Musicianship

Basic tuplet math

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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.

10 ideas for more focused practicing

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It can be difficult to keep practice sessions focused and productive. Distractions, burnout, boredom, and bad habits get in the way of progress. Try some or all of these, see what works well for you, and make the most of your practice time.

12(+) ways to practice a technical passage

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Rhythmic variation is an excellent way to keep some variety and interest in your practicing, even when you’re stuck on a particularly frustrating passage.

Did I play that “right?”

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Execution of musical passages isn’t really about “right” or “wrong,” but rather about degrees of rightness. To borrow an idea from manufacturing or engineering, we might think in terms of tolerances.

Creativity, hard work, and beginning jazz improvisation

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I occasionally teach a university course in jazz improvisation, geared toward beginning improvisers. Sometimes I think prospective students are afraid to sign up because they don’t consider themselves already to be musically creative. On the other hand, I have some students enroll in the class with unrealistic expectations about the results, thinking that they will … Read more

Ornaments are notes

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I think there are some unintended consequences of the way ornamentation is notated in Western music. Often the ornaments are indicated with some kind of abstract symbol or with tiny “extra” notes (like grace notes), located visually outside of the music’s rhythmic structure. This sometimes leads less-experienced musicians to the conclusion, consciously or otherwise, that the ornaments do not have precise rhythms. Sometimes music teachers feed this problem by explaining the rhythmic aspects of ornamentation in a vague or misleading way.

For example, many of my saxophone and oboe students are initially stymied by this moment in the first of the Ferling 48 Famous Studies:

ferling-example.preview

An unclear but common way to explain this is to indicate the pitch pattern of the turn—up a diatonic step and back down, down a diatonic (or maybe half-) step and back up—and then say something to the effect that these notes “steal” time from the F-natural. The grace notes in the next measure can be poorly explained by emphasizing that they go “on the beat.” These explanations aren’t factually incorrect, and make some sense to someone who already understands what the end result should sound like, but leave a lot of unanswered questions for students who are less experienced with ornamentation.

To be clearer about the turn, I think it helps to think through exactly how many notes have to be played in the space of the F-natural (five) and some possible ways to fit them in. Here are a few:

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Music for its own sake

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At least once every few weeks, my social media feeds get flooded with links to the latest article about how kids should learn music because it turns them into excellent businesspeople and scientists and politicians. The latest is an opinion piece from the New York Times.

Condoleezza Rice trained to be a concert pianist. Alan Greenspan, former chairman of the Federal Reserve, was a professional clarinet and saxophone player. The hedge fund billionaire Bruce Kovner is a pianist who took classes at Juilliard.

Multiple studies link music study to academic achievement. But what is it about serious music training that seems to correlate with outsize success in other fields?

It may or may not be true that musical training sharpens math skills and teamwork skills and so forth. But I am irritated by the subtext that music isn’t something worth pursuing on its own merits—it is only valuable as cross-training for making a “real” contribution to society. Nobody ever seems to wonder whether education in mathematics or reading or science makes people into better musicians.

Photo, Bill Selak
Photo, Bill Selak

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