Puzzle for the collective

Scott Murphy

Hello, SMTers! Now with June here and (perhaps) more time on your hands, maybe you are in the market for a little music-theory-related fun, in the form of a music-theory puzzle that extends some ideas from Cohn's "Funky Rhythms" article in MTO. 

About a year and a half ago, I posted something on my blog (https://musicellanea.blogspot.com) that ended with a little puzzle: to solve that one, it helped if you knew some music of the composer Robert Schumann. I shamelessly self-promoted the puzzle on smt-talk, and William Ayers was the first to post the right answer.

To solve a similar (and yet dissimilar) sort of puzzle at the end of my most recent post on the same blog, it would help if you know some music of the band Yes.

Enjoy!

Jeremy Day-O'Connell

Long-Distance Runaround: snare drum from c. 0:42 to c. 1:07

Scott Murphy

Hi, Jeremy -- thanks for playing!

Very cool quintuple snare above quadruple time! Yes, to get a cycle of four members, some pulse stream(s) must be in 5 while (an)other pulse stream(s) must be in 4, and that's exactly the case here.

However, for there to be a cycle, the quintuple pattern cannot start on the first big beat; because, if it does, it cannot start again on a big beat some power-of-2 units later, as 5x ≠ 2^y for any integers x and y. That's what I hear happening here: the snare comes right on, instead of a little after, the first big beat at 0:42.

Let's set the unit to a 1/3 of a second -- the pulse rate in the keyboard part -- and call the big beat at 0:42 timepoint 0. If the snare began at timepoint 2 instead of timepoint 0 -- curiously, a conventional backbeat placement -- but then continued its onsets every 5 units as in Yes's original, then there would be a snare hit right at the change back to tonic harmony at timepoint 32 around "sound" of the lyric "Long time waiting to feel the sound." (I encourage you to play this alternate snare part; I find it to be a bit of fun revisionism.) That's the cycle; it would come full circle again at timepoint 512 (although that's a *lot* of pure duple, even for the most duple of pop). Regarding Cohn's "wave of release and relock," this particlar cycle would be lock (2) - release (4) - release (8) - release (16) - relock (32) - etc.

There are four such cycles for the four integers 0, 1, 2, and 3—let's call this number L for "lock." For N as any non-negative integer, a (re)lock occurs at timepoint 2^(L+4N). My revision of Long-Distance Runaround uses L = 1: (re)lock occurs at timepoint 2 (N=0), timepoint 32 (N=1), timepoint 512 (N=2), and so on.

The example of this in Yes than I am thinking of uses L = 3, and it's on an album after Fragile, on which Long-Distance Runaround appears.

Scott Murphy

Here's my answer: http://musicellanea.blogspot.com/2020/02/yess-our-song.html

-Scott