User:KingHyperio/Thoughts on Harmony in 31edo

So much of xenharmonic theory is focused on mathematical characteristics of different tuning systems, with few people able to make interesting music using them. I wanted to give my thoughts on how harmony in 31edo works because the way that many people learn how to make music in 12edo is by learning the chords and the functions that we give them, so by expanding a 12 tone model to 31, we can use the same logic to bring functional harmony to 31 and make composition easier.

The model of harmony that I'm using here is based on a dualist 12edo model, essentially treating the root and fifth as the most significant notes in a key, and treating the Im as the inverse of the I chord, flipping notes on the P1-P5 'axis' for a negative version. The rules will be applied differently for scales that don't include fifths, which are more common in 31 than in 12. Examples are Greeley[8] and Oneirotonic, and I'll go into those farther down.

12edo Model

In 12, we can separate notes in a key based on how they contribute to the goal of tension and release. I use the names Stable, Modal, Hollow, Unstable, Leading, and Odd for these, describing the P1 and P5, m3 and M3, M6 and m7, M2 and P4, m6 and M7, and m2 and A4 respectively.

Stable notes are the bread and butter, and are where we try to lead back to with our dominant chords.

Modal notes are also possible spots for resolution, as they determine the quality of the root chord, but the resolution isn't as strong, as seen in IV-I or Isus-I movements.

Hollow notes don't serve much of a role on their own, being a whole step away from a stable note, and generally act as additional color when added to a root chord, without adding much dissonance, but they are each a tritone above a modal note.

Unstable notes add tension and take you away from the root, and are instrumental for subdominant chords, being a half step away from a modal note.

Leading notes are a half step away from a stable note while not relating to the other in a dissonant way, so alongside an unstable note, they create dominant chords, and on their own, they create 'mediant' chords.

Odd notes are similar, but while they're also a half step away from a stable note, they're a tritone from the other, so they don't resolve as cleanly, and mainly serve to add extra tension to a chord.

Tonic chords are those that don't have unstable or leading notes, with odd notes creating a type of 'wolf tonic.' Subdominant chords are those with unstable but not leading notes, dominant chords are those with unstable and leading notes, and mediant chords are those with leading but not unstable notes. Mediant here is a relative misnomer, but it describes the role well, often being a smooth transition to a dominant that has a distinctly different character from a subdominant. This is all a personal view of harmony and is inherently subjective, however it does give any triad or tetrad in 12 a useful role that tends to match up with how we traditionally think about chords.

31edo Model

To expand this system, we can split notes in 31 into the six roles depending on what role they tend to serve, allowing us to create functional chords in any given scale.

Stable: Here we still just have the P1 and P5, as they are still the most important notes in the scale and they guide the harmony.

Modal: m3 and M3 are here, while also adding the subminor and supermajor thirds, s3 and S3. They're used for similarly consonant chords, the 6:7:9 and 9/7/6 otonal and utonal triads that are analogous to the standard 4:5:6 major and 6/5/4 minor, and the unstable notes guide to them similarly.

Hollow: The M6 and m7 are joined by S6 and s7, as they are both useful for root chords 4:5:6:7 and 12/10/8/7, similar to I7 and Im6 (or ImM6 depending on notation). They also don't traditionally guide anywhere except to each other, just like M6 and m7. The neutral sixth ~6 and seventh ~7 are generally set here as well, as the 6:7:9:11, 11:14:18, and 8:10:13 are all common root chords that are considered reasonably consonant in their respectively tonal landscapes, and the notes fit this description more than any other.

Unstable: In addition to M2 and P4, the neutral 2nd and superfourth ~2 and ^4 are added, as well as the S2 and v4. They share a similar role, and are just more or less directional depending on the context. The ~3 is also added, as while it's not a second or fourth, root neutral triads aren't consonant in the same way as we'd expect from a modal note. Instead, they act similarly to a sus chord, with the ~3 resolving by small semitone or diesis to a more consonant third in scales like Squares[8] and Würschmidt[10].

Leading: Here we have the m6 and M7, as well as the s6 and S7, for obvious reasons. The s6 and S7 resolve by the more directional semitone, and are thus useful in harmonic major and minor respectively. There's also a neutral third distance between the two, so in a scale that includes both like Squares[8] or [11], dominant chords tend to make use of both.

Odd: Many notes are in this camp. We have the clear cut s2, m2, A4, and d5, and the ^1, v5, ^5, and v8, which are here mainly because they resolve by too small a distance to be very satisfying on their own, but they add tension and can assist the resolution of some other dominant chord, and they can be interesting tensions when used right, a description similar to the m2 and A4 in 12edo.

These aren't strictly rules, as notes like ~2 and ^4 may be used more as odd notes, and instability can come from dissonant but directional intervals like neutral thirds and sevenths instead of just unstable notes. The 'rules' are a basis to work off of, and can't apply without exceptions, but are useful for understanding and using new harmony.

Example Scales

This model is nothing without chords that it applies to, so some important examples are listed. A more extensive list can be found here.

