Don't listen to them suckas when they say you too irrational

[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.

Comments

[antimatter spork]

1.

Personally, as a performer, I would much rather see the excerpt notated with nested tuplets than either incomplete tuplets or irrational time signatures, but I will admit that my preference is largely based on my personal familiarity with notational conventions; I think Kyle Gann's solution makes the most sense, but good luck finding performers who would rather see Gann's solution than nested tuplets.

I don't really like incomplete tuplets, they seem like they would be easily misinterpreted.

[DJA]

2.

The problem with nested tuplets in the Trance example, though, is that it's not at all clear that the 4:6 eighth tuplet nested inside of the 6:4 quarter tuplet are actually equivalent to regular (non-tuplet) eighth notes.

Sure, you could work it out, but I don't see how that's actually easier to read than incomplete tuplets. To me, it looks much harder.

Nested tuplets make sense when you're subdividing further -- like, for instance, if you want to split a quarter-note triplet in three. But I don't know that they work so well in this case.

[roger]

3.

I was really thrown by the irrational time signs at first, but once I realized that I know what a triplet sounds/feels like, then it's no problem to string four, or however many, together and then just jump back and forth between them and the eighths.

simpler to do than explain!

[Dan Schmidt]

4.

Man, do they really call these "irrational" time signatures? The mathematician in me is offended - if it's a ratio of integers, it's rational. The only use of mathematically irrational time relationships I'm aware of is Nancarrow.

In any case, my favorite example of non-power-of-2-denominator time signatures is the song "Pretty Noose" by Soundgarden, which has a measure of 2/4+5/12 (they just completely leave out the last note of a sextuplet). It happens around 1:07 (and other places) here: http://youtube.com/watch?v=wt9XlVqKYhA

[matthew wengerd]

5.

As far as notational convention, I agree that the double nesting (for the sake of cancelling the outside tuplet) is confusing at best. The incomplete signatures, once explained make the most sense.

As far as the click, the click that follows the "irrational time signatures" feels much more natural than the 4:4.

As an aside, even though I enjoy your political posts, I'm glad to see a musical post from you and even more glad to see a rather technical one regarding the problems you face. The large ensemble at the University is playing a Brookmeyer tune you did the copy work on and I find that type of work fascinating - It's interesting to see that side of your work.

Did you have any thoughts on my recent post?

[DJA]

6.

Man, do they really call these "irrational" time signatures?

Yes.

The mathematician in me is offended - if it's a ratio of integers, it's rational.

I agree -- it's stupid. We need to come up with a better, non-innumerate name.

[DJA]

7.

Dan's Soundgarden clip.

Great example. Really underrated band.

[DJA]

8.

Did you have any thoughts on my recent post?

Dude, you're asking me? Ask someone who actually makes money on their gigs.

[John Guari]

9.

When I thought about which of those makes my head explode the least, I initially chose the 4/4 example with the split up triplets, just for a first read's sake. After looking and listening, I think maybe the "irrational time signatures" way might be better in the long run to understand where a specific part would interact with the bassline. After some mileage in rehearsals, I probably would prefer the mixed 2/4+2/3.

Finale is a curious beast.

[mclaren]

10.

No, this is not "really very easy." Almighty bog on a minibike, this is worse than partial differential equations.

At least you can do it in Finale. That's nice to know. But reciting the Gettysburg Address in Sanskrit backwards has got to be easier.

The weird & hilarious thing about all this is that if you're working with a MIDI sequencer, these kinds of rhyhms are absolutely effortless -- just shorten the notes you select by whatever percentage you want. If you need a measure of 4/3, select all 4 notes and shorten 'em to 66% of their normal length.

It's only when we try to notate this stuff with conventional Western notation that it all turns into living hell.

[Javier Ruiz]

11.

Hi, Darcy.
For me a tuplet is an operator that changes the note duration. So an standard triplet is simply a 2/3 operator.
That way there is not too much need to think of incomplete tuplets.

Of course that does not make playing them easier.

Back to the Finale list...

[RES]

12.

Another way of notating this is through metric modulations. Split the measures up into separate tempos; beats 3-4 of ms 1 and the 1st triplet of ms 2 would be split out as a separate measure of 4/4 where the tempo was equal to quarter triplets from the preceding tempo. Then you'd switch back to the preceding tempo for the following 4 8ths in a new 3rd measure in 2/4 that is in the original tempo. This is very easy to set up in finale or sibelius; and easy to read. Of course, if other parts are in other tempo layers, as they appear to be in the complete example, then you need to find a simple common notation that works for them all.

[Corey Dargel]

13.

