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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  • Did I play that “right?”

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    Sometimes my beginner students will play something for me, then ask, “Was that right?” What they generally seem to mean is something like, “Did I use the correct fingerings in the correct order?” A student who is slightly more advanced might ask the same question, but also wonder whether the rhythms were “correct.” A student farther along than that might take into consideration things like accurate observance of marked articulations and dynamics.

    Setting aside creative concerns and looking only at technical matters, when is a student’s playing “right?”

    I try to impress upon my advancing students that 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.

    In other words, for a beginner, a half-note rhythm might be “right” if the half notes are roughly twice the length of quarter notes, and a pitch of “D” might be “right” if it is closer to D than it is to D-flat or D-sharp. For a somewhat more advanced student, it might not be “right” until the rhythms can be played without wandering too far afield of a metronome and the pitches trigger the “in tune” light on a cheap electric tuner. For an even more advanced student, those tolerances wouldn’t be fine enough—we might expect the rhythmic ratios to be accurate to within a few percent, and the pitches to be accurate within so many cents.

    photo, Lewis Meyer
    photo, Lewis Meyer

    At the highest levels of musical technique, we question what tolerances are accurate enough for our audiences, or for someone with even more finely-tuned ears—a conductor, perhaps, or an audition panel, or collaborating musicians, or a record producer. Are my rhythms “right” when they are within a tolerance of a hundredth of a second? A thousandth? A ten-thousandth? Who will hear the difference if my pitches are within a tolerance of ten cents? Five cents? The more I think about it, the more I’m certain I can never really be satisfied, because as my execution gets more accurate, my ears get less tolerant.

    If your ears are currently “tolerating” your level of accuracy when you play, it may be time to listen more closely and critically. Don’t be satisfied with “right”—go for more right.

  • Sorry, but you still can’t bring your reed knives on US flights

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    There has been some buzz (no pun intended) among US reed players about an announcement from the infamous Transportation Security Administration that some knives will be allowed in carry-on luggage starting next month. But make no mistake—your reed knife will still need to go in your checked bag or it will be confiscated at a security checkpoint.

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    Check out these resources for more information:

     Keep those valuable and razor-sharp reed knives in your checked bags.

  • MS Word music hack: Automatic sharps, flats, and naturals (Updated 2019)

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    Ten years ago in 2009 I wrote a blog post about how to set up auto-complete sharp, flat, and natural symbols in Microsoft Word 2007 running on Windows Vista. Here’s an update for Word 2016 on Windows 10.

    Thanks to Ariel Detwiler for calling my attention to the need for an update.

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    Type this (plus space bar) Get this
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    Will it work on Office 365?

    No, the web version of Office doesn’t currently (2019) support this feature.

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    1. Open up a new Microsoft Word 2016 document.

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    3. Highlight the flat. Click the “File” tab, then “Options.” A dialog box appears; click “Proofing” at its left-hand side. Click the “AutoCorrect Options…” button. On the “AutoCorrect” tab, make sure “Replace text as you type” is checked. In the “Replace:” box, type:
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    4. Repeat steps 2-3 for the sharp and natural signs. In step 2, use “266f” for sharp and “266e” for natural, and in step 4 use “-sharp” and “-natural.”

    If you are in the habit of using the pound sign (#) and lowercase b for sharp and flat signs, shame on you for your unprofessional documents. You might be tempted to set up your AutoCorrect to correct those. The problem with doing it that way is that you will have to create a separate AutoCorrect entry for each note (Ab, Bb, Cb, etc.). Plus, it’s a good idea to just get in the habit of typing “F-sharp” so if you’re using a system that doesn’t have your special AutoCorrect set up, you will be left with that more correct version instead of a pound sign or letter b.

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  • More on airline travel with musical instruments

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    I think it’s also worth sharing the musical instrument policies of a few of the major (US) airlines:

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  • Things that aren’t jazz

    ByBret Pimentel
    Photo, frawemedia

    Okay, first of all: what I’m talking about here is the “mainstream” jazz tradition, insofar as such a thing exists (you can make a good argument that it doesn’t, really). “Jazz” is a wide net to cast. To flip it around, if I were going to list things that aren’t “classical” music, I might say “use of the electric bass guitar” or even “microtonality.” Are those things really mutually exclusive with classical music? No. But they are not part of the tradition of the Viennese masters, not what your local community orchestra would play, not what most people think of when they think of classical music.

