12-Nominal Note Naming System

I’ve developed a (kind of) new microtonal note-naming system that works more or less the same way as Kite Giedraitis’s ups and downs system but with nominals (or letters) O through Z instead of A through G.

The basic problem that ups and downs try to tackle is the fact that when naming the notes of certain equal divisions of the octave (EDOs) higher than 12 (the one that the Western world uses mostly nowadays), sharps and flats tend to cross over. What I mean by this is that when using a chain of fifths to name the notes, the sharp of one letter ends up being higher than the flat of the letter above and vice versa. For example, in the 41edo tuning system, where there are 41 equally-spaced notes per octave, D sharp is higher than E flat. The issue is magnified when things like double or triple sharps/flats become involved, where some multi-flats of higher letters become lower than the natural note of the letter below and some multi-sharps become higher than the natural note of the letter above, depending on how sharp the tuning system’s best approximation of the ratio 3/2 (most accurate perfect fifth) is. An example would be E double flat being lower than D natural. This problem doesn’t exist in 12edo, though, because there are only 12 notes per octave, and they make a complete circle of fifths where the sharp of the letter just below is the same as the flat of the letter just above.

Ups and downs deal with this by turning those would-be crossover notes into an up or a down of the adjacent note. For example, if D flat is lower than C sharp and there’s nothing else in between C and C sharp, D flat could instead be called up-C. The consequence of this is that some fifths no longer “add up” correctly, meaning the nominals don’t follow the fifths pattern of F-C-G-A-E-B, if the up or down note name is used instead of the sharp or flat note name. If the flats cross over the sharps in some EDO system, the ups and downs names would just be whatever the next lower nominal is preceded by “up-”. This means that, for that system, there would be a fifth that goes up-A to F, and A to F isn’t the correct order for a chain of fifths.

What having 12 nominals instead of 7 improves is that this pattern-breaking happens less of the time, and the maximum number of ups and downs to use decreases, making for slightly less complicated or messy note names/notation. This is because the difference between 7 fifths and the closest whole number of octaves in just intonation (called the apotome) is relatively quite large; in fact, it’s more than half of a plain whole tone (9/8). This means that even just a single sharp and single flat will cross over in just intonation Pythagorean tuning (which uses a chain of just 3/2 fifths). The rest of the distance to 9/8 is the difference between only 5 fifths and the closest whole number of octaves (called the Pythagorean limma). Using this instead (thus using only 5 nominals) would allow for single sharps and flats to not cross over in just intonation, but would not for double sharps and flats, which would still cross over between steps of a whole tone between nominals (the other step size would be a minor third with this many nominals). The difference between 12 fifths and the closest whole number of octaves, however, is much smaller. Because the step sizes between the nominals are also smaller, this also means that double sharps and double flats often cross over, but the higher number of nominals also help larger EDOs not contain as many ups and downs, as well as lower degrees of ups and downs (e.g. double-up compared to quadruple-up). So, while the idea of a 5-nominal system is something that I also somewhat want to explore, I went with this system to develop first.

For EDOs with a single chain of closest perfect fifths that eventually leads back to the root note (i.e. EDOs where the number of steps of their best approximation of 3/2 is coprime with the number of note per octave they have), it follows the chain of fifths, moving to sharps, flats, double sharps, double flats, etc. when needed, unless it falls within the range of another nominal, in which case, it becomes the next higher or lower degree of up or down of that note (including for sharp and flat notes).

For EDOs with multiple rings of closest perfect fifths, such as 34 or 72, ups and/or downs are added in between the ones for the corresponding EDO with one ring, with the degrees of the ups and downs adjusted to compensate for ones already present in the single-ring EDO if needed.

For EDOs with a best approximation of 3/2 of 720 cents and sharper, I only used 5 nominals, with the fifth pattern Y-T-O-V-Q, and for EDOs with a best approximation of 3/2 of 4 steps of 7edo and flatter (except for 11edo), I used only 7 nominals, with the fifth pattern R-Y-T-O-V-Q-X. For 11edo, I used all the nominals in order except for U.

For now, I’ve decided on note O being equivalent to A, with concert O5 being the same as concert A4 being 440 hz. I’ve also decided the fifths pattern to be P-W-R-Y-T-O-V-Q-X-S-Z-U with U going forward to P sharp and P going backward to U flat in just intonation. This may cause issues since the center of the 7-nominal fifth pattern is D (equivalent to T in my system), but if others don’t think it’ll be much of a problem, then I’ll keep it this way. A mnemonic I came up with to remember the 12-nominal pattern of fifths is: “Possibly, when raving yellow tigers overtly vocalize, quiet xylophones sing zealous undertones.”

Thanks to Kite (who I mentioned earlier), groundfault, and Stephen Weigel, who helped me develop this system. Below is a PDF of all EDOs from 5 to 72 notated in this “dodecanominal” note naming system, as I’ve been calling it, with enharmonic equivalences.

Download or view PDF

I may continue to update this article (and PDF) with new info, corrections, diagrams/graphics, and so on.