The way people learn how to compose often comes from learning a bit of music theory. We learn how different notes and chords interact with each other, in a format taught as functional harmony. Learning how different chords contribute to the overall goal of tension and release allows musicians to gain a deeper understanding of music, and makes composition more accessible to them.
If we want to realize this goal in 31edo as well, we need to develop a working model of functional harmony in this new landscape we have, and make it possible for people to begin making music right away. To do this, I'll be adapting a 12edo model of harmony into one that functions well in 31edo.
In 12edo, I find that chords come down to four main functions. We have tonic chords, which are our home chord to resolve to, or in some cases, a substitute for this home. We have subdominant chords, which are unstable, and point us away from our tonic. We have dominant chords, which are unstable and directional, and lead us back into our tonic. Finally, we have a fourth function, chords that have direction back to our tonic, but without the same instability of a dominant chord. The best example of this is IIIm from the major scale, which is hard to place as a tonic or dominant chord, so I believe it fits better in a separate group. I call this mediant function off of the IIIm, but the name isn't widely used.
We can rigorize these definitions by placing them depending on the function of individual notes. I use six groups of notes for this purpose, which I call Stable, Modal, Hollow, Unstable, Leading, and Odd. In 12edo, stable notes are the root and perfect fifth, the main two points of resolution, and notes which are almost always in our home tonic chord. Modal notes are our minor and major third, deciding the tonality we're in, though not contributing to function on their own. Hollow notes, the major sixth and minor seventh, similarly don't affect function, adding color when on top of a tonic chord, also being a tritone away from one of the modal notes, creating an opportunity for dissonance without affecting function. The unstable notes are the major second and perfect fourth, taking you away from the root, and being right next to the modal notes, being an essential component of a subdominant or dominant chord. Leading notes are the minor sixth and major seventh, a half step away from the stable notes, leading you back to the root. Finally, the odd notes are the minor second and tritone. They're also a half step from a stable note, but a tritone from the other, so the release from them isn't strong enough to be leading intervals. However, they can enhance existing dominant chords.
This is a fairly simplified version of the model, but it allows us to categorize 12edo chords easily. Dominant chords consist of unstable and leading steps, subdominant chords consist of unstable but not leading steps, mediant chords consist of leading but not unstable steps, and tonic chords consist of neither. From here, we can compare chords within the groups based on what roles their other steps have.
To make this system work for 31edo, we need to put all 31 steps into the categories based on how they function in a scale. First off, the stable notes are still just the tonic and perfect fifth. They're the main points of resolution, and as the first four harmonics, they're the most consonant parts of a root triad.
For the modal steps, we still have the minor and major thirds from 12edo, but we can add in the subminor and supermajor thirds as well. These are both tasked with defining the tonality of a scale, and are part of the consonant root triads of 6:7:9 and 9/7/6 respectively.
The hollow steps of M6 and m7 are joined by S6 and s7, as they share much of their function, with 4:5:6:7 and 12/10/8/7 chords similar to the dominant and minor major 6 chords that define hollow steps in 12edo. Additionally, we can add the neutral sixth and seventh, as they're important for root chords such as 11:14:18, 8:10:13, and 6:7:9:11, and also don't point anywhere specific.
The unstable M2 and P4 are joined by the n2, S2, s4, and S4, all within a semitone of one or more modal notes, and all capable of providing instability to a chord. Additionally, we have one more step to add, and that's the n3. The neutral third isn't modal like the other thirds, as its root triad of 18:22:27 is significantly more dissonant and hard to stabilize on. Instead, its distance from the more stable thirds places it in the same group as the seconds and fourths.
For leading steps beyond the minor sixth and major seventh, we just add in the subminor sixth and supermajor seventh. These are the more directional equivalents of the m6 and M7, and are used in generally different contexts, while still providing direction to our stable root and fifth. Additionally, the interval between the s6 and S7 is a dissonant neutral third that expands outward by chromatones to the fourth between the P5 and P8, making them very useful tools for tension and release.
Finally, for odd steps, we have the s2, m2, A4, and d5, all for the same reason as the 12edo m2 and A4. Additionally, we add in the S1, s5, S5, and s8, as they're close to a stable step, but without the ability to drive direction into them, due to the gap being so small. In this way, they also add dissonance and tension to existing chords, but without the ability to provide much function on their own.
It's important to note that the function of some steps may differ depending on context. No model is perfect, and if our ears hear something a certain way, a model should aim to explain it. For example, the superfourth is an unstable step, but when placed atop a harmonic seventh chord as an up eleventh, it functions closer to an odd note, like a sharp eleventh. These functions are a basis to work off of, and are useful for understanding 31edo harmony, but can't apply without exceptions.
