[WARNING: This is a post about irrational time signatures, incomplete tuplets, and the intricacies of music notation software. If your eyes just glazed over, you should probably skip the rest of this.]
Kyle Gann has been busy working out how to notate Henry Cowell-inspired time signatures in Sibelius, like 2/3 and 5/6. Commonly called "irrational time signatures" (even though they aren't, strictly speaking, irrational), these meters have been around for years. People like Ferneyhough use them but they've never quite caught on, even in new music circles.
Everyone's first reaction, upon first encountering a time signature like "2/3" is "What the fuck is a third note? And how the fuck I am supposed to count that?"
"Well," (Cowell's thinking goes), "if a quarter note is 1/4 of a whole note, and a half note is 1/2 of a whole note, then it follows that a third note is 1/3 of a whole note -- in other words, one triplet half note. So, in a time signature of 2/3, a measure is the length of two triplet half notes." Similarly, 5/6 is the length of 5 triplet quarter notes, and so on.
Of course, these time signatures only make sense in the context of a more standard meter. You can't start in 5/6 -- you have to be chugging merrily along in 4/4 or whatever, and then switch gears to go into 5/6 for a measure, and then go back to a "rational" time signature.
As you might imagine, this is really, really hard to play accurately. Take a look at the examples Kyle Gann posted (from his I'itoi Variations):
Once you understand how to count 2/3 and 7/12, etc., the above no longer totally incomprehensible. But with time shifts coming every bar, and no common underlying pulse, it's still bloody daunting -- at least for a jazzer such as myself who is used to hearing rhythms in relation to a steady pulse.
Kyle also posts some excerpts from scores by Bang on a Can composer Michael Gordon's work. As Gann points out, Gordon doesn't use those Cowell-inspired "irrational" time signatures. Instead, he uses incomplete tuplets within conventional meters (like 4/4). This seemed much easier for me to wrap my head around than time signatures like 3/2 and 5/6.
Here's an example of how Gordon uses incomplete tuplets to create the illusion of "irrational" time signatures being superimposed over a regular beat. This is the bassline from Gordon's Trance (originally posted by Gann):
Now that looks like something I can relate to. Kyle disagrees, saying "I can't imagine maintaining a 4/4 beat through the Trance example above." But, at least to my usual way of thinking, the steady underlying pulse is exactly what's needed in such a situation. Even if, in measures 2-4, the only part of the bassline that lines up with the pulse is the downbeat, that still intuitively seems way easier than alternating bars of 2/4 and 2/3 (the notation Kyle prefers).
Anyway, because I am a giant dork, I wanted to see whether I could recreate the Trance example in Finale (my weapon of choice) in such a way that it would play back accurately. No one seemed to know how to do this. I mean, if all you care about is how it looks on the page, notating the figure is trivial -- but where's the fun in that? I wanted to hear it. This would also allow me to test to see whether it was easier to feel in the original "incomplete tuplet" notation, or when renotated into "irrational" time signatures.
It took some doing but I finally figured out how to get this to work in Finale. The first measure is easy, of course, but as soon as you run into that incomplete triplet in m.2 things get tricky. Finale won't let you enter an incomplete triplet. And it also won't let you enter an "X in the space of Y" triplet where "Y" is a quarter note triplet.
I thought about trying to approximate the duration using sixty-fourth notes -- I mean, 11 sixty-fourth notes is almost the same thing as one quarter note triplet, and it would be really easy to create a tuplet that was "1 quater note in the space of 11 sixty-fourth notes" in Finale. But that was kludgy and awkward, not to mention inaccurate -- I figured there must be a better solution.
Here's what I eventually worked out -- for those measures with incomplete tuplets, just make the entire measure a tuplet.
Here, I'll show you what I mean:
First, just enter the notes as non-triplets -- you'll have to turn off "Jump To Next Measure" in Speedy, and Finale will warn you about "too many notes" -- just ignore that for now. When you're done, you'll get this mess:
Next, for measures 2-4, you need to use the Tuplet Tool on the first note of each measure, with these settings: "6 quarters in the space of 4 quarters." This creates a tuplet that spans the entire measure, turning every quarter note in the measure into a quarter note triplet (or, a "sixth note," if you insist). (Epic!) But it also turns all of the eighth notes in the measure into eighth notes triplets. (Brutal!)
But not to worry -- it's actually very easy to turn those triplet eighth notes ("twelfth notes," ugh) back into regular eighth notes, by applying a nested tuplet that lengthens them by the same amount we've already shortened them -- effectively canceling the tuplet.
In this example what we need to do is click on the first eighth note in measures 2, 3, and 4, and apply a "4 eighth notes in the space of 6 eighth notes" tuplet. Here's what we end up with:
I set the tuplet numbers to display ratios and note values for clarity's sake, but you don't have to do that -- in fact, we're going to hide them anyway.
Right-click or control-click on the handle of each tuplet in mm.2-4 and UNcheck "Show." Then create a "3" text expression that matches the tuplet number font (by default, it's Times bold italic 10 pt.), apply as needed, and you are good to go. When you're done, it should look like this:
Looks right, and plays back.
Or, if you want to get really fancy, you can create graphic incomplete triplet brackets instead of all the little 3's. It's a bit tricky (too fussy to explain here) and takes a bit longer, but I actually like this notation better:
So... after all that, how does this sucker sound? And is it even playable by humans?
Here is the Finale playback of the Trance bassline (click to listen/right-click or ctrl click to download) -- I added a quarter note click track to help you orient yourself.
And, as promised, I made a version with a different click track -- one that would correspond to the conducting pattern if the figure was notated in mixed 2/4 and 2/3 à la Kyle Gann. Here is the notation:
And here is the Finale playback with the mixed 2/4+2/3 click track. (click to listen/right-click or ctrl click to download)
Huh. I'll be damned.
I am curious which version everyone else finds easiest to feel.
Finally, for context (because how you feel the bassline obviously depends very much on what everyone else is doing), here's the first two minutes of the Icebreaker recording of Trance:
MP3: "Trance 1" (excerpt). (click to listen/right-click or ctrl click to download) Composed by Michael Gordon, performed by Icebreaker.UPDATE: I have a followup post which talks about using compound meter (i.e. 12/8) as an alternative to both "irrational" time signatures and incomplete tuplets.