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Stalefleas

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= 22edo chord and interval notation =
 
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.  
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.  


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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.
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 ===
= Intervals =
Using a "sounds-like" system of interval naming, we can identify 22edo as having the following intervals:
Using a "sounds-like" system of interval naming, we can identify 22edo as having the following intervals:
{| class="wikitable"
{| class="wikitable"
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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.
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 ==
= 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.
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.


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=== Temperament Notation ===
=== 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.
{| class="wikitable"
!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 major
|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.
{| class="wikitable"
!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:
{| class="wikitable"
|+
!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:
{| class="wikitable"
|+
!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.
= Additional resources =
Video of the tertian triads of 22edo:
https://www.youtube.com/watch?v=fMeRlKlreV8
= Contributors =
Official members of TOTY 2025, as identified by their discord handles at the time:<blockquote>Nick Vuci (project lead)
stalefleas (project lead)
Sevish
pailiaq
maily
Wrenharmonic
Domin
Ebooone
bartonius
Colonizor48
ground
x e n o [ i ] n d e x
dogwithabome
YoVariable
54edo Pajara
CompactStar
Samikata
MidnightBlue
j87j]oi:bu
recentlymaterialized
MR. LEAKY 666
Romeolz
don page commata
The Universe of Music
Alexei</blockquote>
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