# Notation for Equal Division of an Interval?

In an interview, Jacob Collier talks about microtonal voice leading (dividing a minor 3rd into 4 equal parts):

June Lee notates this with a horizontal bracket and a description. Is there any other way to notate this sort of thing?

Some more specific questions:

• What if I want to continue down using the same microtonal interval after I hit the target note?
• What if I want to skip some notes in the division?
• How would I notate a passage of notes that isn't a scale (for example, starting on Bb, going ~66 cents down from that, then ~33 cents up from that, then landing on A, for a half tone split into 3 equal parts)?
• How would I notate a series of chords on a keyboard-like instrument where only one of the voices is using microtonal voice leading over time?
• How would I notate a chord where the chord members are equal divisions of an interval? For example, a chord constructed of an octave split into 5 equal intervals? What if I want a chord that is an overlay of that chord and another chord using it's own equal division of intervals (for example, a 5-split octave and a 7-split octave)?
• How would I notate a combination of the last 2 bullet points, where some voices are related to the a vertical division of an interval and other voices are related to the horizontal division of an interval, where the 2 intervals aren't necessarily the same?
• What if I want to sub-divide an interval created by dividing an interval?

There is probably a point where an "equal division of an interval" notation should just be put aside and Sagittal notation would be a better choice. And quarter-tones can be used where they fit. But for simple cases, like Collier's demonstration, I think an "equal division of an interval" notation makes more sense.

For clarification, when I use the term "equal division of an interval," I mean cent-wise, not frequency-wise.

This is my first post on this SE site, so feedback would be greatly appreciated.

• Typically it is a good idea to ask one well defined question rather than a whole set that is trying to get at an idea. Also, why does it matter that "equal division" is described in cents rather than frequency? They represent the same thing in different units. – ggcg Mar 25 '20 at 20:46
• @ggcg When I specified cents I was thinking in terms of a difference, not a ratio. It slipped my mind that intervals were defined in terms of ratios for frequencies, so I can edit that part out. As for the sub-questions, should I edit them out or summarize them or something else? – sporklpony Mar 25 '20 at 20:49
• It is not clear to me what the true nature of your question is. Is it about the half step and walking from one note to another in equal 1/2 steps, or 1/4 steps, etc. Or is it about the use of equal temperament in general. Or is it about the difference in step one gets by dividing an interval equally versus using existing intervals, e.g taking what you get from a 24 micro tonal octave versus dividing a 1/2 step into 3 equal pieces, etc. Or something else. What about non equal temperament? One can voice lead by a succession of decreasing changes. – ggcg Mar 25 '20 at 20:57
• @ggcg My core question is how to notate an a/b step, where a and b are arbitrary whole numbers. Actually, that's a slightly different question than what I asked above but is another way of thinking about it, I think? I'm new to microtonality so there is the chance that what I'm asking just doesn't make sense. – sporklpony Mar 25 '20 at 21:02
• Okay. I am sure you'll get some answers. – ggcg Mar 25 '20 at 21:15

## 1 Answer

Terminology-wise, for equally-divided octaves, there's, well... Equally-Divided Octave (wow, you theorists are great at naming things, huh). This generally will be used to describe alternate tuning stsyems like 17-EDO, where the octave is divided into 17 equally (logarithmically) spaced intervals.

Another way that tuning systems will generally handle this is Equal Temperament, which means the interval is divided equally (again, logarithmically). For tuning systems, that interval is nearly always an octave because of octave equivalency.

Now, those terms clearly don't really apply to the melodic uses Collier fancies, where an interval smaller than an octave is divided equally for voice-leading purposes. June Lee is probably one of the world's best Collier transcribers, and I like his system you describe: bracket with performance text (Collier's works usually have plenty of 'bracketeering' with all the hard-to-notate things he's doing). Why? It's really clear what's meant, even if you're not an expert in microtonalism. Notationwise, that's the main thing sheet music should strive to be: easy to interpret.

A specific idea I had: the bracket thing is cool (it reminds me of polyrhythm notation, which isn't too big of a conceptual leap from microtones). One clear way to write the performance text could actually be to use a form of slash notation where the interval to be divided is shorthanded on top, and the partitions into which it shall be split are numbered below, like a fraction (m3/4, for example). This would go above or below the bracket.

• What if I want to skip some notes in the division?

You might be able to get away with using the same notation above and using the positions of the notes on the staff to show that one note has been omitted, but that's likely to be misinterpreted except in the best of cases.

• How would I notate a series of chords on a keyboard-like instrument where only one of the voices is using microtonal voice leading over time?

You're going to want to make sure every voice in those chords is separately written, and make sure that the one microtonal line is isolated enough visually that it's clear that only that line is to be voice-led under the semitone. How you're going to get a keyboard instrument to execute that is another matter entirely...

• What if I want to continue down using the same microtonal interval after I hit the target note?
• How would I notate a passage of notes that isn't a scale (for example, starting on Bb, going ~66 cents down from that, then ~33 cents up from that, then landing on A, for a half tone split into 3 equal parts)?
• How would I notate a chord where the chord members are equal divisions of an interval? For example, a chord constructed of an octave split into 5 equal intervals? What if I want a chord that is an overlay of that chord and another chord using it's own equal division of intervals (for example, a 5-split octave and a 7-split octave)?
• What if I want to sub-divide an interval created by dividing an interval?
• How would I notate some voices being related to the vertical division of an interval and other voices are related to the horizontal division of an interval, where the 2 intervals aren't necessarily the same?

I'm not going to get into any specifics, because that's some seriously crazy stuff, but the general principle to follow is that if the concept you're trying to notate is complicated enough to make it hard to convey in sheet music (and I promise you that all of the above is FAR too complex for reasonable sight-reading), then for the love of god please write it out in words in performance text. Don't make people guess at what your weird notation is supposed to mean - not only will people invariably get it wrong, they'll despise you for it, too.

Be aware that xenharmonic melodies are already pushing the limits of standard musical notation. The staff was not designed to accomodate anything besides 12-TET, and consequently even the simplest of your/Collier's ideas require exceptional notation. Be also aware that this kind of music is complex enough to not draw many human performers; what good is making sure your sheet music is 100% accurate if no one will play it but you? Be also aware that even the most knowledgeable musicians among your audience will have a very hard time distinguishing intervals that are not multiples of a semitone; you could fret for hours about whether to divide that perfect fourth into 13 or 17 notes, but it's very likely that the two will have the exact same effect on almost everyone.

Two more systems that have merit: If (m)any of your notes happen to fall exactly 50 cents above or below a regular note, you can just use the half-flat and half-sharp symbols (and the three-quarters versions) taken from 24-TET accidental notation.

Finally, one of the best methods is just to work out exactly how many cents above or below a 12-TET note you want your melody note to be, and just write + or - however many cents you want above the notes. That's completely intuitive and is the most precise way of doing things, and I think a lot of your more complicated ideas would require this notation to have any chance of human performance. I've seen Lee himself use that system in analyses before, if I remember correctly.

• "The staff was not designed to accomodate anything besides 12-TET": actually, it was originally designed for diatonic music, which is why the development of chromatic music led to the invention of accidentals. It could be extended further, as with the extended accidentals of 24-TET. But pitch notation basically reached its current form a couple of centuries before equal temperament became widespread; classic staff notation actually specifies a twelve-tone system without saying anything about temperament. – phoog Mar 26 '20 at 3:09