Moment of Symmetry Scales

A common method of scale construction is by taking a single interval and stacking it over and over to make a scale, keeping the notes within the octave. When you get a scale with only two different step sizes, it's called a Moment of Symmetry or MOS scale.

The diatonic scale is an example of this. For instance, the major scale has step sizes of Whole Whole Half Whole Whole Whole Half, and is "generated" by the perfect fifth, meaning that you stack the perfect fifth to get all the scale steps.

This works in 31edo as well. For instance, if we take a root note and stack 6 perfect fifths up, as shown in the diagram by the straight lines, we get a perfect fifth, major second, major sixth, major third, major seventh, and augmented fourth. These intervals are 18, 5, 23, 10, 28, 15 steps of 31, which when combined with the root, give us our scale.

This is the Lydian mode of the diatonic scale, the same one that we'll get in 12edo, though with slightly different, and generally more in tune, interval sizes, due to the slightly flatter perfect fifth "generator". We call an interval that we repeatedly stack to get a scale our generator, reaching new intervals in a "generator chain". If we were to instead stack 5 fifths upward and 1 downward, we'd get the major scale.

The set of major modes is collectively known as Meantone[7], as they're the seven note scales generated by the perfect fifth or fourth. Meantone refers to the temperament with the perfect fifth as a generator. In any given temperament, you take some generator and "temper" out the difference between select intervals, declaring them to be the same.

Interval	Temperament	Scale Sizes
1\31	Slender	2, 3, 4, 5, ..., 29, 30
2\31	Valentine	2, 3, 4, 5, ..., 15, 16
3\31	Miracle	2, 3, 4, 5, ..., 10, 11, 21
4\31	Nusecond	2, 3, 4, 5, ..., 8, 15, 23
5\31	Tutone	2, 3, 4, 5, 6, 7, 13, 19, 25
6\31	Slendric/Mothra	2, 3, 4, 5, 6, 11, 16, 21, 26
7\31	Orwell	2, 3, 4, 5, 9, 13, 22
8\31	Myna	2, 3, 4, 7, 11, 15, 19, 23, 27
9\31	Neutral/Mohajira	2, 3, 4, 7, 10, 17, 24
10\31	Würschmidt	2, 3, 4, 7, 10, 13, 16, 19, 22, 25, 28
11\31	Squares	2, 3, 5, 8, 11, 14, 17
12\31	A-Team	2, 3, 5, 8, 13, 18
13\31	Meantone	2, 3, 5, 7, 12, 19
14\31	Joan	2, 3, 5, 7, 9, 11, 20
15\31	Tritonic	2, 3, 5, 7, 9, 11, ..., 25, 27, 29

With Meantone specifically, stacking two fifths gets us our 9/8 major second, but it's also a 10/9 major second, as two of these seconds gives us a 5/4 major third. This is what meantone is, it's the temperament with the perfect fifth or fourth as a generator that "tempers out" 81/80, the difference between 9/8 and 10/9, 27/16 and 5/3, 81/64 and 5/4, etc.

Each interval in 31 is a generator for some associated temperament, with a list on the right of the most common names, and the meanings of the names listed here. These temperaments exist outside of 31edo, but they're easy to examine this way. The important thing to know is that you stack the generating interval up or down until you reach a scale with two step sizes, which we call a MOS, including a set of intervals in each scale that you use to make chords and melodies.

Some of these specific scales are more useful than others. Specifically, familiar examples include Meantone[5], the pentatonic scale, Meantone[7], the diatonic scale, and Meantone[12], the chromatic scale. Other examples include Tutone[6], the whole tone scale (though 31 isn't divisible by 6 so one of the whole tones is slightly larger), which includes a 4:5:7:9 chord on the tonic, and two such chords in Tutone[7], and Slendric[5], close to an equally spaced pentatonic scale.

