In 12edo, the standard triads are made from a stack of two thirds, those being Diminished, Minor, Major, and Augmented.

With five thirds, we now have twenty five possible options, five of which span a perfect fifth. These are the five basic triads. Four are consonant root triads, those being 6:7:9, the subminor triad, 6/5/4, the minor triad, 4:5:6, the major triad, and 9/7/6, the supermajor triad. 18:22:27, the neutral triad, is an important harmonic option, but is noticeably less consonant.

The twenty other standard triads include important options like multiple varieties of augmented and diminished triads. An example is the harmonic diminished triad, a 5:6:7 chord, made from a subminor third stacked on a minor third. The simplicity of the ratios gives it a more consonant sound than a standard diminished triad, made from two stacked minor thirds. A similar option is a squares triad, with a neutral third stacked on a supermajor third, for a 7:9:11 chord with a more consonant sound than the traditional 16:20:25 augmented.

But really, why stick to triads?

Tetrads

For the most part, tetrads attempt to be natural extensions of some triad, and they tend to accomplish this in one of two ways. They may take the triad and add an extension that doesn’t increase the harmonic complexity much, if at all, such as with the Harmonic Seventh Chord (4:5:6:7) or Undecimal Tetrad (6:7:9:11), or by stacking another interval that fits with those already used, in an effort to preserve the overall sound and allow the internal intervals to be consonant, such as with the Major or Minor Seventh Chords.

Subminor Seven	Undecimal Tetrad	Minor Seven	Utonal Tetrad	Major Seven	Harmonic Seven	Supermajor Seven	9-over Tetrad	Neutral Seven
P1	P1	P1	P1	P1	P1	P1	P1	P1
s3	s3	m3	m3	M3	M3	S3	S3	n3
P5	P5	P5	P5	P5	P5	P5	P5	P5
s7	n7	m7	S6	M7	s7	S7	m7	n7

There are nine specific tetrads of this type that I'll talk about, as I consider them to be the most important harmonic resources. First, the subminor triad can be extended with a subminor or neutral seventh. The subminor seventh keeps the overall sound of the chord similar, with a 12:14:18:21 chord, and the neutral seventh adds a more complex sound while keeping the harmonic complexity of the chord low, going from a 6:7:9 triad to 6:7:9:11 tetrad.

The minor triad can be extended similarly. The minor seventh is the most standard extension, similar to a minor seventh chord is 12edo, with the representation 10:12:15:18. An alternative is a supermajor sixth, with utonal (undertonal) representation 12/10/8/7.

The major triad can be extended with a major seventh just like in 12edo, though with a more relaxed sound due to the flatter, more consonant third and seventh. However, whereas in 12edo the dominant seventh would be the other standard option, 31edo has a better alternative, that being the harmonic seventh. The subminor seventh is a very natural extension to the major triad, due to being the next highest harmonic for the simple triad. This is called the harmonic seventh chord, 4:5:6:7, and has a clear consonant sound due to how pure the 5th and 7th harmonics are. It's useful for how rock music tends to employ dominant seventh chords as consonances.

The supermajor triad has a corresponding supermajor seventh that can be added, with a bright piercing sound. An undertonal extension is also possible, that being the minor seventh. This 9/7/6/5 chord is a second type of dominant seventh chord, with a distinctly different quality, and a useful place on the fifth degree of a minor scale. Similar to the harmonic seventh chord, it's a consonant type of dominant seventh that can find use in rock and some blues styles.

While the neutral triad is more complex and dissonant than the other basic triads, it has a very natural extension, that being the neutral seventh, creating a 18:22:27:33 chord with a distinctly cold sound.

Tetrads that add notes other than extensions, such as the subminor add4 (6:7:8:9) or major add2 (8:9:10:12), add character and a feeling of complexity without increasing the size or harmonic complexity of the chord much if at all.

Otonalities

JI Ratio	JI Cents	31edo Cents
3/2	701.96	696.77
5/4	386.31	387.1
7/4	968.83	967.74
9/8	203.91	193.55
11/8	551.32	541.94

An important class of chords in 31edo is those directly from the harmonic series. The harmonic seventh chord, 4:5:6:7, is a very common example, but the chord can be extended to 4:5:6:7:9:11 while keeping harmonics that fit well together. The 3rd, 9th, and 11th harmonics are all slightly flat, with the 9th and 11th flat by almost exactly the same amount, so the chord comes out sounding very close to its JI counterpart.

Subsets of this chord come up in multiple common scales, with three 4:5:6:9s in Meantone[7], one 4:6:7:11 in Orwell[9], one 4:6:7:9:11 in Squares[11], and two 4:6:9:11s in Neutral[7]. The prescense of these important chords is part of what makes these scales good, and is a key strength of other common scales.

