Basic tuplet math

ByBret Pimentel

A young music student with some basic competencies might be comfortable with these kinds of rhythms:
tuplet-mathBut these are a little trickier to pull off well:

tuplet-math-1 tuplet-math-2

Divisions of the pulse into twos and threes is simple enough conceptually, but in most cases we really learn those kinds of rhythms better by ear—we just learn what eighth notes or triplet eighths sound like against a quarter note pulse. A division of the pulse into fourths, like sixteenth notes against a quarter note pulse, is something that could ostensibly be derived: you could start from a duple subdivision, then make the mental shift to hearing the subdivided pulse as the new pulse, then subdivide that. But the quadruple subdivision is common enough that I think most of us ultimately learn to play it by ear, too.

The quarter- and half-note triplets in the first two bars of the example above are a little harder to place accurately, I think, because the “extra” pulses are hard to ignore. If I can somehow mentally block out the pulses (audible or implied) on beats two and four in the first measure, then I’m playing three notes per pulse again, and that’s really no different than, say, eighth-note triplets in 4/4. Same thing for the second bar: if I can tune out the pulses on beats two, three, and four, then I’m just playing that same triple subdivision.

One approach to this problem is to simply learn the less-familiar subdivision by ear. Music notation software is the perfect tool for this; with a little skill you can usually enter rhythms as complex as you like, and hear them played back with the utmost precision against a pulse of your choice. But here are a couple of other useful tricks.

Trick #1: Subdivide long-duration triplets

To play triplets against a duple or quadruple pulse, you can derive them from triplets played against a single pulse. For example, this rhythm…

tuplet-math-1

…can be derived in this way:

tuplet-math-3

Trick #2: Approximate complex tuplets with duple/triple subdivisions

“Tuplets” that are prime numbers greater than three can be difficult to audiate on the fly. But in many cases (not all cases) there is some room for complex tuplets to be somewhat less than perfectly even. For example, a composer might write a scalar “run” that contains a certain collection of pitches that doesn’t divide neatly into the number of beats allotted, and use a tuplet to make them fit into the score in a legible way. In such a case, it might be appropriate for the run to accelerate or decelerate a bit. If so, the tuplet can be reimagined as a series of duple, triple, or quadruple subdivisions.

This, for instance…

tuplet-math-2…could be approached like this (with a slight acceleration effect on each tuplet)…

tuplet-math-4…or like this (with a slight deceleration effect on each triplet:

tuplet-math-5Rewriting the rhythms into duple/triple/quadruple subdivisions of the pulse makes it easier to practice these rhythms methodically and consistently with a metronome, and allows for some anchoring which ran really solidify longer runs. Even if the intention is eventually to play the tuplet with more exact evenness, practicing them this way can help to shore up technique in the early stages.

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