There are clearly too many of these MOS scales to go into all of them, but there are certain features that distuingish them. For one, the size matters pretty significantly. In 12edo, the pentatonic scale is a great melodic device due to its small number of notes, the diatonic scale is enough to supply interesting chord progressions, and the chromatic scale includes almost every piece of music you've listened to. In 31edo, we can use these size references to find suitable scales to work with.

Secondly, we want scales with good intervals for making chords. A MOS like Meantone[7] is good in part because it has 6 perfect fifths, giving us 6 notes that can all be the roots for simple chords. In the diagram, we have a mode of Squares[8], a scale generated by 11\31, the supermajor third. This scale has 4 perfect fifths, with each one allowing useful triads and tetrads to be built. For instance, the tonic of the scale can be the root of a neutral or supermajor triad, and the fourth note 11\31 can be the root of a subminor, neutral, or supermajor triad.

Unlike in the diatonic scale, each scale step with a perfect fifth can be the root of two or three basic triads, providing a very different harmonic landscape than in major, where each scale step tends to only be the root of one useful triad. Squares[8] includes three subminor triads, three supermajor triads, and four neutral triads, each of which have an unfamiliar sound to a listener used to 12edo.

Finally, we differentiate scales based on how large their large steps are, and how small their small steps are. In the 12edo major scale, the whole steps are twice the size of the half steps, giving us a 2:1 ratio between the large and small step sizes. In 31edo, the major scale has a 5:3 ratio between the two, but there exist numerous scales on very different ends of the spectrum. This ratio between the large and small step sizes is known as the scale's hardness.

Hardness

In 12edo, the diatonic scale is the basis, and its hardness is as simple as it gets, 2:1. In 31, the hardness can vary widely from scale to scale, with Squares[8] specifically having a hardness of 7:2, much larger of a difference than the diatonic scale. This leads to the scale feeling dramatic and intense, with the small semitones in the scale providing useful direction. On the other hand, we have a MOS like Orwell[9].

Orwell[9], as described by Sevish, a microtonal musician and composer, is a bittersweet scale with beautiful melodies. These melodies come in part from the "soft" ratio between the step sizes, 4:3. While the neutratones are larger than the diatones, and it is possible to orient yourself within the scale, the difference between them isn't as clear as in a scale like major. Orwell[9] has two fifths, and like Squares[8], gets much of its harmonic material from subminor and supermajor triads. However, with a drastically different hardness, the scales have distinctly different melodic properties, with the clear direction from the small semitones in Squares[8] not found in Orwell[9].

On the tonic of the principle (first) mode of Orwell[9], we have both of these subminor and supermajor triads, a subminor seventh chord, and a 4:6:7:11 harmonic subset chord. However, without perfect fifths on most steps of the scale, we rely on other types of chords to fill in the harmony. The premier example of this is the Orwell Tetrad, created by stacking three of the subminor third generators on top of the root note. This gives us a chord made up of a root, subminor third, super fourth, and minor sixth, with an unfamiliar but strangely consonant sound, reminiscent of a diminished seventh chord, but without much of the dissonance that one would bring. However, if we want more perfect fifths, we have that option in what's known as a MODMOS.

MODMOSes

A MODMOS is a modified moment of symmetry scale, with a small number of the generators or steps altered slightly by a chroma to give a different final product. A MOS's chroma is the difference between the large and small step, like the chromatic semitone in major. An example is Harmonic Minor in 12edo, where one of the fifths is flattened, and another is sharpened, giving us a major seventh instead of a minor one. In this case, we're sharpening the minor seventh by Meantone[7]'s chroma to a major seventh.

This same idea can be applied to other scales. For instance, we can flatten the second and fourth notes of our Orwell[9] principle mode by a diesis, giving us a scale known as Graham Orwell for microtonalist Graham Breed.

