Here’s a short piece I’ve just finished writing for guitar; maybe it will become part of a series. The style here is romantic and the texture is more homophonic than contrapuntal, a departure from the keyboard-oriented canons I’ve been working on recently. It feels good to now be writing for the instrument I actually play. This is in fact my first “composed” piece for guitar — my past guitar work has been improvisational and it’s taken me some time to move from a spontaneous to a planned approach to working with the instrument. One thing I’ve learned is that writing for guitar involves a constant interplay between abstract musical thinking and a nut-and-bolts examination of the instrument’s possibilities and tendencies. I suppose that’s true for any instrument, but it’s especially so for the guitar because the guitar is polyphonic, but not in the rational, orderly way the keyboard is polyphonic; instead, in a quirky, limited way, where the fingers of the left hand can quickly become immersed in a nightmarish game of Twister if the composer isn’t careful. But one nice thing about being the author of the piece you’re playing is that if a certain note gives you trouble you have the authority to change it, and I did that several times in the course of practicing this! In the clip, you’ll hear me playing my 2009 Connor guitar. Feedback is welcome.
Since a math olympiad question from Singapore went viral a few days ago, there have been lots of explanations cropping up all over the net. I thought I’d chime in with a restatement of the scenario in which the participants, Albert and Bernard, think out loud as they arrive at the answer. Here goes.
Albert and Bernard ask their friend Cheryl when her birthday is. An onlooker observes the interaction.
Cheryl says, “I’m not going to tell you exactly when my birthday is, but I’ll give you all a clue. It’s one of these ten days:
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
Now let’s see if you can narrow it down.”
Albert and Bernard realize they have no way of deciding between the ten options, so they ask for another clue. Cheryl says, “OK, I’m going to whisper the month in Albert’s ear and then I’m going to whisper the day in Bernard’s ear. I’m going to ask Albert and Bernard to keep what I tell them secret and not share the information with each other, but maybe they’ll still be able to figure it out?” Cheryl goes ahead and whispers the day in Albert’s ear and the month in Bernard’s ear.
Albert says: “I still can’t tell when Cheryl’s birthday is. She did tell me the month, but within that month — in fact, within all the months provided — there’s more than one possible day it could be. The information Cheryl gave me does allow me to draw one conclusion though: I’m absolutely certain that Bernard doesn’t know when the birthday is either!”
Bernard: “So, Albert, you’re using your knowledge of the month to conclude that I couldn’t possibly know the answer? How can you be so sure of what I know and don’t know?”
Albert: “Well, Bernard, you were only told the day, not the month. The only way you could know the answer from the day alone is if Cheryl had told you 18 or 19. If she had told you 18 you’d know it’s June 18 because that’s the only one of the ten possible birthdays where the day is 18. Similarly if she had told you 19 you’d know it’s May 19. What I’m saying is that I know Cheryl didn’t tell you 18 or 19, therefore I know you have no way of determining the birthday. For any other number besides 18 and 19, you still have several options to choose from, with no way of narrowing it down further. For example if Cheryl had told you 14, you wouldn’t be able to decide whether it’s July 14 or August 14.”
Bernard: “But how do you know Cheryl didn’t tell me 18 or 19? Aha, the only way you could know Cheryl didn’t tell me 18 or 19 is if she told you a month where there are no possibilities on those days. So she must have told you a month other than May or June, which means her birthday must be in July or August. And with that smaller set of options, I now know exactly when it is!”
Albert: “Really, you’ve figured it out? Well then, I can gather that the day must not be 14, because if Cheryl had told you 14 you’d still have no way to decide between July 14 and August 14. Bernard, you yourself ruled out May and June based on what I said before, and now I’m ruling out July 14 and August 14, so the only remaining possibilities are July 16, August 15, or August 17. And considering those three options together with what Cheryl told me, I know the answer too!”
Onlooker: “Albert and Bernard, I’ve been listening to you talk through this whole thing, and now I also know the answer. If Albert narrowed the possibilities down to July 16, August 16, or August 17 and then declared that he knew the answer, I know Cheryl must have told him the month July. If Cheryl had told him August, then Albert would still not be able to decide between August 16 and August 17. So, now we all know Cheryl’s birthday is July 16!”
