Microtonal Music Composition

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By Victor Mercola

Microtonal music, by definition, is a type of music that uses intervals between notes smaller than a traditional semitone or half-step. However, it has become an umbrella term for music that doesn't use the standard 12-note tuning system. For my MU2300 final project, I created one original song and made a cover of another song, both of which did not use the standard tuning style.

Introduction

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. However, the JI isn't used in modern music because the ratios between notes are not preserved when transposing sound. It also is mathematically impossible to tune a piano with a single rational interval such that the octave's 2/1 ratio is preserved.

For starters, traditional Western music breaks down the octave into 12 logarithmic equally spaced intervals. This tuning method is called 12 equal divisions of the octave, or 12edo. Two notes that are adjacent on a keyboard have the irrational frequency ratio of 2(1/12)/1. 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, and the natural (♮) cancels out other accidentals in a measure. Double-sharps and double-flats do exist, but are rarely used. A traditional Western diatonic scale consists of five whole steps and two half steps. For example, the standard C Major diatonic scale consists of the notes C, D, E, F, G, A, B, and C in that order. Other scales do exist (pentatonic, octatonic, etc), but the major scale is the most important.

To better describe intervals, the semitone can be split into 100 equally spaced steps, or cents. Two notes a cent apart have a frequency ratio of 2(1/1200)/1, and there are 1200 cents equally distributed in an octave. This interval is too small for the human ear to differentiate, because human ears can only detect about a 5-10 cent difference between two notes. Using cents is useful in determining how close two notes are to each other. Solving the equation below determines how many cents the rational interval a/b spans. For example, The 3/2 interval is an interval about 701.96 cents. 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 22edo, 31edo, and 53edo) can sometimes provide much better approximations of harmonic series ratios than 12edo. Microtonal music does exist historically in non-Western music. For example, some types of Indian music unequally divides the octave into 22 unequal divisions of the octave.

(a/b) = 2(X/1200)

22edo, in MuseScore

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Instead of breaking the octave into twelve equally spaced notes per octave, let’s break it into 22 equally spaced notes per octave. The ratio of two adjacent notes on this keyboard would be 2(1/22)/1, or about 54.55 cents. Since 22edo has almost twice the amount of notes as 12edo, new accidentals have to be introduced to denote the interval changes. One way to do this is to introduce "up" (↑) and "down" (↓) to denote shifting the base note up or down by one step, respectively. Sharps and flats can be repurposed to denote increments of three steps. We can combine these two types of accidentals together to get a range of accidents from -4 steps to +4 steps. Just like double-sharps and double-flats in 12edo, sharp-ups and flat-downs might not be used as much as the other accidentals.

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

The standard 12edo scale has the pattern long-long-short-long-long-long-short, which can be denoted as "5L2s". 22edo has a similar scale with the exact same interval pattern, except long steps span 4 notes, and short steps span 1 note. This scale pattern can be generated with the circle of fifths. Letter names can be given to the notes in this scale, and the other notes' names can be derived with accidentals.

Now, the scale has these 22 notes:

C, D♭/C↑, C♯↓/D♭↑, C♯/D↓,

D, E♭/D↑, D♯↓/E♭↑ 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.

Linear comparison between 12edo (top) and 22edo (bottom) scales, with corresponding note names.

The 22edo diatonic scale isn't the only scale that can be made with these 22 notes, though. Similar to 12edo, the starting point of the scale can be changed to make the Major, Dorian, Lydian, Mixolydian, Minor, and Locrian modes. The 22edo Porcupine[7] scale has one long step and six short steps in the pattern "long-short-short-short-short-short-short", where the long step spans four notes and the short step spans three notes. Porcupine is denoted as "1L6s" - its modes are named "Chinchillian", "Badgerian", "Zebrian", "Dingoian", "Gazellian", "Lemurian", and "Pandian" [citation/fact-check needed].

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 apart 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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Score

72edo, in Ableton Live

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 72edo 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 72edo's six separate scales.

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Final Thoughts

Just like in other forms of art, you have to learn the rules before you can learn how to break them. When I first found microtonal music by accident, I had this weird revelation when I heard a chord change that couldn't exist in standard 12edo tuning. Honestly, it's fun trying to come up with new scales, new tuning methods, and new chords based on how the notes sound good together. The tuning system, scale, and mode all affect how the music sounds, just like how the colors of a piece change how it looks.

Sources

Further Viewing

  • Terpstra Keyboard Web App: An online web-based application that you can make custom hexagonal keyboards with. It is based on the Terpstra Keyboard. Open this link in Chrome - for some reason, it doesn't work in Firefox.
  • The Rational Keyboard by Fritz Obermeyer: An "infinitely" large just intonation keyboard that shifts what keys you can see based on what intervals sound good together.
  • REAPER - I didn't have time to experiment with this DAW, but I read that it has microtonal piano roll capabilities.

Tools used in this project

  • Ableton Live Suite - Proprietary DAW used for 72edo.
  • MuseScore - Open-source sheet music program with plugin capabilities, used for 22edo.
  • Inkscape - Open-source SVG editor, used to make the diagram for this project.
  • 22/31 EDO Retuning Plugins Suite by GitHub user "eubwah". This QML plugin takes a MuseScore file and retunes its notes into 22edo and 31edo based on accidentals.
  • ZynAddSubFX and ZynFusion: Two open source microtonal synth VST/LV2 plugins with microtonal capabilities (up to 128 notes per octave)
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