This modified scale has four perfect fifths, on the first, second, third, and fourth steps. It has three supermajor and three subminor triads, one of which extends to a 6:7:9:11, as well as one minor triad and a major triad on the tonic that extends to a 4:5:6:7:11. The melodic properties are quite similar to standard Orwell[9], with the same 4:3 hardness, but the slightly altered harmonies provide a different view to compose with.

An additional type of modified MOS is a MOS Cradle, which takes a MOS, such as Squares[5] mode 5, with a step pattern of 9 9 2 9 2, and splits one of the step sizes into two smaller sizes, well dispersed just like the steps of a MOS. For example, the Xenosquare Cradle splits the 9-edostep into 5 and 4, giving us a scale with steps 5 4 5 4 2 5 4 2, with a 4:6:7:9:11 root chord, and access to the neutral sixth to perfect fifth resolution present in Squares MOSes. The scale's name comes from Squares and the atomic number of Xenon, 54, as well as xen being a common way to refer to microtonal scales.

World Music

Other cultures often have very different musical traditions than our own, and this is no more true than when it comes to tuning. For instance, Arabic music makes use of scales involving neutral intervals, usually close in tuning to 24edo. However, 31edo shares these neutral intervals, and as such, can be used for a similar purpose. There are numerous Arabic scales that can be represented in 31edo, many of which are MODMOSes of Neutral[7].

Rast and Bayati are two of these scales. Rast is similar to the principle mode of Neutral[7], but with the fourth flattened to a standard perfect fourth. One way of using it is by interpreting it as a standard pentatonic scale with the minor thirds split into two neutral seconds, and this is indeed the way that King Gizzard and the Lizard Wizard used the mode starting on the second scale degree in their song Rattlesnake.

The way that scales are constructed in the Arabic musical tradition is different than in the west, with this idea of a MOS absent. Instead, scales (called Maqamat) are traditionally made from segments, generally tetrachords and pentachords, known as Jins. The tuning in the Arabic tradition is notably different than in 31edo, though as approximated in 31edo, Rast is constructed from a Rast and Upper Rast Jins, with Rast being a whole tone, neutratone, neutratone, and whole tone, and Upper Rast being a whole tone followed by two neutratones.

Many of these Jins are well approximated in 31edo, with Maqamat such as Rast, Bayati, and Saba all roughly represented, with diatonic modes and MODMOSes representing Ajam, Nahawand, Hijaz, and Kurd.

The Indonesian scale Slendro is tuned differently depending on location, as instruments are not nearly as standardized as in the west. However, the Slendro scale is roughly similar to 5 equal steps to the octave. The amount by which any given instrument will deviate from this framework differs, though Slendric[5] is a decent approximation. The MOS Slendric[6], while not used for this purpose, adds a diesis that breaks the symmetry and allows a listener to orient themselves, while adding to the hazy sound that Slendric[5] has been associated with.

However, it's important to note that these, and any other generalizations made of world music here, do not capture the full picture. Cultural influence is a powerful thing, and the western manner of learning music does not appropriately capture the regional context and differences in tuning that Indonesian and Arabic music include.

Nonoctave Scales

An interesting characteristic of 31edo is that, due to how small the step size gets, notes next to an octave sound very octave-like, unlike the major seventh and minor ninth of 12edo. These are the suboctave and superoctave, 30\31 and 32\31. While they're far more dissonant harmonically, scales can still use them as a point of resolution if an octave is lacked.

Importantly, 30 and 32 are composite, unlike the prime 31. 31 being prime is useful in many ways, as any interval of 31edo will reach every other interval when used as a generator. However, having these composite near-octaves gives us more tools to work with. 30 is divisible by 5, and there's a class of scales using a fifth of the octave as a period instead of the octave itself. These are called Blackwood, and 31edo can utilize them.

