A Flexible Standard

Microtonal music has been notated in a chaotic variety of ways for hundreds of years; and in the last century, notation systems have proliferated even more. In many ways, that’s beautiful. However, it represents a challenge for performers interested in this repertoire. Therefore, Edition Zalzal has chosen a single, flexible standard for notation: Helmholtz-Ellis Extended Just Intonation accidentals, or HEJI. (Here is the most current document summarizing this notation.)


This notation system is arguably the most common microtonal system in use, aside from quarter-tone accidentals (which were mostly standardized in practice between about 1990 and 2010, quite recently!). It was chosen for its clarity and for its compatibility with standard accidentals, including standard quarter-tone accidentals. That is one pillar of Edition Zalzal’s notational approach: compatibility with existing norms. Other systems, like Sagittal accidentals, take a more radical graphical approach; but that is arguably less appropriate for a large-scale documentation project such as this.


Another pillar of our approach is that all information must be available: nothing must be assumed. For that reason, we have provided cent deviations for the pitches in every piece; although for larger equal divisions of the octave (edos), occasionally only a representative subset is given. Even when the music uses only standard accidentals, such as in 19-edo, we explicitly provide their pitch as it is re-defined in that system. In addition, with rare and intentional exceptions, every piece is provided with either tablature or thorough editorial fingerings.


From one piece to the next, a single notational element should represent the same thing. In standard music publishing, this is trivially obvious, but it is important to emphasize in microtonal music. Despite accidentals being slightly redefined between microtonal systems, their function remains consistent, to the extent that this is possible (as detailed below). In addition, we have standardized various composers’ approaches to indicating harmonics, string numbering, coda marking, and so forth. In all cases, we have attempted to defer to normative notational practice for the classical guitar.

However, harmonic notation is a special case. In that case, we have decided to adopt a consistent usage for harmonic symbols based on bowed strings’ “touch harmonic” notation. That is: For open string harmonics, the string is indicated, and an open diamond notehead represents the touch point on the string. For artificial harmonics, the fretted note is shown by a standard notehead, and the touch point above that is shown by an open diamond notehead. In all cases, if the sounding pitch differs from the touch point, the sounding pitch is provided above in parentheses. This can create awkward-looking notation, but it is absolutely unambiguous.

In rare cases, such as contrapuntal passages, black diamond noteheads can be used to prevent rhythmic ambiguity (chiefly between quarter and half notes). In even rarer cases, such as when the touch point is physically very close to the bridge (and thus unable to be represented on the staff), we must fall back on notating the sounding pitch of the harmonic with a small circle above the note. But this latter notation is avoided whenever possible; and where it must be used, explicit text instructions are given for how to locate the touch points.

Adapting HEJI

Taking a page from the Sagittal playbook, our guiding philosophy is that each accidental symbol can undergo slight changes to its pitch definition, as long as its function in a given system is preserved. For example, in HEJI, standard sharps or flats change a pitch by 113.7 cents. However, in 12-edo, they of course change a pitch by 100 cents; and in 19-edo, they change a pitch by only 63.2 cents. The changes are clearly defined in each piece’s preface, and the result of the consistent notation is that triads and other familiar structures retain their simple appearance.

The structure of standard staff notation is deeply, historically rooted in the circle of 5ths. This approach is preserved in Edition Zalzal. The primary method of representing any edo is by taking its closest approximation of the perfect 5th interval, chaining it, and using standard accidentals to represent the resulting pitches. But where earlier theorists had assigned double or triple sharps and flats to the outer reaches of large circles of 5ths, we take a secondary functional approach, and borrow modified accidental symbols from HEJI. (The discussion below is necessarily somewhat technical, and assumes the reader does not need definitions for advanced terms.)

The most common modified accidentals used are quarter tone accidentals and standard accidentals with attached up or down arrows. These are applied to intermediate pitches in edos, i.e. those which cannot be represented by standard sharps or flats alone. In general, arrow-attached accidentals are used to represent smaller intervals (such as one step of a larger edo), and quarter-tone accidentals represent pitches roughly between the “semitones” as defined in each system.

However, some systems with particularly sharp or flat 5ths necessarily break this pattern. For example, in 22-edo, a single step of 54.5 cents is represented by an arrow accidental rather than a quarter-tone accidental, despite the fact that it equally divides a semitone. This choice is due to the fact that alterations by a single step of 22-edo tend to provide close approximations of 5-limit harmonies such as the Just Intonation major 3rd, rather than quarter-tonal or 11-limit harmonies. In HEJI, arrow accidentals represent the 5-limit, and quarter tone accidentals represent the 11-limit; and so 22-edo is written with the accidental that best corresponds to the most common function of a single step, rather than the size of that step. In practical terms, this means that the best approximation of a JI 4:5:6 chord in 22-edo is written C-E(arrow down)-G, rather than C-Ed-G. The second option would be very misleading.

Of course, the functional approach is not perfect. Intervals in every edo approximate JI ratios of a variety of limits, and it would be impossible to notate the function of each individual case. Arrows in 41-edo, for example, typically indicate 5-limit and 7-limit intervals equally often. It would be confusing and arbitrary to mark the 5-limit with an arrow and the 7-limit with a hook symbol (as in HEJI proper), because then two different symbols would be used for the same alteration (1 step of 41-edo). We have endeavored to find the best compromise in every case—or rather, the compromise that best aligns with our priorities.

In some cases, a piece has a recognizable JI “underlying structure,” but was written to be performed in an edo. In these cases, the JI version is notated, because it is more informative about the functions of pitches; and instructions are provided to the player on how to execute it in the edo. This is less complicated than it sounds—typically, the “instructions” are actual tablature.

In some legacy systems such as 24 and 31-edo, there is an established notational practice that is not significantly improved by substituting strictly HEJI accidentals. In these cases, the established practice is left in place. In other cases, such as Turkish music, the established practice deviates confusingly from HEJI, as for example some of the same symbols have very different definitions. In these cases, two versions of the score will be offered: one in the original notation and one in HEJI-based notation.

Finally, there are some pitch systems that do not lend themselves to adaptation to either HEJI or any edo—or else their notation as an edo, following a circle of 5ths rule, would be so misleading as to be useless. (Typically, the reason for this might be a particularly “bad” perfect 5th, which distorts the appearance of intervals further along the chain.) In these cases, such as Thai sueng tuning and Balinese Pelog Selisir, the pitches are notated using their closest approximation in 72-edo. The reason is that this edo builds from the familiar pitches of 12-edo, while maintaining a sufficiently fine-grained resolution to be informative.


Any suggestions for improvements to the notation can be directed to zalzal@untwelve.org. This is a work in progress—just because we have thought about this problem in detail does not mean we are absolutely correct! However, we are likely reasonably close.

Yours in clarity,

Robert Lopez-Hanshaw

Series Editor