Tracy Numbers

Here’s a story about African rhythms and cancer and combinatorics. It starts a few years ago when I was taking a class in Afro-Cuban rhythms from Russell Shumsky, with whom I’ve studied West-African drumming for many years. Among the basics of Afro-Cuban are the Bell Patterns, which come straight out of Africa. The most basic is the “Standard Pattern”, commonly accompanying 12/8music. “12/8” means there are four clusters of three notes and you can count it “one-two-three two-two-three three-two-three four-two-three”. It feels like it’s in four, particularly when played fast.

Here’s the standard bell pattern in music notation. Instead of one 12/8 bar, I’ve broken it into four 3/8 chunks. Let’s call those “mini-measures”; I’ll use that or just “minis” in the rest of this piece.

Bell patterns are never played in isolation, but circularly on fast repeat, so the first note immediately follows the last.

In the sound sample, I’m playing a background beat on a conga, emphasizing the beginning of the 12/8 measures. The actual bell pattern is on the high “child” bell of a Gankoqui, an African dual-cowbell set.

That’s my Gankoqui. I bought it off someone on Etsy who imports them from Ghana. It came with that little thin stick that sounds nice, but sometimes I use a regular drumstick when things get loud.

The problem · Russell’s a good teacher and the standard pattern isn’t that tricky, but I just couldn’t get a grip on it. It’s a little harder than it looks what with cycling it really fast, and then you’re playing it against complicated music with other instrumental voices. I probably would have got there, but the lessons ran out of gas in the depths of Covid.

Introducing Tracy · She was Russell’s long-time partner, a good person and good drummer too. When you were struggling with a complex rhythm it was helpful to watch Tracy’s hands, because she was always on the beat.

Tracy lived with stage four metastatic cancer for many years and braved endless awful rounds of therapy while remaining generally cheerful. She could be morbidly funny; I bought her congas (you can hear one behind the beat in the samples) when she had a storage-space problem. She told me she was carefully planning her finances so she’d run out of money just before the cancer got her.

I always enjoyed any time I spent with her. Then, a dozen years into her cancer journey, this last summer it got into her brain and it was pretty clear her end times were upon her.

The hospice · Tracy’s last months were spent at St. John Hospice in Vancouver’s far west. I can’t say enough good things about it. If you’re near Vancouver and your death becomes imminent, try to be there if you can’t be at home. It’s comfortable and the staff are expert and infinitely kind. The rules that apply at hospices are different from those at hospitals; for example, Tracy’s cat joined her in residency and had the run of the place.

I (and other fans of Russell and Tracy) visited the hospice a few times. My last visit was just days before her death and, while she was fatigued and spaced-out, it was still Tracy. I wasn’t close enough to call her a friend, but I miss her.

We got to talking about Afro-Cuban music and I laughed at myself, saying how I never could get that damn bell pattern down. Said Russell: “Oh, you mean the standard 12/8 pattern? Tracy, let’s show him” and on the second try, they were doing it together, just voices, ta ta ta-ta, ta ta ta.

Driving home from the hospice, I told myself that if Tracy could manage the bell pattern in her condition, I could bloody well learn it. So I studied the details and used a metronome app and after a while I thought I had it down pretty well.

Sounds cool · I go to a weekly by-invitation African drum jam where I’m on the weaker end of the skill spectrum. The first time a 12/8 came along after I thought I’d learned the pattern, I had to summon up courage and then I fluffed the first few bars. But after a while I was grooving along and smiling and thinking the bell sounded pretty cool against the thunder of all the djembés and dununs.

And, even played amateurishly, it does sound cool. Let’s have another look at the music.

West-African drumming often tries to achieve rhythmic tension, where a given note could fit in multiple ways and your ear is not 100% sure what’s going on. The standard pattern does this, twice.

Remember, I said that 12/8 sounds like it’s “in four”, especially if you hit the first beat of each of the four mini-measures. But two of the four minis here go around the first note, weakening the 4/4 feel. Especially on that third mini; you can feel the beat slide by the missing “one”.

Also, the last three notes are evenly spaced two beats apart, so six of them would fill the 12-beat pattern, suggesting that this might be in triple time, not 12/8.