Our main blackwood option is the scale that starts with a neutratone, then stacks a chromatone, and repeats this pattern until we get to the suboctave. This gives us a scale of P1 n2 S2 M3 s4 d5 P5 n6 S6 M7 s8. We see a major triad or major seventh chord on the tonic, then a minor triad or minor seventh chord on the n2, and this pattern also continues and repeats. So, you can start on any note in the scale, and alternate major and minor chords as you fall back to the root. And, by sacrificing the octave, the tuning of the chords is significantly better than in any more traditional blackwood tuning.

A similar option exists by stacking alternating whole tones and dieses, with the scale alternating supermajor and subminor chords instead. Either of these scales only can exist due to 30's five-fold symmetry, and we can get similarly interesting scales using the three-fold symmetry.

In 12edo, the augmented scale alternates stacking minor thirds and half steps to get a six note symmetric scale. In 31edo, we can similarly alternate minor thirds and chromatones, giving us the scale P1 m3 M3 P5 M7 s8.

The other neighbor of the octave has four and eight fold symmetry, with the first giving us an interesting diminished scale option. We alternate chromatones and supertones, giving us the scale P1 s2 m3 M3 d5 P5 s6 m7 S8. We can utilize the eight-fold symmetry for scales involving many neutratones and diminished seventh chords, but there's another use that the superoctave has.

This near-octave can also act as a period. There's a type of scale that is usually generated with a fairly flat perfect fifth, called Mavila, sometimes associated with the Indonesian Pelog. It's known for having swapped major and minor triads in its "diatonic" scale, with interesting melodic properties to boot. However, there are two ways to generate it. Instead of using flat fifths, jeopardizing the tuning of our triads, we can use a sharper octave. Our 32\31 superoctave is perfect for this purpose, so by stacking perfect fifths and resetting at the superoctave, we get the "anti-lydian" scale P1 n2 m3 s4 P5 n6 m7 S8.

It's important to note that, with these "nonoctave" scales, they don't continue above the octave as normal, they use their near-octave as the period of equivalence. So, the second octave of our anti-lydian scale is S8 M2 n3 P4 S5 M6 n7 A8. This ensures that standard triads still work as we'd expect at any point in the scale. The same minor triad that we find on the tonic, we find it on the second and the fifth, and the major triad that we find on the third is on the sixth and the seventh. There's also a corresponding "anti-major" with a superfourth instead of the subfourth we saw. These scales, while similar in harmonic structure to major modes, just without octaves, have very different melodic properties. They're built from neutratones and supertones instead of the more traditional whole tones and diatones, giving us spicier, more alien-like harmonies than we're used to.

We can swap the generator and period of common temperaments to achieve new types of nonoctave scales. For instance, using 31\31 as the generator and 7\31 as the period gives you a variant of Orwell[9] where every step has an orwell tetrad, and every other step has a perfect fifth. This comes out to P1 n2 s3 S3 S4 P5 m6 s7 M7 S8. Doing the same for Squares[8], we get P1 s3 n3 S3 P5 s6 n6 S7 P8 A8.

Key	Eb	Bb	F	C	G	D	A	E	B	F#	C#	G#
Minor Second	s2	s2	s2	s2	s2	m2	m2	m2	m2	m2	m2	m2
Major Second	M2	M2	M2	M2	M2	M2	M2	M2	M2	M2	S2	S2
Minor Third	s3	s3	s3	m3	m3	m3	m3	m3	m3	m3	m3	m3
Major Third	M3	M3	M3	M3	M3	M3	M3	M3	S3	S3	S3	S3
Perfect Fourth	s4	P4	P4	P4	P4	P4	P4	P4	P4	P4	P4	P4
Tritone	A4	A4	A4	A4	A4	A4	d5	d5	d5	d5	d5	d5
Perfect Fifth	P5	P5	P5	P5	P5	P5	P5	P5	P5	P5	P5	S5
Minor Sixth	s6	s6	s6	s6	m6	m6	m6	m6	m6	m6	m6	m6
Major Sixth	M6	M6	M6	M6	M6	M6	M6	M6	M6	S6	S6	S6
Minor Seventh	s7	s7	m7	m7	m7	m7	m7	m7	m7	m7	m7	m7
Major Seventh	M7	M7	M7	M7	M7	M7	M7	S7	S7	S7	S7	S7
Octave	P8	P8	P8	P8	P8	P8	P8	P8	P8	P8	P8	P8

