# Unsquaring the Meter: the first entry in an examination of new theoretical concepts in music

Note: I'm sure this has been written about before, but I haven't seen it.

The vast majority of Western music notation is dominated by meter, the common of these being 4/4: 4 quarter notes, making up a measure. Meter, of course, is simply a notation for how many notes are in a measure, so let's drill down through that to examine notes. Note values are all, at present, based on divisions of the "whole note" (aside from breves which are a multiplication). Common practice note values go from a whole note, to a half note, to a quarter note, to an eighth note, et cetera: in hard numbers this is equal to [1, 1/2, 1/4, 1/8, ...] or can be represented as the range of the function 1/(2^n) where n is any integer. I'll refer to that result set as K.

We have a good deal of freedom in this system as we can adjust the granularity of our meter as we please: technically, any measure of 4/4 can be notated as 1/1 or any equivalent fraction, and this is equally as accepted as a measure of 7/8 or 21/16 or 59/64. We can also as demonstrated use any value in the numerator position. However, I think we still limit ourselves unnecessarily within this system, as if one were doing divisory math with only denominators in K we would immediately run into trouble – we can't divide by anything other than powers of 2!

Now, the common practice solution to this problem for note values is to use tuplets. Using tuplets, we can divide bar further by any value we please: the most common of these is a triplet, a division by 3. I don't want to get too in-depth on tuplets because they're very messy to define – they are by nature a hack, a mod, a workaround for a system which does not allow true division. The structural issue with tuplets is that they only work for note values in a measure and no interface exists for the expansion on their divisory possibilities into meter. The solution to this lies upstream; let's go back to "limiting ourselves unnecessarily".

In math, we can divide any number by any other number; why not in music? For this I propose lifting a simple limitation, that being that "a note value must be in the set K". A note value should be allowed to be any fractional value in the range of the function 1/n. (Sidebar: the 1 is kept because it refers to the whole note. Any math system needs a universal '1'.) I'll elaborate on this now through its relation to tuplets and use that to elaborate on meter as well.

So, tuplets: we'll work through this with an example. The below measure has one note in it.

If we want four notes in that measure, easy: we just use four quarter notes (4 * 1/4 = 1).

Now, what if we want only three notes to take up the duration of that measure? Using tuplets, we'd use a half note triplet with three half notes.

Now here's the important part: what do we call an individual note in that measure? The answer in common practice would be "A half note in a triplet" (more specifically, "in a half note triplet"). I think that the complexity of such an answer is self-proving, but here's a much more complex example.

What's the third note in that measure called? "A half note in an eighth note quintuplet in a quarter note triplet in a half note triplet in a half note triplet". We're not getting paid by the word.

Let's simplify the "half note in a half note triplet". First, let me define a triplet as a 3:2 ratio – 3 notes in the space of 2. This is equivalent to the fraction 2/3. So "a half note in a half note triplet" = 1/2 * 2/3 = 1/3. The second example is 1/2 * 4/5 * 2/3 * 2/3 * 2/3 = 16/135. That half note is equal to sixteen 135th notes, and the former half note is equal to one 3rd note. So a half note triplet = 3rd notes, a quarter note triplet = 6th notes, an eighth note triplet = 12th notes, and so on. Instead of having some unit have a "thirteen-tuplet" or a tredecuplet, it's just 13th notes. It's easy! No latin prefixes!

So since note values aren't limited to K values anymore, neither are meters. Instead of having to notate a measure in 12/8 and include a "dotted quarter note = whatever bpm" marking, just say 4/4 and use 12th notes. 6/8 is 2/4 with 12th notes. This then extends itself to the use of these notes for partial tuplets – below is a measure from my arrangement of Blue Rondo à la Turk where I want to use eleven 12th notes, or in the common practice, three full eight note triplets followed by an eighth note triplet cut short by one note.

Now, this is obviously a hack using the current system – the triplet markings are redundant, I mostly just kept them visible to help the player. In addition, it doesn't function in playback. However, this is almost exactly what I'd want printed on the part. I say "mostly" previously because if I wanted this to be actually correct I'd need a way to notate a 12th note, and this is where I just have to shrug and say "I don't know". The K-based system helpfully only has 4 unique symbols, one of which is extensible:

(taken from John Steffa)

Breve, whole note, half note, and quarter note, with all values smaller than a quarter note being a quarter note divided by how many 'flags' they have – 16th note has two flags, 1/4/2/2 = 1/16. So what symbols do we use for a 21st note, Eden? Well, I have no idea. But it exists even if we have to use weird workarounds to describe it. I'm all written out for the morning so I'll see y'all next time.