JI Ratio	JI Cents	31edo Cents
13/8	840.53	851.61
17/16	104.96	116.13
19/16	297.51	309.68

While 4:5:6:7:9:11 is the most notable extended otonality, including the essentially pure and slightly flat harmonics, 4:5:7:13:17:19 is similarly available, with the pure and sharp ones. The included harmonics beyond the 4:5:7 harmonic shell voicing are fairly high in the series, so they're less relevant, but subsets of this chord are still important resources in lesser used scales like Miracle[10] and A-Team[13].

JI Ratio	JI Cents	31edo Cents
11/9	347.41	348.39
17/13	464.43	464.52
19/13	656.99	658.06
19/17	192.56	193.55

Interestingly, in the same way that 11/9 is essentially pure due to the 9th and 11th harmonics being flat by the same amount, 17/13, 19/13, and 19/17 are all represented within about a cent of their JI counterparts, with 13, 17, and 19 all about two quarter commas sharp. This contributes to 4:5:7:13:17:19 subset chords being reasonably consonant when voiced well, despite the higher harmonics.

Progressions

Standard chord progressions that work will in 12edo will naturally work well in 31, though there are some interesting movements that work due to specific characteristics in 31edo, which I'll discuss here. There are obviously far more than these two, but these are particularly interesting to me.

The first example is the Spiral Progression.

Major Triad, Spiral Up

A	A	Ab	Bbb
C#	C	Db	Db
E	F	F	Fb

We'll start with any major triad. A similar idea can be achieved with other basic triads, but major is the standard. For this example, I'll choose A major.

Major Triad, Spiral Down

A	G#	Gx	Gx
C#	C#	B#	Bx
E	E#	E#	Dx

You can "spiral" upwards or downwards, with one being the reverse of the other, so we'll choose upwards. We'll take the third of the chord down two dieses, or one chromatone, and the fifth up three dieses, or one diatone. This gives us a chord with notes A, C, and F, for an F major triad.

Next, we do the same thing again, getting notes of F, Ab, and Db, for a Db major triad. If we do it one more time, we get Db, Fb, and Bbb, for a Bbb or At major triad, one diesis above our starting A major triad. This can continue indefinitely, slowly spiralling upwards.

Minor Triad, Spiral Up

A	Ab	Ab	Bbb
C	C	Db	Dbb
E	F	Fb	Fb

This works because in 31edo, three major thirds don't make an octave like in 12edo, instead ending one diesis below an octave, at the suboctave. In 12, this progression cycles every three chords, ending at the same spot, and is a version of the common jazz progression the Coltrane Changes.

Minor Triad, Spiral Down

A	G#	G#	Gx
C	C#	B#	B#
E	E	E#	Dx

The second example is more abstract, and makes use of standard 12edo triads as its basis.

The idea is to choose root notes from far apart in the meantone generator chain. These are notes that'll differ by some unfamiliar gap, generally a neutral interval of some sort. From here, you just choose standard familiar chords on the roots, such as major and minor triads and basic seventh chords.

While the gaps between the roots, and indeed every voice in the chords, are unfamiliar, the chords themselves will sound pleasant to any listener who's used to 12edo. The progression sounds both alien and familiar, achieving a sense of newness without sacrificing any consonance.

You can hear progressions of this type in 24edo here from Hidekazu Wakabayashi, a very talented microtonalist and composer.

Chord Notation

We can start with a note, like F, or a scale step, like I. For our chords, we can name the base triad or seventh chord first, with additions and changes after. For example, F major seven is FM7, and I supermajor minor seven (a 9/7/6/5 chord) is ISm7. s, m, n, M, and S are short for subminor or sin, minor or min, neutral or neu, major or maj, and supermajor or saj. For augmented and diminished chords, we can use + and ^o.

Harmonic and Orwell chords are used enough that they get their own names, such as Ih7 for a 4:5:6:7 on I, and FO for an Orwell Tetrad on F. We can add in extensions at the end, like Fh7t11 for a 4:5:6:7:11 chord.

Personally, I find it useful to employ a few extra symbols, though these aren't standard. Shell voicings don't have a well recognized symbol, as it's often implied in the music, or up to performer discretion. Here, it's a useful symbol to have, so I use a $ sign. Additionally, a symbol for a Joan triad, or 8:11:15, can be handy, so J is used.

List of Chords

This is a panel with a simple list of 31edo chords that I consider to be important. It includes the chord name (if one exists), the overtonal or undertonal JI representation, notes in the chord on the root note C, chord notation, its complexity in certain temperaments, and potentially chord scales that work well over it. It is by no means comprehensive, that wouldn't even make sense.

Triads

Tetrads

Pentads

Hexads+