The Things You Can Steal from Students

As you may know, I love using Sibelius to generate wacky rhythms, but one of my students, Ben Raker, showed me some in a piece of his today (50 minutes long!) that I’d never tried. For some reason I’ve generally shied away from tuplets-within-tuplets, but Ben had come up (accidentally, he admitted, by punching the tuplet button twice instead of once) with a scheme for a quasi-irrational but actually elegantly geometric acceleration and ritard:

Hear the result here.
I quickly realized you could get more gradual patterns with larger numbers:
Hear the result here.
– not to mention more complex quasi-irrational sequences:
Hear the result here. Looks like it’s back to the Disklavier for me. Oh, Henry Cowell, had you only lived to hear this.


  1. says

    It probably makes me some kind of super-dork for knowing this off the top of my head, but nested tuplets are in the Sibelius manual, and are the accepted method for accel/ritard when using feathered beams.

  2. mclaren says

    Ah, you’ve discovered that one. Taking it farther, have you noticed that you can approximate infinite series that approach a limit by changing the numbers in the tuplets?
    For example 11 with 10 atop it then 9 atop it then 8 atop it, etc. This infinite continued series approached phi, 1.6180339. Now, one of the fun things you can do is to look up various continued fraction denominators and figure out that you can get different terminating values depending on the sequence of numbers — you can approach asymptotically any value you want to with the right sequence of tuplets. 13 with 11 atop it with 9 atop that with 7 atop that, and so on down to 1, or 81 with 27 atop that with 9 atop that with 3 atop that, and so on. Unlimited opportunities for rhythmic exploration here.
    Even more fun, but completely unnotatable: different types of broken tuplets in the same sequence. At this point we’ve not only broken completely with Western notation, we’ve shattered the entire concept of meter.
    For example:
    Suppose you have a sequence of durations like Q Q Q 5 5 Q Q 7 7 7 Q Q Q 9 9 Q Q 11 11 11. What meter is that in? (Q = a regular quarter note, 5 = a 5 in the time of 4 quarter note, 80% length, 7 = a 7 in the time of 6 quarter note (6/7 the length of a regular quarternote), 9 = 8 in the time of quarter (8/9 of the length of a regular quarternote), 11 = an 11 in the time of 10 quarter note (10/11 of the length of a regular quarter note).
    Now suppose you overlay that rhythm with an entirely different set of durations made up different broken tuplets — say, Q Q 3 3 Q Q Q 4 4 4 Q Q 6 6 Q Q Q and so on. (Where Q = a regular quarter note, 3 = a triplet quarter note, 4 = a 4 in the time of 3 quarter note, length 75% of a regular quarter note, 6 = a 6 in the time of 5 quarter note, 5/6 of the length of a regular quarternote.)
    The result teases the ear with a non-cyclic but constantly repeating pattern aking to a Penrose tiling sequence, but in rhythm rather than visually.
    The only drawback of such wholly unnotatable rhytms involves the fact that they must be entered by hand, as step sequences in MIDI using clock ticks (480 ticks per quarter note works well).
    When you combine this with varying levels of accelerating acceleration and decelerating deceleration, lotsa fun. (To get an accelrating acceleration speed up the tempo linearly in a steady sequence of notes, then play it back from one MIDI sequencer into another one, then perform the same speedup on the second recorded MIDI sequence. This can be repeated to get third derivative accelerations, fourth derivatives, fifth derivatives, and so on.)
    KG replies: The Q Q 3 3 Q Q 5 5 5 Q Q and so on idea was suggested by Cowell in New Musical Resources, and I used that one in my Folk Dance for Henry Cowell:
    It was easy in the old Encore program, I don’t know a way to do it in Sibelius.