The table below includes functions for chords in many common/important scales, free to use and mess around with.
| Tonic | Subdominant | Dominant | Mediant | |
|---|---|---|---|---|
| Major | I, VIm | Isus4, IIm, IV | V, VIIo | IIIm |
| Orwell[9] 1 | Ih11(no3,9)*, Is, IS, dbIIIS | dIIO, dbIIIs | dbIIIO, tIVO, bVIO | VIIO |
| Squares[8] 5 | IS, tIIIS(#5), dVIS(#5) | In**, dVIn, dVIS | dbIIsn7, dIIIS(#5), tIIIs, dbVIS(#5), dVIs, tVIIS(#5) | IS(#5), dbIIn, tIIIs, tIIIn |
| Dylathian | I(v5)***, I(x5), IIIO | Isus2(v5)****, Isusv4(v5), IIsus2(v5), IIsusv4(v5), dVsS6(^5), tVIsS6(^5), tVIsusv4(v5) | IIsS6(^5), dIV(v5), dIVsus2(v5), dIVsusv4(v5), dVIO, tVIIsS6(^5) | tVIsus2(v5) |
| Harrison Major | IS, VIm(^5)***** | Isus4, IIm, IV | IImS6, VS, tVIIs(vb5) | tIIIs |
| Graham Orwell | Ih11(no9), Is, dbIIIS | bIIS(x5), dbIIIs | bIIS, dbIIIO | IO, bIIs, IIIm, IIIS |
| Neutral[7] 1 | Ih11(no3,7) | In, IIn, dIIIh11(no3,7), dIIIn, Vn, dbVIIn | ||
| Mothra[11] 1 | IS9, dVs7(bb5), tVIs7 | IIs7, tIIS(add2), dIVS6, VIs7(bb5), dIs7 | VSs7, tVIIs7(bb5) | tIIIs(add4) |
| Nusecond[8] 8 | IsM6n7$******, Is(vb5), VIm(vb5) | dbIII+(#3), tIIIo | tVm(vb5), dVIIsM6n7$, dVIIs(vb5) | bIIo |
| Mode 8 | Ih7 | IIn(v5), tIVm(v5), Vs(add4) | V, VIIm(v5) | IIIm |
*: The notation here is a bit strange. no3 says that there's no 3 in the chord, not no 3rd harmonic. This only comes up for the 3rd and 5th harmonics; for the rest the harmonic name matches the position in the chord. Also, in many of these extended chords, the super eleven will act as an Odd, not Unstable interval, like a #11.
**: This chord acts sort of like a sus chord, "resolving" to a sub or super chord on the same root.
***: These chords are called Delta Rational chords, and chords of this variety make up a significant portion of Oneirotonic chords. Specifically major v5 chords often have an add2.
****: These sus chords approximate (within about a cent) 19/17/13 utonal and 13:17:19 otonal chords.
*****: This is a "wolf tonic", with all tonic components but a superfifth in a scale that has mostly perfect fifths, so it prevents tonicization of the sixth, and makes it clear that there's still movement to go to get to the real tonic.
******: $ here signifies a shell voicing. no5 can also be used, though in long chord names like this $ is cleaner. Additionally, in this scale, the b2 acts as a leading tone due to the absence of any remote fifth, and the ~6 generally acts as an unstable note.
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 sd5 or 7/5 in the 7-limit. These are the M2, S2, S4, A4, s6, and m7. The d5 can also be put here.
Exotic Dissonance: This is a category that includes mainly intervals that are off of a consonant fundamental interval by a diesis. This gives them a warped, hazy sound. These are the S1, S5, s8, and S8. The s5 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 n3, s4, d5, s5, n6, and n7, and sometimes n2, S4, A4, S5, and n9.
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, n2, M7, and S7, including minor ninths.
You may have heard that, in the major scale, the placement of the tritones and semitones allow for maximal dissonance when resolving to the root. This basically means that the tritone, one of the most dissonant intervals in 12edo, resolves by the most directional interval, the semitone, to the consonant major third, creating a ton of tension for a great resolution.
The major scale still has this property in 31edo, but some other scales do as well. For instance, in Squares[8] mode 5, there are two of these cases. First, there's a neutral seventh between the subminor second and supermajor seventh, resolving by directional chromatones to the octave. Additionally, there's a neutral third between the fifth and octave, resolving outwards in the same way.
These neutral intervals have the ability to sound consonant in some contexts, but in a primarily 7-limit context, they have a dissonance to them that begs to resolve, in the same way that in 12edo, tritones are used to great effect in jazz chords, but the interval on its own has a clear dissonance to it.
The principle Oneirotonic mode is a similar example, with a dVIO-IM(x5) resolution that feels similar to a V7-I. The superfourth between the S7 and s4 resolves by chromatones to the rooted major third, with the n6 staying around for the following chord.
The other scale using orwell tetrads for resolution is Orwell[9], with a tIVO-Is resolution in its principle mode that has a similar sound to a diminished seventh chord resolution. The m6 and M7 resolve outwards to the P5 and P8, with the S4 and n2 able to resolve to the S3 or s3 respectively.