Music

Escher’s Drum

My aim in this piece was to create a rhythmic tapestry that visits eight different rhythmic tiling canons. A rhythmic tiling canon is a composition in which each player repeats the same pattern, with each player beginning at a different time.  The pattern and entry points are crafted so that once all the players have begun to play, there will be exactly one player striking every beat.  That’s to say, every beat is covered by one of the players but no two players ever coincide on the same beat.

This piece visits all eight possible rhythmic tiling canons where the cycle consists of twelve beats and the entrances of the players are equally spaced. Before writing this piece, I created a visualization of these eight possibilities where they were shown in sequence, with gaps in between them. My goal in the current piece was to visit the same eight possibilities in a seamless way, where there would be no gaps. My idea was that in progressing from a tiling canon based on rhythm A to one based on rhythm B, there should be a connecting passage where rhythm A is mixed with rhythm B. And so the current piece consists of the eight tiling canons (in which all three voices play the same rhythm) together with connecting passages where different voices play different rhythms.

My aim was to arrange the canons so that, if canon B follows canon A, the mixture of B and A sounds interesting. In order to do this, I tested all possible mixtures of the eight rhythms to find the ones I liked best. There were more than (8 choose 2) = 28 pairs to consider, because each rhythm consists of twelve beats (three bars in 4/4 time), and when two rhythms are superimposed there can be a skew of 0, 1, or 2 bars.

At first I tried to structure the piece so that no rhythm would ever be revisited, but I abandoned this goal after finding certain rhythms to be “dead ends,” so to speak. That’s to say, I might progress from rhythm A to rhythm B and, in the transition period, might enjoy the mixture of A and B. But then, while playing rhythm B, I might not be able to find any rhythm C that hadn’t already been played, where I could transition from B to C with a good-sounding B/C mixture. In such cases I had to return from rhythm B to another rhythm previously visited, so that I could then step from that previous rhythm to a new one.

In some cases I tried to make transitions from one rhythm to another smoother by a process of subtraction. If moving from A to B, I might gradually strip notes away from A until it began sounding like it might be a stripped-down version of B. These subtracted rhythms notwithstanding, a notable aspect of this piece is that it is based entirely on eight rhythms — no more than eight. Another notable aspect is that throughout the piece, the three players never coincide on the same beat — neither when they are playing a canon nor when they are playing transition material between canons — that is by design.

My experience in writing this piece was one of constant remembrance: thinking of African rhythms I had heard before. It fascinated me that these eight tiling canons and their combinations can be derived mathematically, but they sound so reminiscent of rhythms I’ve heard musics from the African continent, rhythms that I imagine must have evolved through a process of experiment and enjoyment. Incidentally, though I’ve been seeking out good records of African percussion for much of my life as a record enthusiast, the number of albums I return to is fairly small — I’d list these as the main ones: “Olatunji, Drums of Passion,” “Drummers of Dagbon,” “Drummers of Burundi,” and “Yoruba Drums from Benin, West Africa.” Let me know about any others I should listen to!

3 thoughts on “Escher’s Drum

  1. These rhythmic tiling canons are very interesting, I’m imagining applications where pitch is involved (generating a melody from such “tiles” under transformation).

    More generally I’m really glad I found your work, as a fan of counterpoint it’s been rewarding so far. Once I’ve worked trough all (nearly 100 now!) canons I’m sure I’ll have some questions for you ;).

    You might enjoy https://musescore.com/groups/counterpoint-and-fugue as with all communities it’s mostly beginners but there are some really nice canons to be found:
    By Bernard Greenberg: https://musescore.com/user/1831606/sets/1270096
    I also like writing canons myself: https://musescore.com/boblucassen/inverted-complete-augmention-modulation-canon (but I’ve not been as productive!)

    I don’t dare adding more links for fear of your spam filter. Keep writing and perhaps we’ll talk more.

    1. Thanks for your comment, Bob. I’m glad to hear about the canon and counterpoint community on musescore.com. I will definitely listen to your works there as well as those by Bernard G. Please do send me any comments or questions you have on any pieces of mine that you hear; I’d be happy to discuss and exchange ideas about counterpoint more generally too. FYI, there are extended descriptions of some of my canons online, but they’re currently scattered in different places (some in blog entries, some in liner notes to albums on bandcamp, and some in former Facebook posts that have not yet been migrated to a new home) — best to just ask me for more info if anything piques your interest.

      About rhythmic tiling canons and melody, this is such an interesting area of conjunction. I’ve had the Escher’s Drum piece recorded by a percussionist who happened to be a gamelan expert. I asked him to try performing the piece with gamelan instruments and he made a version where each of the three percussive “voices” in the piece gets assigned to a different pitch. The result really delighted me in that I could hear coherent, seemingly intentional melodic fragments coming out of the piece even though I hadn’t written them in. You can hear what I’m describing in the second track here: https://rudiseitz.bandcamp.com/album/eschers-drum

      A related effort, my Canon 86 is based on a rhythmic cycle that’s almost a rhythmic tiling canon (not exactly one because the cycle has a “extra” shared hit in the middle). The rhythm came first in the piece and the melodic ideas emerged from it. https://rudiseitz.com/2020/05/15/canon-86-tigers-eye/

      One more thing that comes to mind, I have a post about how to create a tiling canon “by hand”: https://rudiseitz.com/about/how-to-build-a-rhythmic-tiling-canon-draft/

      Rudi

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