  3. Jeremy Beck says

    This might be a little dim of me, but wouldn’t it be easier to either write everything with feathered beams, or put the irrational-y figure in a box with “accel” and “rit” arrows over the relevant portions? This certainly is the most clever way I’ve seen to actually get feathered beam effects to play back in MIDI, but if anyone put this score in front of me to play I’d claw my eyes out.
    Just saying, it all sounds vaguely cadenza-like to me anyway, so why not notate it that way?
    KG replies: The reason for not using accel. and rit. is so you can have a measured acceleration or deceleration against a steady beat, which was an idea that Cowell proposed and one that Nancarrow loved using. And by doing it this way, you can have more control over several accelerations or decelerations at once. It’s not for live performance, but for MIDI playback. I hate it when performers claw their eyes out, gives me the creeps.

  4. Luk says

    acc/rit against a steady beat is what Chopin is supposed to have used for his rubato playing, right?
    KG replies: Allegedly, but not having a Disklavier, he couldn’t have been able to get it mathematically precise. :^D
    UPDATE: And on further reflection, what attracts me about the first two examples is that they *don’t* sound free and cadenza-like. I can hear that the first and last notes are triplet halves (or the quintuplet quarter), I can hear the steady triplets in the middle, and the progression in the second one is too even to sound improvisatory. I love those examples because they sound like some strict process that I can’t listen quickly enough to catch. Replacing them with performer freedom wouldn’t do it for me. (The third I don’t care for as is – something more interesting could be done there.)

  5. says

    Regarding feathered beams, with Sibelius 6, nested tuplets are no longer required for that.
    However, in older versions Sibelius was wrong to encourage the use of them for accelerando beams because though the playback is “correct,” it resulted in the notes being spaced closer together as they get faster — which violates standard accepted engraving principles for feathered beams (in which the notes should be evenly spaced).

  6. luk says

    regarding your upated comment on my previous post –
    “I love those examples because they sound like some strict process that I can’t listen quickly enough to catch. Replacing them with performer freedom wouldn’t do it for me.”
    Now that is a very interesting comment to me, a performer. I listened to your excerpts and it was precisely the way the relation between upper and lower staff was “executed” too strictly that made me think. I would guess that most performers would “make” the examples sound more fluent than they are written out. You would then have to make sure that they understood this was specifically not desired.
    I’d be interested in knowing what your expriences are in this regard. This type of music might necessitate a performance practice that is not yet part of a tradition.
    KG replies: Well, LOL as they say, I guess my experiences are spending four years analyzing all of Nancarrow’s player piano studies, and then writing ten of my own.

  7. says

    Wow. I was put off notation software years ago because it didn’t want to let me do anything fun. I generally don’t write for human beings so I’ve avoided using notation software anyway, preferring instead other programs which can give you more direct control over your data.
    Now you’re telling me I wasted all those years punching in numbers and writing scripts in a MIDI editor when I could have just been drawing cool shapes in Sibelius?
    KG replies: I’m telling you you’ve wasted all those years your way, when you could have been wasting them *my* way.

  8. Ciprian says

    Here’s a link to an interesting article written by Steve Vai on tuplets (the second half is more extreme):
    P.S. 1. I don’t know if this article would actually interest you (I’m a beginning composer), but I hope I’ll be able to give something back to you as thanks for all the interesting things I’ve learned from your blog. Thanks for writing! Been listening to “Composure” lately, wonderful stuff!
    2. English is not my native language, so excuse me if I sound weird.
    Ciprian (from Romania)
    KG replies: Thanks for the interesting article, Ciprian. Your English is better than that of many American musicians.

  9. says

    Fun post.
    Oddly, the shareware I use, Harmony Assistant, allows for “folk dance style” tuplets but not nested ones.
    That Folk Dance piece was terrific. I have played around the idea of splitting tuplets myself, (e.g. QQ 55555 QQ 555 QQ 55 where 5’s are quint 8ths) but didn’t really get off the drawing board.

  10. says

    3rd notes and 5th notes, as HC correctly noted around 1916 or so, could also be notated specifically by determining different shaped noteheads, an idea he seemingly tossed off at the time, but was in reality radical. Of course, Cowell never made use of these ideas in his own music, but as Kyle points out, other composers since his time have.

  11. Joe Kaplan says

    This is actually exactly how Sibelius Reference suggests you do Feathered Beams. (At least before Sib. 6) Do a bunch of a nested tuplets then hide all the brackets and add in the beams yourself.