Secrets Of Microtonal Notation

Hi all! You might remember my previous post on Timothy’s site about transcribing microtonal music and what it could entail. Here, I’d like to describe some best practices for notating equal temperaments (my favorite kind of tuning). Part of the reason it’s so important to articulate these is because they’re not commonly used, so people invent their own systems and ideas. They will also often independently discover the same solutions (the ones outlined here).

Due to the prevalence of Western notation (with the standard 5-like staff), current best practice is to use that staff, with the same letter system, and then try to notate as closely to 12-TET as possible. Since the entire system is based on the diatonic scale, that means we want to notate the tuning system’s diatonic scale the same way as 12-TET’s, with different proportions for the whole and half step, and to provide extra accidentals if needed. Enharmonic equivalents, thus, will necessarily be completely different in non-12-TET tuning systems; in other words, C-sharp and D-flat will not occupy the same pitch class. In tuning systems where the perfect fifth is sharper than 12-TET’s (such as 17, 22, 27, 29, 32-TET) C-sharp will be higher than D-flat. In tuning systems where the perfect fifth is flatter than 12-TET’s (such as 19, 26, 31, 33-TET) C-sharp will be lower than D-flat.

In many academic environments, the easiest approach to microtonality has been to further divide 12 into smaller pieces. Thus, you get the term “nth tones” depending on how small you go. 24-TET is quarter tones, 36-TET is sixth tones, 48-TET is eighth tones, 60-TET is tenth tones, and 72-TET is twelfth tones. Modern notation for quarter tones has settled on Stein-Zimmerman accidentals, with skilled performers quite familiar with them. Then, any in-between notes can be notated with extra arrows attached to accidentals. For these, I use Gould arrow accidentals, which is also what I use to represent “ups and downs.” I have seen each of the nth tones in use in others’ music except for tenth tones, which is ironic considering how it could unify certain non-diatonic equal tunings that are multiples of 5 (15-TET, 20-TET, and 30-TET).

It is tempting, and naive, to think that the best solution to microtonal notation in different TET’s is to simply use 12-TET with cent number alterations on every single note. This method may be effective for complex tuning systems and/or those that are extremely hard to read, but can lead to significant errors and missed estimation. Understanding a tuning system’s diatonic scale in its own terms instead allows the listener to use the same categories that they have built up over time, but with slight alterations, instead of constantly having to add and subtract different amounts, something we want to avoid having to do when processing music. For example, notating an E major scale in 19-TET and giving the directive “major thirds and fifths are just a little on the flat side” is more effective than messy cent deviation notation.

There are two consistent methods for notating equal temperaments outside of cent deviations - and these are “native fifth notation” and “subset notation.” Native fifth notation is what we are focusing on this post, since it works best with diatonic tunings. Tunings that work best with native fifth notation tend to have fifths that are close to a 3/2 frequency ratio (in other words, as close to the interval of ~701.955… cents as possible). To notate a tuning using its native fifth, we must find its closest approximation to the perfect fifth of 3/2, notate that “C-G,” and then keep constructing a chain of fifths until we have exhausted all the notes. We may also need to add microtonal accidentals to re-spell in-between notes, like we would in 12-TET. 19-TET is a very nice example, containing no new microtonal accidentals - if you look at all of its intervals within the octave, you can see that 11\19 of an octave, or 694.74 cents, is closest (only 7.215 cents away). Microtonalists purposely use a backslash when they are describing intervals of an equal temperament, because forward slashes so often denote frequency ratios, and we wouldn’t want to get the two confused, though it’s often obvious from context which is which.

Equal temperaments will always have a diatonic scale to use if they are 36-TET or higher, a rule that comes from this equation for the major scale, (5 * whole step size) + (2 * half step size) = Equal temperament. The whole step size and half step size in this equation must be integers to give you an equal division of the octave. Easley Blackwood states this rule in his theoretical treatise “The Structure of Recognizable Diatonic Tunings.” For reasons we don’t have time to get into here, the <36-TET tuning systems that are diatonic, and that we should notate using native fifth notation, are: 17, 19, 22, 24, 26, 27, 29, 31, 32, 33, and 34-TET. Other equal temperaments within this range are best notated as subsets of an “nth tone” tuning (12 * an integer), built on 5-TET or 7-TET circles of fifths, or, in the rare case of 11, 13, and 23-TET, best notated as subsets of a different xenharmonic diatonic tuning (22, 26, and 46 respectively).

Here are some ways I’m contributing: 

Firstly, I’m trying to provide more material in the form of transcriptions so that the wider musical public can form solid opinions on readability. In some tougher choices between notation systems, we simply don’t have enough data to make a broad claim about preference for a particular notation scheme. 15-TET is the best example of a difficult choice because there are serious advantages to using both native fifth (Easley Blackwood set a precedent and it mimics syntonic comma behavior) and subset (of 60-TET, or “tenth tones”) notation. The more eyes we have on xenharmonic music the better.

Secondly, I provide the entire chromatic scale in Dorico and via a PDF, with all the enharmonic equivalents you would need. It’s wise to have a chart of the equal tuning in front of you and count intervals along when first starting to get used to one.

Thirdly, as a microtonal performer who is connected with other musicians - I often know what they want to see and read! They do not want to see 16-TET in native fifth notation, for example, which renders sharps going downwards and flats going upwards in an absurd scheme, but simply as a subset of 48-TET. Closeness to 12-TET or diatonic systems will always trump microtonal theory ideas about how things could work. The slight disadvantage of subset notation is that it doesn’t have a system of “naturals” on the white keys of a hypothetical piano - but this is only a small technical issue for the composer, not a real hurdle to performers, as they simply treat microtonal scales as chromatic, and/or learn them regardless of whether they are accidental-laden.

Hopefully this piques your interest! I recommend trying to create a 17 or 19-TET circle of fifths, and then checking my chromatic scale pdf or the xenharmonic wiki to see how you did.