Standardizing Ups and Downs Notation

Published 4 Oct 2024
Last Modified 6 Oct 2024

There are different ways to notate microtonal music. For just intonation, there’s Helmholtz-Ellis notation, Ben Johnston notation, and of course, the Functional Just system. There is also Sagittal notation which can notate both equal temperaments and just intonation, and things like Maneri-Sims which were made for a specific tuning system (in this case, 72-tone equal temperament).

But what I’m a fan of is Kite Giedraitis’s ups and downs notation. It’s meant to notate equal divisions of the octave (EDOs), which includes the aforementioned 72-tone equal temperament, or 72edo, as well as rank-2 or “two-dimensional” temperaments such as porcupine and miracle. The thing that makes it so great in my opinion is its simplicity and its compatibility with standard musical notation. Deviations from a chain of fifths are notated with ups for going up in pitch and are notated with downs for going down in pitch. Drops and lifts can be added for even smaller pitch increments in the case of large numbers of notes in an octave. Sharps, flats, naturals, clefs, staff lines, etc. all stay the same.

However, I noticed a problem with this system. For those higher EDOs, there’s no universally agreed-upon number of scale degrees (edosteps) that an up or a down should represent. For example, 270edo is a very accurate (yet not particularly practical) tuning system where, in theory, one could choose how many scale degrees to assign to an up or a down—one person might choose to notate an up as 5 steps, but another may choose 6 steps. Furthermore, one person may be used to seeing it one way and read it incorrectly or more inefficiently when they read the work of another composer who another number of steps to an up or a down, even when it’s specified at the beginning of the sheet music. I’m aware that such a large number of steps in an octave cannot reasonably be played on basically any existing instrument without some form of major modification, but 270edo is just an example, and it can be loosely followed on fretless instruments when using it as a model of just intonation. This issue can also apply to lower EDOs, which are often more physically playable on musical instruments.

So I wanted to come up with a formula to set the number of edosteps in a “standard” up and down in each EDO, and combine this with other rules to form a “standardized” ups and downs notation system. After discussing my ideas with different microtonalists online, I settled on these rules, which I propose to form the universal standard for ups and downs notation:

It might seem rather complicated at first, but I actually designed the rules this way so that the intervals that ups and downs correspond to are defined simply with few exceptions. I chose 66edo as a general cutoff point for drops and lifts since 64edo is the last EDO to map 81/80 to a negative number of edosteps using the closest approximations of 3/2 and 5/4, and it wouldn’t matter for 65edo since it maps 81/80 to 1 edostep in this way anyway.

I wrote a script in C to generate the note names for different EDOs as a demonstration and implementation of this standard. I did add some other stylization that is optional and based on preference, but handy for certain musical elements and scenarios. This includes half-accidentals for when the number of edosteps in the closest approximation of 3/2 is even, and only having 5 note letters/nominals for EDOs 5, 8, 10, 13, 15, 18, 20, 25, and 30, even though the other nominals could technically be added as alternative names for other nominals or out of their usual ascending note order depending on whether the EDO is divisible by 5 or not. The script can be found here. I may update the script with a feature to show note names without half-accidentals, which would be better for the established system of chord names in ups and downs notation.

I hope you found this article interesting, and if you’re a microtonal composer (especially if you work with sheet music and higher EDOs), feel free to share your thoughts and criticism of the rules in this article.