Sorry, Darcy, but I come down on Kyle's side... The "irrational" time signatures make more sense to me as a performer. However, in the age of MIDI, Finale, and Sibelius, I think it would be compulsory for composers who write these kinds of rhythms to provide a MIDI realization for the performers to listen to before beginning rehearsals. It may also be useful to provide each performer will a full score PDF and a MIDI realization of the full score. I am, of course, fully aware that most performers will not bother with such things before the first rehearsal, but in an ideal situation... I also acknowledge that giving performers an easy-looking score is almost a guarantee that they will not go through it beforehand. On the other hand, I think one should always notate as neatly, concisely and unobfuscatedly (is that a word?) as possible. Those are my thoughts.

[andrea]

14.

i had a big argument with an ACA fellow about this whole thing a year ago. i thought there had to be a way to notate this stuff in a more standard meter and that would ultimately be easier to read and perform for the standard classical wanker. she won me over (don't argue about rhythm with recorder players well-versed in really old music, really new music, and Karnatic music): the point was that it's ultimately cleaner and easier to read weird time signatures than a slew of metric modulations or tuplets; the weird time signature preserves the metric phrasing, e.g., going from 4/4 to 4/6 results in a nice tempo shift that could be notated in a myriad of ways, but the 4/6 retains the 4-ness of the music. the argument resulted in this open instrumentation piece from me, which you are free to download and enjoy (or despise?) at your leisure. it's nothing earthshattering; just working through these kinds of ideas musically. i plan to keep messing with these things.

[DJA]

15.

Hey Andrea,

Thanks. Cool piece -- do you have a recording?

BTW, for some reason, my comment answering your question over at Kyle's place ("How do you do multiple simultaneous time signatures in Finale so that the barlines don't line up?") got deleted. If you didn't see it before it vanished, email me for details.

But basically it's the same process I used to create the 3/2 bars in the example above -- use long-ass measures and put the time sigs and barlines in as expressions. If you get the auto-positioning on the expressions right, and include "Note Expressions" in your spacing options, it's not a complete pain the balls.

[Kyle Gann]

16.

Sorry, when Arts Journal shifted my blog to its new location, the most recent comments - those added after the secret move and before publication - got lost. I've added back in as many as I could find, and if I missed one, let me know and I'll retrieve it.

[andrea]

17.

hey darcy, i don't have a recording, yet. and thanks for the finale tips. it does sound like it's just shy of a complete pain in the balls; i may just have to go through with it, though.

[Tim]

18.

Really nice post Darcy. But here's the thing I wonder about - notation isn't really about writing down the easiest way to play something, it's about writing down what comes closest to what the composer wants. And in that respect there's a very distinct difference between the version in 4/4 and the version that switches between 2/4 and 2/3. It's a moot point which is easier to read, but they both affect differently how you read - which I sense in your own comment "for a jazzer such as myself who is used to hearing rhythms in relation to a steady pulse". The 4/4 version carries with it a secondary set of rhythmic implications (to do with pulse, groove, "the one", etc) that are buried by the irrational time signature version (as well as clarifying the cross-cutting of two conflicting beat patterns). The irrational version, however, will suggest a different set of interpretative values to a performer. Personally, I think your nested tuplet version is the most interesting (it looks much more fluid and 'swingy' to me), but Gordon presumably has his reasons for his choice; part of the interpretative process is to consider why he chose that (new) system over the (already available) alternatives.

On the MIDI question, a novice asks: if you have to type in a percentage in order to reduce/increase note length (as per mcclaren's comment above), isn't it impossible to really accurately do tuplets, which are 33.3333... %? (And I suspect it's something like this that is the origin of the "irrational" nomenclature.)

[Pat]

19.

Darcy,

This is interesting, but... I'm wondering if/how this relates to notating in our jazz/big band/what have you world. Clearly some of the rhythmic ideas Maria and her disciples are using now (all the across the barline triplets) could be notated, perhaps more clearly in this paradigm. But even if you get through the notation issues, would it be worth the time and hassle, particularly in already too-short rehearsals, to use it?

[g]

20.

I admit to being bewildered by those time signatures in the 1st example, but, you know, I'm used to 6|8 meaning 2 pairs of triplets and so forth.

But anyway, remember "Live in 2 2/3|4 Time?" Seems to me something like that would be less confusing, say just give the 1st bar 4 and a third beat but keep everything over 4.

[g]

21.

Sorry, "3 2/3|4 Time."

[g]

22.

Sorry, sorry, I guess I meant "4 and two thirds beats," in addition to the Don Ellis title typo. All done now.

[DJA]

23.

Hi Tim,

But here's the thing I wonder about - notation isn't really about writing down the easiest way to play something

But that's a very important practical consideration! As Pat says, rehearsal time is always short and you don't always have the luxury of spending half an hour explaining what your nonstandard time signatures mean.

