Steps in 31edo	Cents
Super Unison	38.71
Augmented Unison	77.42
Minor Second	116.13
Neutral Second	154.84
Major Second	193.55
Supermajor Second	232.26
Subminor Third	270.97
Minor Third	309.68
Neutral Third	348.39
Major Third	387.1
Supermajor Third	425.81
Sub Fourth	464.52
Perfect Fourth	503.23
Super Fourth	541.94
Augmented Fourth	580.65
Diminished Fifth	619.35
Sub Fifth	658.06
Perfect Fifth	696.77
Super Fifth	735.48
Subminor Sixth	774.19
Minor Sixth	812.9
Neutral Sixth	851.61
Major Sixth	890.32
Supermajor Sixth	929.03
Subminor Seventh	967.74
Minor Seventh	1006.45
Neutral Seventh	1045.16
Major Seventh	1083.87
Supermajor Seventh	1122.58
Sub Octave	1161.29
Perfect Octave	1200

Steps in 31edo

With 31 steps to an octave now, how do we make sense of them all? Well, in 12edo, we often learn the smallest building blocks first, those being the semitone and whole tone, or minor second and major second. In 31edo, we have more of these building blocks, the Super Unison, Augmented Unison, Minor Second, Neutral Second, Major Second, and Supermajor Second.

As in 12edo, the Augmented Unison is the Chromatic Semitone, the Minor Second is the Diatonic Semitone, and the Major Second is the Whole Tone. We'll shorten these names to chromatone, diatone, and whole tone for convenience. The Super Unison is often called a diesis, but fifthtone works as well, due to being a fifth of a Whole Tone. Finally, the Neutral Second and Supermajor Second can be shortened to neutratone and supertone respectively. This is a naming scheme from Freddi Sturm and Rami Olsen of Hear Between the Lines, a fantastic Youtube channel and microtonal music duo.

Enharmonic Equivalence

A fact you may notice is that the chromatone and diatone are no longer the same size, though they are in 12edo. In fact, all enharmonic equivalencies from 12edo are different here. C# and Db are now distinct notes, with C# being enharmonic to Ddb, or D down-flat, and Db enharmonic to C#t, or up-sharp. This is a consequence of the diatone being 1.5x the size of the chromatone, so the whole tone is split unevenly in half between the two.

The two semitones can each be used for direction. For example, in the key of F, E would be the standard major seventh, a diatone below the octave. However, we also have the option of Et, the supermajor seventh, which is a chromatone away, as Et and Fb are enharmonic. This leading tone has a distinctly different quality, primarily useful in a diatonic context for minor scales, where there aren't other major intervals to warrant a major seventh. Ed and E# can function similar to leading tones as well, though the direction is noticeably weaker.

Ideal Leading Tones

In George Secor's 17-tone Puzzle, he notes that a maximally directional semitone is in the range of 60-80 cents, peaking at approximately 70 cents with the 17edo semitone.

In my experience, getting significantly smaller makes the semitone far less directional, with dieses and quartertones too small to provide meaningful direction. Getting larger than 70 cents also makes it less directional, but slower, as a minor seventh can still lead into the octave in a meaningful way. A way to quantify this is by taking the step as a fraction of the octave, and giving it a score based on how far its reciprocal is from 17. With a score of 1.5, the 31edo chromatone is very directional, while the diatone has a score closer to 7, finding use where a weaker, more familiar resolution is desired, as it's close to the 12edo semitone's score of 5.

Because the just major seventh of 15/8 is 1088 cents, but a more directional major seventh would have a size of about 1130 cents, tuning systems have to find a way to compromise on these harmonic and melodic ideals. 12edo has a semitone between the two, satisfying each goal decently well. The Carlos Alpha scale has a major seventh of right around the pure 15/8 ratio, instead lowering the octave to allow the leading tone to function in each role well. 31edo just has a different leading tone for each goal, making it versatile without the need to alter other notes in the scale.

List of Steps

The table below is a panel, with the buttons containing information about each of the 31 steps, including important JI ratios it represents, the type of consonance or dissonance it acts as, temperaments and scales it generates, roles it has, its solfege and contracted note name, how it functions in a scale, and its description as an interval independent of one.

0\31	1\31	2\31	3\31
4\31	5\31	6\31	7\31
8\31	9\31	10\31	11\31
12\31	13\31	14\31	15\31
16\31	17\31	18\31	19\31
20\31	21\31	22\31	23\31
24\31	25\31	26\31	27\31
28\31	29\31	30\31	31\31