This piece is a follow-up to Escher’s Drum, my earlier exploration of rhythmic tiling canons. I had set out to surpass Escher’s Drum when I began working on Fifteen Beats, but I came away with an appreciation for the special chemistry that had transpired in the earlier piece. Fifteen Beats is more ambitious in some ways but less ambitious in others. It’s more ambitious in that it explores sixteen different rhythmic tiling canons built from rhythmic cycles of fifteen beats. In contrast, Escher’s Drum explored only eight different canons built from shorter twelve-beat cycles. For Escher’s Drum I found the individual canons in a paper by Hall and Klingsberg, whereas in Fifteen Beats I found them in an enumeration by Harald Fripertinger.
In the mathematical literature on tiling canons, rhythms are presented as a series of note onsets. For example, this is the pattern that Fifteen Beats begins with: [000000101010101]. In that representation, 0 indicates a rest and 1 indicates a hit. From a composer’s standpoint, there’s a lot of work to do in turning [000000101010101] into something that sounds good. First of all, you’ve got to decide where the cycle should begin, and usually it will make more sense to the ear if it begins with a hit than with a sequence of rests. So our pattern would best be played as: [101010101000000]. Next you’ve got to decide where the accents go – that takes lots of listening and experimentation – and in the end it’s a matter of taste. I chose to accent the fifth beat: [101010101000000]. And then of course you’ve got to decide what sonorities to use and what tempo to play at. I’ve been working with two drums and a bell. When a rhythmic cycle is played by three voices in a canon, it can come out sounding very different depending on whether the bass drum enters first, then the treble drum, then the bell, or whether the bass drum enters, then the bell, then the treble drum. That’s another choice to be made. And finally of course, you have to decide how to arrange the individual canons in sequence to form a meaningful progression.
That last point is where Fifteen Beats takes a less ambitious approach than Escher’s Drum. In Escher’s Drum, my aim was to create interludes between each of the canons, where the rhythm from the earlier canon would be mixed with the rhythm from the upcoming one. This required testing lots of rhythmic combinations to see which ones sounded the most interesting, and then arranging the canons in an order so that each canon’s rhythm would mix well with the one coming next. And to create smooth transitions I sometimes had to do more than just mix rhythms: in some cases I transformed one rhythm into another by subtracting notes or shifting the accent pattern. In Fifteen Beats, I decided to simply juxtapose the canons, so that each canon would be played for four cycles, and then the next canon would begin immediately, starting with the bass drum announcing the new rhythm without any remnants of the earlier rhythm playing in the other voices. There was still a lot of experimentation to be done in finding a pleasing order for the canons, but there were fewer restrictions to work with since the rhythms from adjacent canons were never directly mixed.
When I finished Escher’s Drum I had the feeling that I’d arrived at something of an optimal solution for the constraints I was working with, whereas in finishing Fifteen Beats I feel this is really just one of many ways the rhythms here could be presented. That said, it’s a way that I enjoy hearing, and I hope you will too.
The visualization was made with MIDITrail by Wada Masashi. ■