Orwell[9]: One of the most popular alternative scales in 22 and 31 will obviously be an important one to consider. We see that the principle mode of P1 ~2 s3 S3 ^4 P5 m6 s7 M7 P8 has 2 stable, 2 modal, 1 hollow, 2 leading, and 2 unstable notes, with the ^4 sometimes acting as an odd note, particularly when used as a tension, as it has a character similar to a #11. So, we have tonic chords like Is, IS, and dbIIIS to choose from, reminiscent of a natural minor scale. Two key subdominants are a subminor triad on the s3 and an orwell tetrad on the ~2, including different modal notes each. The main dominant here is the orwell tetrad on ^4, making use of the m6 and M7 leading notes, with unstable notes on either end that lead into your choice of modal note, allowing a very easy change from subminor to supermajor tonality mid composition (aided by the fact that the inverse of the scale is the same but with a S6 instead of s7).

Squares[8]: A much harder scale than Orwell[9], Squares has a similar structure. Using the fifth mode, we have P1 s2 ~3 S3 P5 s6 ~6 S7 P8. Here, we again have two leading notes, an important tool for the scale. The main tonic is the IS, but a IS with a doubly augmented sixth, known as a squares triad as it's just two supermajor thirds stacked, is also an option. Subdominants all utilize the ~3, with supermajor or neutral triads on the ~6 both being important options, and a root neutral triad as a sus chord option. The main dominant here is a 6:7:9:11 or subminor neutral seven on the s2, mimicking a tritone substitution by expanding the outer seventh by half steps to get the octave, with an unstable third and downwards leading fifth. However, here the outer interval is a neutral seventh, and all the resolution is by small, directional semitones, making for a great resolution. Other options exist as well, like a squares triad on the P5 or S7, or a subminor triad on the ~6.

A scale like the Oneirotonic Dylathian, with intervals P1 M2 M3 v4 v5 ~6 S6 S7 P8, doesn't have a perfect fifth, but can use a 8:10:13 root chord to similar effect. Here, we have a superfourth between the seventh and fourth, perfect for dominant chords like an orwell tetrad on the ~6, with a similar idea to a V7. The approximately equal octatonic scale Greeley[8] has a similar problem with no fifth, but it deals with it differently. The mode P1 m2 s3 S3 A4 ^5 M6 ~7 P8 for instance has a 6:7:10:11 root chord, which is its main tonic. However, with no leading notes, its best option is the m2, as without a fifth, the m2 avoids what makes it a bad leading tone in a diatonic context, with no tritone distance from a root tone. A dominant option here is a 6:7:10:11 on the ~7, making up for its lack of unstable notes with an odd ^5 and still relatively unstable ~7.

Consonance and Dissonance

Consonance in 31 largely comes from the harmonic series, but it depends significantly on context, specifically whether the consonances being utilized are mainly of the 5-limit, 7-limit, or 11-limit. A 7/5 tritone will be a dissonant sound in the 5-limit, but in the 7-limit it's a characteristic consonance that shows up in the harmonic seventh chord, among many others. I find it useful here to roughly split the intervals into categories, named as follows: Perfect Consonance, Imperfect Consonance, Exotic Consonance, Ambisonance, Exotic Dissonance, Potent Dissonance, and Sharp Dissonance. The names here aren't important, just serving to categorize interval based on general sound. This is also just how I personally think of the sounds.

Perfect Consonance: These are the basic fundamental consonances, all of the 3-limit, sounding pleasant in essentially any context they're in. These are the P1, P4, P5, and P8.

Imperfect Consonance: Here we have the traditional 5-limit consonances, the m3, M3, m6, and M6. They also tend to be consonant anywhere in isolation, but they're not quite as fundamental.

Exotic Consonance: This is a pretty bad name for the group but it's what I've used for a couple years so it stuck. This includes septimal consonances that also don't have an especially harsh sound in pental harmonic contexts. These are the s3, S3, S6, and s7.

Ambisonance: This is a more subjective category, including intervals that, depending on context, have a very different sound and sometimes function. A characteristic example is the A4, which is a key dissonance in the 5-limit but takes on the role of vd5 or 7/5 in the 7-limit. These are the M2, S2, ^4, A4, s6, and m7. The d5 can also be put here.

Exotic Dissonance: This is a category that includes mainly intervals that are off by a more consonant fundamental interval by a diesis. This gives them a warped, hazy sound. These are the ^1, ^5, and v8. The v5 may also fit this description depending on context.

Potent Dissonance: These are mainly those intervals that have a strong desire to resolve, usually by semitones. This can include tritones and neutral intervals mainly, including ~3, v4, d5, v5, ~6, and ~7, and sometimes ~2, ^4, A4, ^5, and ~9.

Sharp Dissonance: These are more harsh than the other dissonances, with a sound that isn't necessarily very directional, but has a lot of bite to it. These are the s2, m2, ~2, M7, and S7, including minor ninths.