Septimal Meantone

As discussed before, Meantone as a 12-note tuning system hasn't been used recently due to some keys having out-of-tune intervals. However, in 31edo, these other intervals aren't badly-tuned like in standard meantone, but are instead just 7-limit intervals. For example, in 31's Meantone[12], 8 keys have standard, near-pure 5/4 major thirds. However, the 4 other keys don't just have unusable thirds, they're fairly consonant supermajor thirds with a JI ratio of about 9/7. 9 keys have standard 6/5 minor thirds, while 3 have 7/6 subminor thirds.

We pick some note, generally Eb, and keeping going up perfect fifths until we get our twelve-note "gamut", or set of notes. The interval between the last note, in our case G#, and the starting note Eb, is a superfifth, adding variation.

The result is that every key is tuned slightly different, but all are usable. Only one key, G#, doesn't have a perfect fifth on it, but in exchange we get a system where the key you play in matters. Some have subminor and major tonic triads, some have minor and major, and others have minor and supermajor. You can play harmonic seventh chords in some keys, and consonant minor seventh chords in others.

This proves that 31edo doesn't have to be an unwieldy system to utilize. Retuning an instrument to this septimal meantone scale provides generally far better in tune intervals, a wealth of new triads and tetrads to play due to the 7-limit intervals, and variation to unlock new emotional and sonic landscapes by changing the key, all without increasing the number of notes used. This is, in my opinion, just a straight up improvement to 12edo.

We can retune traditional 12edo scales depending on which key of this Meantone[12] scale we choose. For instance, harmonic minor is the standard minor scale with a major seventh, however, we can sharpen the seventh to a supermajor seventh, as we would see in the keys of E, B, or F#.

There are reasons to pair subminor and major or minor and supermajor intervals together in this way. For one, the major seventh already doesn't connect to any other note in the scale by a perfect fourth or fifth, so we can use the more directional option without having to worry about the supermajor seventh ruining the tuning of any chords.

More generally, however, intervals of these types just pair well together. Subminor and major intervals generally have the higher prime harmonic in the numerator of the ratio, with the same holding for minor and supermajor in the denominator. So, a chord like 4:5:6:7, with a major third and subminor seventh, or 12/10/8/7, with a minor third and supermajor sixth, works well.

Additionally, many intervals can be thought of as representing subminor and major JI ratios at once, such as 15\31 being a 45/32 5-limit augmented fourth and 7/5 septimal diminished fifth, or 20\31 being a 14/9 subminor sixth and 25/16 augmented fifth, allowing it to serve as an extension to a 4:5:7 harmonic shell voicing, 5/2 from the major third and 16/9 from the subminor seventh. So, simple chords like 4:5:6:7, 6:7:9:10, 12/10/8/7, and 9/7/6/5 all appreciate these as chord scales.

I call scales like this YB and GR scales for Yellow/Blue and Green/Red respectively, referencing the Kite Giedratis Color Notation names for major, subminor, minor, and supermajor.

This idea is shown to full effect in YB Altered, a retuning of the altered scale from 12edo. The intervals are P1 s2 s3 sd4 (or vd4, for subdiminished or downdiminished fourth) sd5 s6 s7 P8, or P1 A1 A2 M3 A4 A5 A6 P8. Here, a 4:5:7 harmonic shell voicing has access to all the altered tensions you could hope for, just like the traditional altered scale, but they're tuned such a way that they work well over both the major third and subminor seventh. The A4, for instance, is the standard sharp eleventh, which is common on a dominant shell voicing. However, because it's also a 7/5 diminished fifth, it sounds good over the subminor seventh.