The 4/4 version carries with it a secondary set of rhythmic implications (to do with pulse, groove, "the one", etc) that are buried by the irrational time signature version (as well as clarifying the cross-cutting of two conflicting beat patterns).

That is true, to an extent, but it's not quite as important as you might think. Most sensitive musicians have gotten pretty good at finding the accents in the music, regardless of where the barlines fall.

I may do a follow-up post about my piece "Ferromagnetic," which started out as alternating bars of 4/4 and 5/4 over a half-time 4/4 rock beat. The horn players hated that (and it was hard to conduct), so I re-notated it as 3 bars of 4/4 plus a bar of 6/4. (Then I got grief from the piano and guitar, since their figures lined up better with the 4/4 + 5/4 pattern!) But it ultimately made virtually no difference to how the piece sounds -- except that the entrances were more solid in the 3 x 4/4+6/4 version because the ink was easier to read.

The "cross-cutting of two conflicting beat patterns" in Trance only really becomes clear once the other instruments come in (IMO). Then you can really hear the steady 4/4 grinding against the incomplete tuplets.

And I suspect it's something like this that is the origin of the "irrational" nomenclature

But 33.33333... isn't irrational. Pi is irrational. Numbers with an infinitely repeating decimal aren't.

[DJA]

24.

Clearly some of the rhythmic ideas Maria and her disciples are using now (all the across the barline triplets) could be notated, perhaps more clearly in this paradigm.

If I ever have time, I may steal that idea for a follow-up post.

[DJA]

25.

g, the "fraction over a fraction" thing is a bit weird, but certainly people have done it that way. They have also put note values in the denominator, i.e., 5/[image of a quarter-note triplet].

[Tim]

26.

That is true, to an extent, but it's not quite as important as you might think. Most sensitive musicians have gotten pretty good at finding the accents in the music, regardless of where the barlines fall.

Fair enough, but I still wonder whether building that sort of double think ("Oh, I see - it's actually in 4/4...") into the score has implications for performance (and rehearsal?). I accept that I'm taking an idealistic position here that's probably unrealistic in terms of rehearsing a band, but I still think it's worth considering the musical as well as practical differences between the two. One then makes a choice based on balancing those competing factors.

But 33.33333... isn't irrational. Pi is irrational. Numbers with an infinitely repeating decimal aren't.

Oh, I realise that - what I meant to suggest is that perhaps somewhere along the line musicians (mistakenly) started equating "infinitely recurring" with "irrational".

[g]

27.

I had just assumed he meant that this approach was "irrational" in the same sense as, say, John McCain's foreign policy.

Ha.

[g]

28.

OK, I regret that comment. I should have just thought it and let passersby wonder why I'm giggling maniacally, like I usually do.

[robert f. jones]

29.

Or you could notate the whole thing in 12/8:

Bear with me here; in the following example:
4 = quarter note
8 = eighth note
8. = dotted eighth note
/ = bar line

8. 8. 8. 8. 4 4 4 /
4 8. 8. 8. 8. 4 4 /
4 8. 8. 8. 8. 4 4 /
4 4 8. 8. 8. 8. 4 /
4 4 4 / (this last bar is in 6/8, of course)

This is at least easy to input into Sibelius or Finale. If you beam together the dotted eighths it also looks rational and easy to read - when you go to try to play it you find out that it isn't easy at all. If I had to play the piece, I think I'd prefer
Gordon's original version. If your only concern is hearing what the rhythm sounds like, I think my 12/8 solution is the easiest to deal with.

[DJA]

30.

Or you could notate the whole thing in 12/8

I already did -- see my followup post.

[Phil Fried]

31.

The bass line mentioned would work very well with irrational measures as it includes only two different durations;eighth notes and quarter note triplets. really-two different tempos So;

2/4, 4/quarter triplets, 2/4, 3/quarter triplets,
2/4, 4/quarter triplets, 2/4, 4/quarter triplets(sorry no good symbol on keyboard)

The bass part will sometimes align with the "uber beat" 4/4, but not always. Thats enough to upset the rhythmic apple cart.

Phil Fried

[DJA]

32.

Phil,

This is, in fact, exactly what Kyle Gann proposed, and exactly what I did in the final example in the original post.

As discussed in my followup post (and elaborated on in the comments), the original notation aligns with the 4/4 grid provided you are subdividing in 3.

[Phil Fried]

33.

oppsey--thats what happens when you don't follow up!
Great go'in Kyle, but in my own defense I might point out that there might be a difference between 2 measures 2/4, 2/triplet quarters-- and single combined measure [2/4 plus 2/3 quarters].
oh forget it!!!

[Philipp Blume]

34.

Thank you for this excellent explanation. I think I'll enter some of my handwritten scores into Finale after all...