 22edo. We have many non-tertian triads including suspended chords, chords with no thirds, chords with no fifths, etc. We can adopt the same conventions from standard chord notation and apply them to 22edo.

C - F - G

would still simply be a Csus4 chord. But of course in 22edo we also have options like

C - ^F - G

which we might write as Csus^4.

While this is not standard practice, one could borrow the convention of writing powerchords with a "5" to notate chords without thirds. So for instance

C - G - vB

might be major or minor depending on the context, but devoid of such a context (or in a situation where specificity is desired) this could be written as a C5M7. Or, one could follow the existing convention of writing this as CM7(no3).

Some chords are difficult to write, for instance, quartal chords like

C - F - Bb

which is often notated as C7sus4, and could be done here. Or, one could opt to write this as Fsus4/C.

Highly complex chords can be notated using polychord notation. For instance

C - Db - E - F - Gb - G - B

could be written as DbS(7)/CS.

Of course, there are instances where notating a chord might be difficult, if there are large clusters of notes, for instance. Non-standard labels might need to be referred to or invented. Or, once could simply use standard sheet music notation, which can express any number of notes with absolute clarity.