When soloing over a specific chord, it's useful to have scales for that chord on hand. In 12edo, the options you have are generally fairly simple. There are the diatonic modes, modes of harmonic and melodic minor, and some rarer scales like diminished and double harmonic. These scales all exist in some form in 31edo, but the abundance of scales gives us new and interesting chord scales to play with.
Major triads and seventh chords are some of the most foundational chords out there, so scales involving them are generally useful. Options include Lydian and Major Pentatonic, as in 12edo, but we have new options in the form of Graham Orwell, which we've seen before, Mode 8, which we discuss later, and a scale known as Iceface.
Iceface is originally defined as the chromatic scale on C with every "black key" or accidental tuned up by a quartertone. The idea is to combine major harmonies with "cold", "shimmery" neutral melodies. This is a 24edo scale, but it can be adapted to 31edo for similar effect. We take a meantone chromatic scale on C, with all flat accidentals. Then, we sharpen each flat note by a diesis, getting a scale of C Dd D Ed E F Gd G Ad A Bd B C.
The scale comes from Hidekazu Wakabayashi, whose music shows off its beautiful melodies, using neutral steps as passing tones between the major scale. By rendering it in 31edo, the underlying major harmonies are warm and consonant, making for a very pretty scale.
For a more traditional approach to a major chord scale, we have the option of YB tunings of dominant scales like Phrygian Dominant or Diminished. These have the advantage of working well over jazz chords such as harmonic seven downflat nine, or sharp nine downflat 13 (P1 M3 s7 s9 and P1 M3 s7 A9/s10 s13).
For minor triads and seventh chords, we similar options in GR scales, but some other interesting options come from the traditional minor pentatonic.
For one, the blues scale still works well here. For the added tritone above the tonic, we have the option of a b5, with a more directional chromatone distance from the fifth, or a #4, with a 7/5 ratio that feels consonant over a 6/5 minor third or 9/5 minor seventh.
We can instead split the minor third gaps into neutratones, adding color to melodic runs over the basic minor pentatonic skeleton. As mentioned previously, this scale was used by the band King Gizzard and the Lizard Wizard, and is a mode of 31edo's rendering of the Arabic Rast.
A similar Arabic Maqam that can be used here is 31edo's rendering of Bayati. It's essentially a minor scale with a neutral second above the tonic instead of a major one. Similarly, 31edo's rendering of Maqam Saba is Bayati with a flat fourth and usually flat octave, with a sound that's described as heartbroken and sorrowful.
For subminor and supermajor chords, we have options in MOSes from Orwell and Squares, as the two temperaments generate these triads quickly, with useful extensions in sixths and sevenths. Squares brings subminor and neutral sixths and neutral and supermajor sevenths, while Orwell brings minor and supermajor sixths and subminor and major sevenths. In addition, we have the harmonic series scales and septimal diatonic variants.
These scales take a segment out of the harmonic series and use it as a scale in its own right. They're named Mode + the harmonic they start on, and span an octave. For instance, Mode 8 is essentially the chord 8:9:10:11:12:13:14:15:16 played as a scale, with steps P1 M2 M3 S4 P5 n6 s7 M7 P8.
This specific scale is great as a chord scale for harmonic subsets like 4:5:6:7, as well as major chords of any sort. The gaps between the steps get smaller farther up the scale, ending with a diatone resolution to the octave.
Scales of this type are useful for all sorts of chords. The undecimal tetrad of 6:7:9:11 can use Mode 6 as a simple chord scale, and Mode 5 works as a sort of modified blues scale, with a minor sixth as opposed to a fourth or fifth.
We can also build inverses of these scales, taking a selection of harmonics and flipping it around, for an undertone series scale. I name these as Mode n-over, and a useful example is Mode 9-over, a good chord scale for the 9/7/6/5 supermajor dominant seven chord.
These are a useful way to build scales in support of large harmonic subset chords, but we also have other scales for these larger chords. While small harmonic subsets are represented in scales like Orwell[9], Meantone[7], and Neutral[7], larger MOSes can fit in chords like 4:5:6:7:9 and 4:6:7:9:11. To understand these, we'll discuss complexity.
A chord's complexity in some temperament is the amount of generators it takes to include it. For instance, the complexity of a perfect fifth in meantone is 1, as it's the generator. The complexity of the major triad is 4, as four perfect fifths stacked include every note in the chord.
In general, the number of a chord you'll find in a MOS scale is the number of notes in the scale minus the complexity. So, in the diatonic scale, we have three major triads, as there are seven notes, and the complexity is four. When a scale has low complexity for some set of chords, those chords will be fundamental in the harmony you can create within that scale.
| Chord | Temperament |
|---|---|
| 4:5:6:7:9 | Meantone: 10 |
| 4:5:6:7:11 | Orwell: 10 |
| 4:5:6:9:11 | Mohajira: 8 |
| 4:5:7:9:11 | Tutone: 9 |
| 4:6:7:9:11 | Squares: 10 |
| 5:6:7:9:11 | Squares: 12 |
There are six 5-note harmonic subset chords in the 11-limit, those being 4:5:6:7:9, 4:5:6:7:11, 4:5:6:9:11, 4:5:7:9:11, 4:6:7:9:11, and 5:6:7:9:11. Each has a temperament that generates it most quickly, meaning that it has a lower complexity in that temperament than any other. These are listed to the right.
The five temperaments listed are five of the most important in 31edo, in part because they have relatively low complexities for these important chords. The MOSes of Meantone[12], Orwell[13], Mohajira[10], Tutone[13], Squares[11], and Squares[14] are all useful scales to utilize these chords, all with smaller MOSes to use for melodies.
I name these chords based on their associated temperaments besides the last, the names being Meantone Harmonic, Orwell Harmonic, Mohajira Harmonic, Tutone Harmonic, Squares Harmonic, and Over-5 Harmonic because squares is such an overachiever.
It's useful to see low-complexity chords in each temperament in order to make music, so I've created a panel below to do just that. The chords used are from this list, which is not comprehensive, so this panel won't be either.
| Slender | Valentine | Miracle |
| Nusecond | Tutone | Mothra |
| Orwell | Myna | Mohajira |
| Würschmidt | Squares | A-Team |
| Meantone | Joan | Tritonic |