Today is not April Fool’s Day. I can prove this in several ways:
Proof #1: Was yesterday April Fools’ Day? No. Was the day before yesterday April Fools’ Day? No. Hence, by induction, we may conclude that today is not April Fools’ Day.
Proof #2: April Fools’ Day only happens once a year. So, the probability that today is April Fools’ day is 1/365. That’s almost 0.
Proof #3: If today were April Fools’ day, there would be unanimous consensus about the fact, but there isn’t, because I disagree.
The 33rd piece in my canon album is an invertible crab canon. If we use a sequence of numbers to represent the progression of a musical phrase, one line in the canon would be of a form like this:
1 2 3 4 5 5 4 3 2 1
Having reached its midpoint after 1 2 3 4 5, the line reverses and plays its own first half backwards, 5 4 3 2 1. At the same time, the accompanying line plays the backwards or retrograde version followed by the forward version:
5 4 3 2 1 1 2 3 4 5.
So at any given time, you’ll be hearing 1 2 3 4 5 played alongside 5 4 3 2 1, with those fragments exchanging position — moving between the top and bottom lines — midway through the piece. (There’s no significance in the way this example goes to 5 and back; any number could have been used to make the point.)
The structure could be visualized like this:
I first learned about crab canons as a teenager reading Douglas Hofstadter’s Gödel, Escher, Bach. As much as I loved Hofstadter’s treatment of this and so many other subjects, I maintained a sense that crab canons are rather artificial as musical structures. I’m interested in writing canons because I enjoy listening to them, and part of that enjoyment depends on an ability to follow a canon’s structure as it is heard. But crab canons have always seemed to me a bit beyond the capacity of a mortal ear to follow. The idea of playing a musical line in reverse is simple to describe, but not a very natural thing to do. It can be hard for the ear to discern the relationship between a line and its retrograde, and if the relationship isn’t apparent to the ear, what’s the point of basing a piece on it? Now having worked on some crab canons of my own I can respond to that question by saying, first, that the relationship may not be blatant but it may still enter into perception in subtle ways, and second that it does become more apparent when you listen to a crab canon many times and begin to memorize the lines. The highly restrictive nature of crab canons is also valuable in that it demands, and hence inspires, creativity.
Canon 33 is the second crab canon I’ve written. The first one, Canon 31, is a non-strict kind of crab canon where the durations of some notes are allowed to vary between the forwards and backwards versions of the theme. In contrast, Canon 33, is strict crab canon where the note durations are identical in both directions. Another difference between Canon 31 and Canon 33 is that the first is based on the whole-tone scale and has somewhat unconventional harmonic design, while Canon 33 sticks to fairly straightforward tonality. There’s one structural liberty taken in Canon 33: there’s a certain note instance that’s played sharp when the line is moving in one direction but natural when the line is moving in the other direction. Here’s a visualization of the entire Canon 33 in piano roll notation:
In the clip below, you can hear each line of the canon played separately. First you’ll hear the bottom line; then, after a pause, you’ll hear the top:
Now listen to both lines played together in the official track on Bandcamp:
And here’s a video of it:
This piece is a follow-up to Bam!, my earlier exploration of rhythmic tiling canons. I had set out to surpass Bam! 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, Bam! explored only eight different canons built from shorter twelve-beat cycles. For Bam! 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: . 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  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: . 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: . 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 Bam!. In Bam!, 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 Bam! 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.
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!
This video provides a visualization of the some of the rhythmic tiling canons described in the paper “Asymmetric Rhythms and Tiling Canons” by Rachel W. Hall and Paul Klingsberg. A rhythmic tiling canon is a composition in which each player repeats the same rhythmic pattern, with each player beginning at a different time. The rhythmic 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. The video illustrates all possible tiling canons where the rhythmic cycle consists of twelve beats and the entrances of the players are equally spaced.
It fascinates me that these rhythms can be derived through a purely mathematical process and yet they sound so good — some of them remind me of various rhythms I’ve heard in musics from the African continent but I’m not well-versed enough to be able to pinpoint a specific region or style. I should add that the rhythms don’t sound good in all contexts: they seem to work best when played on three percussion instruments with distinct timbres. If each voice were played using C in a different octave on a piano, for example, the distinction between voices becomes less clear as the ear tends to hear melodic patterns formed between voices. If the universe is willing, I’ll be posting a follow-up with some original compositions built using these canons as a rhythmic foundation.