Microtonal Accidental Systems

Introduction by Timothy Gondola: Unlocking the World of Microtonal Notation

Microtonality can seem like an intimidating subject, especially if you’re used to the familiar twelve-note scale in Western music. However, at its core, microtonality is about exploring the pitches between the standard notes of the piano. To perform and compose in microtonal systems, musicians need specialized notation to represent these in-between pitches. In this post, Stephen, an expert in microtonality, breaks down key microtonal notation systems, making the complex world of alternative tunings more accessible. We’ll explore equal temperament notations, Just Intonation (JI) systems, and other innovative approaches that composers and theorists have developed over the years. 

Western Microtonal Notation

Western microtonal notation seems like a thorny issue, and it’s certainly been the subject of much debate. But there are solutions for notating each equal temperament (ET) in a readable way, as well as for Just Intonation (JI). Often when people work with microtonality for the first time, they independently arrive at similar notational solutions, though they may swap out certain accidentals for others that serve the same function. In this post, we’ll only cover notations using a standard five-line staff. 

Equal Temperament (ET)

Now, I know what you’re thinking — “Won’t this get exhausting with so many different accidental systems? How many symbols do we even need?” Despite what some might assume, you only need two kinds of microtonal accidentals to notate any ET up to 72 (with the exception of 68 if you want to fully encapsulate 34-ET). These two types are Stein-Zimmerman accidentals (used for “quarter tones”) and arrow accidentals. Using them together is extremely effective, reducing the number of “overall” symbols needed to represent notes in the tuning. 

Stein-Zimmerman Accidentals (Quarter-Tone Notation)

Stein-Zimmerman accidentals are today’s standard for notating quarter tones, allowing a chromatic semitone to be split in half. In a tuning where C to C♯ is evenly divisible, you can insert C half-sharp between them. The classic example is 24-ET (quarter tones), where C to C♯ is 2\24 of an octave. (2\24 of an octave is equal to 1\12 of an octave). These accidentals should be used in 17, 27, 31, and 34-ET, and can also be found in other systems. 

Arrow Accidentals (“Ups and Downs”)

Arrow accidentals, sometimes called “ups and downs,” adjust pitches slightly up or down. Composer Easley Blackwood used circles with arrows through them for these, though that graphic is rarely used today (I have never seen it outside of discussing the context of Blackwood’s own work). Gould arrow accidentals, commonly used for sixth tones or eighth tones, are cleaner and more widely accepted. 

Combining Accidentals

When using Stein-Zimmerman and arrow accidentals together, consider these two best practices:

1. Avoid attaching arrows to Stein-Zimmerman accidentals if possible.

2. Ensure unique spelling for each pitch on a particular letter. For example, you’d want to avoid D half-sharp and D double-up denoting the same pitch class within the same tuning system.

These two rules solve most spelling issues, especially in systems like 17 and 31-ET, which don’t require arrow accidentals at all.

It’s easy to see why a system like 19-ET, with its clear diatonic structure and a few extra quirks, is most easily notated with its own dedicated system. However, more complex systems like 71-ET might be easier to manage using 72-ET notation with some performance instructions, since distinct characteristics of higher-numbered ETs become harder to preserve. Examples of more comprehensive systems include Aaron Hunt’s Tonal Plexus notation (for 205-ET) and Dolores Catherino’s Polychromatic Notation. 

Just Intonation (JI)

JI uses pure frequency ratios rather than equal steps, requiring its own notational systems.

Color Notation

Created by Kite Giedraitis, Color Notation simplifies JI by replacing complex math with color-coded symbols. It’s a brilliantly designed system for naming intervals and chords with consistent, intuitive labels. 

HEJI Notation (Helmholtz-Ellis-Just-Intonation)

HEJI is the contemporary standard for JI notation, championed by Thomas Nicholson and Marc Sabat. It starts with Pythagorean tuning for natural notes (based on pure perfect fifths) and uses accidentals to adjust notes by precise frequency ratios. A syntonic comma, for example, is indicated with an accidental arrow. An in-tune major chord might be notated C - E↓ - G, with the arrow showing that E is tuned slightly lower than in 12-ET. 

Johnston Notation

Ben Johnston’s system uses a five-limit Ptolemaic diatonic scale for its naturals, and the symbols it uses to represent higher limits visually encapsulate the numbers. This makes accidental symbols easier to remember, but also less like actual accidentals. The Kepler Quartet’s performances of Johnston’s string quartets have cemented this system’s artistic legitimacy

HEWM Notation (Helmholtz-Ellis-Wolf-Monzo)

HEWM notation expands on HEJI, using ASCII-compatible accidentals for digital notation. It allows fluid respelling of notes without altering pitch, a flexibility not generally found in HEJI or Johnston notation. 

Sagittal Notation

Sagittal Notation, developed by a team of microtonal theorists, is perhaps the most comprehensive system. It’s designed for ultimate flexibility, allowing notation of any tuning system with varying levels of precision. ASCII compatibility and a symbolic system called “Sagispeak” make Sagittal unique. 

Two key approaches include:

• Mixed Sagittal: Keeps standard accidentals while adding Sagittal symbols.

• Pure Sagittal: Replaces standard accidentals entirely with Sagittal symbols like the “Spartan sharp and flat.”

The system even supports both equal temperament and JI versions of the same progressions, thanks to its logical accidentals. Notable examples of its use include Jacob Barton’s “Sagittal Songbook.” 

Final Thoughts

Hopefully, this post provides helpful exposure to different microtonal notation systems. The most broadly accepted and readable systems today are:

• For Equal Temperaments, use a combination of Stein-Zimmerman accidentals and arrow accidentals.

• For Just Intonation, use HEJI notation for clarity and precision.

With the right notation, even the most intricate microtonal music becomes playable and expressive.