    One misconception that many classical musicians seem to have about jazz is that since it has strong improvisatory elements, it must be very free and unstructured. I think the opposite is true: improvisation, at least in the mainstream-bebop/hard-bop-influenced style, requires fairly strict structural underpinnings. None of the following works well when someone is trying to improvise:

    • Free forms. Form in jazz is much stricter than in classical music. Most common jazz tunes have one of two forms. The first is precisely 32 bars in 4/4 time, with four eight-bar sections: AABA. The second is precisely twelve 4/4 bars, a “blues” form. Anything that doesn’t fit exactly into one of these categories is most likely very closely related to one of them.
    • Free harmony. Bebop and hard-bop jazz are extremely tonal. While improvisers may play pitches that fall “outside” the chords, it is almost always within a tonal framework: the notes function as upper extensions of the chords, or at least they are used in a sort of polychordal way that has reference to the underlying harmony and will ultimately resolve back to it. The idea that “there are no wrong notes” isn’t exactly true; notes can definitely sound very wrong if they aren’t properly contextualized with regard to the harmony.
    • Free or changing meter. Jazz makes frequent use of polymeter and syncopation, which can give the impression of shifting meter, but most jazz doesn’t actually change meters, especially in improvisatory sections. The polyrhythms and syncopations work because they are overlaid on an unchanging metric pulse, and resolve to it.
    • Rubato. Tempi also tend to quite strict in jazz playing. A musician might “lay back” in the beat or stretch a rhythm, but the underlying pulse doesn’t change. (In my experience, jazz players generally have much steadier internal metronomes than classical musicians.)
    The freedom that improvisers do have is to create melodies over these fairly rigid structures. The rigidity gives the improviser a predictable framework to work with (or perhaps against). At a more detailed level, there are certain melodic characteristics that classical musicians tend to associate with jazz, that I don’t hear when I listen to jazz or play with fine jazz musicians: Read More “Things that aren’t jazz”

  • Staying challenged

    ByBret Pimentel

    I teach a small woodwind studio at a small university. That means that sometimes especially talented and hardworking students find they don’t have a lot of competition for ensemble placements, awards, and other things. Here’s what I suggest to students in that position, who want to stay motivated and challenged but have bumped up against the ceiling in terms of those typical measures of achievement.

    photo, Brad.K
    • Find inspiration (and some friendly competition) at conferences, festivals, or “clarinet days” (or whatever). Surround yourself by like-minded achievers. Going to a national/international conference can be expensive and disruptive to your semester, but is probably worth it if you can make it work. If not, consider regional events that happen within a few hours’ drive and often over a weekend.
    • Listen to music every day. Spend a few hours scouring a store, library, or online music service for players and repertoire for you instrument that you aren’t familiar with. Cue them up into a playlist so you can listen for five minutes while you get situated in a practice room or walk between classes. Form opinions about them. Next level: add to this some daily listening of music not for your instrument, something completely unfamiliar. Think outside the Western world, too.
    • Record yourself often. Listen back and take notes (the note-taking is important). What do you find embarrassing or unsatisfactory about it? Ask your teacher and see what other resources you can find for ideas on fixing the problem. Keep adding to your list of things to improve, and re-prioritizing as you do improve them.
    • Seek out opportunities that take you outside your comfort zone. Consider entering a competition or taking an audition (even one you know you won’t win), starting a chamber group, tackling repertoire that scares you, joining a rock band, or something else that musicians you admire do, but that seems a little scary and hard.
    • Think about the things you are doing that you feel you have maxed out—maybe you’re first chair in all your ensembles, you’re getting straight As in your lessons, you have won the top scholarship. Now ask yourself: what would it take to really surprise everybody at the next audition, lesson, etc.? What would set a new standard? What would people still be talking about years from now? What would multiply your achievement by two, or ten?

    Have other ideas? Please share in the comments section.

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