The effect, to my ears, is of the bell, higher-pitched than the drums, shifting against the rhythm, or even dancing across it. At the drum jam, at almost any given moment it won’t be just drums, one or more people will have clave sticks or rattles or tambourines or cowbells weaving through the beat.

Mixing it up · After I felt confident playing the standard pattern, it still sounded cool, but I wanted to branch out, not just go around and around the same seven notes. So the first thing I did was start mixing in a few of these.

This repeats the second bar through the end of the phrase. In the sound sample I mix it up with the standard pattern. It’s got less rhythmic tension but on the other hand flows along smoothly with the drum thunder. Also you don’t have to think at all, so you can enjoy listening to what the other people are playing.

Then I got a little more ambitious and reshuffled:

The mini-measures are the same as in the standard pattern, just in a different order. Anyhow, this kind of thing is fun.

Combinatorics · Then one evening I was lying in bed, thoughts wandering, and wondered “How many bell patterns are there?” A little mental math showed that of course there are eight possible arrangements of tones in a 3-note mini-measure. Here they are:

Why are the minis numbered in that order? Every computer programmer looking at this already knows, but for the rest of you: If the notes are ones and the rests are zeroes, they are the eight binary numbers between zero and seven inclusive. So each number’s binary bits show where the drumstrokes are. By the way, numbers four through seven have a note on the one beat, zero through three don’t.

Is it weird to have a zero i.e. silent mini? I don’t think so, sometimes spaces between the notes really matter.

Patterns · Anyhow, the original question was about the number of different bell patterns. Each has four mini-measures with 8 possible values. So the answer is 8 ⨉ 8 ⨉ 8 ⨉ 8, which is 4,096.

And each of them can be identified by four little numbers, ranging from T0000 (I can hear the bandleader yelling “gimme zeroes for the sax break”) to T7777, a flurry of eighth notes that you might use in the big encore-number finish designed to leave the audience yelling as you walk off stage. The standard bell pattern is T5325; in binary “101 011 010 101” and the 1’s are drumstrokes. The first variation above is T5333 and the second is T5253.

The “T” in front of each bell pattern number is for Tracy.

If you go look at the Wikipedia Bell-pattern article, they emphasize that there are lots of different patterns. Now they all have numbers! The article makes special mention of T5124, T5221, and T5244.

But why, Tim?! · I’m a computer programmer with a Math degree, and an amateur musician. Anyone who thinks that these are disjoint disciplines is wrong. And, I think the notation is (on a very small scale) kind of pleasing.

But the work has actually helped me. Now that I’ve considered each mini-measure and its personality. I find all of them sneaking into my Gankoqui excursions, which have gotten noticeably weirder, for example T5635. Nobody’s threatened to kick me out of the jam, so far.

Also, this has given me a real appreciation of whoever it was that, probably thousands of years ago and certainly in Africa, picked the “standard” pattern as, well, standard. Because it’s great.

What’s missing? · You may have noticed that Gankoquis have two bells and I’ve been ignoring that fact. Normally you’d play these patterns on the smaller “child” bell, but sometimes bringing the big parent bell in for a couple of strokes works well. Here’s an example (h/t Russell).

Also, this discussion has been limited to 3/8 minis in 12/8 measures. There’s another whole universe of 4/4 rhythms that also have bell patterns (but everything exists in the shadow of the clave rhythm). In that world a pattern has four measures, each of which can have sixteen possible values, so there are 65,536 different ones.

And I could repeat the numbers construction above for 4/4. But I’m not going to, because the rewards feel smaller. In my experience, 4/4 rhythms lope smoothly along and everyone knows where the one is even when there’s no note on it, so there’s less ambiguity to work with. Anyhow, any neophyte (like for example me) can play a pretty smooth bell line against 4/4; just start with clave and add variations (or don’t) and you’ll be fine.

Useful? · These numbers are just elementary mathemusical fun. If anyone else wanted to use them that’d be a pleasant surprise. If “anyone else” is you, go ahead, but they have a name and you have to use it. These are called Tracy Numbers.

Colophon · Music fragments by MuseScore Studio. Sound samples facilitated by GarageBand, a Shure MV51, and PSB Alphas.