Microtonal Music Composition

Microtonal music is music that uses intervals that are smaller than a traditional half step. In a nutshell, Microtonal music is made with sounds between the notes on a standard keyboard. For my MU2300 final project, I created one original song and made a cover of another song, both of which must not use the standard 12-note tuning method.

Introduction

For starters, traditional Western music breaks down the octave into 12 equally spaced intervals. This tuning method is called twelve-tone equal temperament or 12TET, 12ET, or 12edo (equally divided octave). We call these twelve notes C, C♯/D♭, D, D♯/E♭, E, F, F♯/G♭, G, G♯/A♭, A, A♯/B♭, and B. The interval spanning two notes is called a whole step or a whole tone, and the interval spanning one note is called a half-step or a semitone. The sharp (♯) and flat (♭) accidentals are used to denote differences of a half step. A traditional Western Major scale follows the interval pattern whole-whole-half-whole-whole-whole-half. For example, a C major scale is the pattern C, D, E, F, G, A, B, C.

The semitone can be split into 100 equally spaced steps, which are called cents. Two notes a cent apart have a frequency ratio of 2^(1/1200):1, and there are 1200 cents in an octave. This interval is too small for the human ear to differentiate, but is good for tuning instruments.

22edo

Now, instead of breaking the octave into twelve equally spaced notes per octave, let’s break it into 22 equally spaced notes per octave. Each note is about 54.5 cents apart.

Comparison between 12edo (top) and 22edo (bottom) tuning systems.

Since 22edo doesn't function in similar ways to 12edo, new accidentals have to be used. "Up" and "down" can mean shifting the base note up or down by one step, respectively; "sharp" and "flat" can mean shifting up the base note up or down by one step, respectively. Then, we can combine these to make "sharp up", "sharp down", "flat up", and "flat down", which shift the base note by +4, +2, -2, and -4 steps, respectively.

Symbol	Steps Up/Down
♭↓	-4
♭	-3
♭↑	-2
↓	-1
♮	0
↑	1
♯↓	2
♯	3
♯↑	4

Now, the scale has these 22 notes:

C, D♭/C↑, C♯↓/Db↑, C♯/D↓,

D, E♭/D↑, D♯↓/Eb↑ D♯/E↓,

E, F, G♭/F↑, F♯↓/G♭↑, F♯/G↓,

G, A♭/G↑, G♯↓/A♭↑, G♯/A↓,

A, B♭/A↑, A♯↓/B♭↑, A♯/B↓, and B.

In 22edo, there are two main types of major scales: the Superpyth scale. and the Porcupine scale. The superpyth scale works similar to the standard 12edo scale: seven notes following an interval pattern of long-long-short-long-long-long-short. In the 22edo case, long intervals are 4 steps long, and short intervals are 1 step long. For a standard C scale, this should be C, D, E, F, G, A, B, C. Just like 12edo, the distances between short intervals is only one step; however, unlike 12edo, the long intervals have three notes in between instead of one. The Porcupine scale sounds much different, however: its interval pattern is long-short-short-short-short-short-short, where the long interval is 4 steps long and the short interval is 3 steps long. I personally dislike the porcupine scale, but it's a scale type that does exist.

One major notation difference when composing with these note names is that accidental notes that are equivalent in 12edo are not equal in 22edo. For example, A♯ and B♭ are equivalent in 12edo, but are two steps different in 22edo.

This is an etude I made in A↓ (A-down) minor in MuseScore using the 22edo plugin made by GitHub user eubwah. The PDF file for the score can be downloaded here: File:Etude-in-a-down-minor-score.pdf.

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72edo

Equal divisions of the octave that are divisible by 12 are easier to use because they can be represented by multiple piano tracks. This is what I did with my cover piece of "Toward the Continuum" by Dolores Catherino, who was one of my inspirations for this project. The original version of "Toward the Continuum" is in 106edo; however, I couldn't perfectly fit that into Ableton Live, so I reduced it down to 72edo. The sheet music for Catherino's works are available online - this is what I used to create the cover.

The octave is grouped into 6 sets of 12 scales, each 1/6 of a semitone apart (~16.67 cents) in ascending order: red, orange, yellow, green, blue, violet. The scale starts from C-red, goes across the C's of the other scales, then returns to D-red, and so on.

Using the Simpler tool in Live, I converted an excerpt of me playing piano into a single instrument, used the tuner in Live to tune it to A-432 Hz (as instructed on the sheet music), then adjusted it as closely as possible to the cent difference required for 72edos scales.

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What's the point?

Recall that music is fundamentally based on ratios of the harmonic series (1, 1/2, 1/3, 1/4, etc.) and that an octave has the frequency ratio of 2:1, a perfect fifth has a ratio of 3:2, and a perfect fourth has a ratio of 4:3. This method of tuning an instrument based on fundamental frequencies is called just intonation, or JI.

12edo splits the octave into twelve equal, irrational ratios of 2^(1/12):1. A fifth (C to G) in 12edo has the ratio of 2^(7/12):1; this ratio is about 2 cents flat of the ratio 3:2. The harmonic seventh (C to B♭) has a ratio of 2^(5/6):1 in 12edo, which is 31 cents sharp of its just intonation counterpart, 7:4. More divisions of the octave means that the higher precision for specific intervals. Higher divisions per octave (like 31edo and 53edo) provide much better approximations of harmonic series ratios than 12edo.