Jan 11

Tuplets and polyrhythms

A section of John Cage's "Living Room Music" which features quintuplets

How do I do this accurately at a slow tempo?

5 for 4 over a full bar of 4/4 at quarter=60? How do you do anything but fake that? All it takes is a little math! Slow tuplets can be a real killer, and if you try to perform them like you would an eighth note triplet (probably by feel), you’ll be all over the place. The least common multiple is your friend! In the case of our example (from John Cage’s Living Room Music, if you’re playing along at home), that would be 5×4 = 20. This number is the number of even pulses the bar is divided into that can accomodate both the quintuplet and the normal quarter note pulses. We could also have come to this from another angle: if the rules and standards of notation are preserved (tuplets over 3 beats need to have dots. Very few composers follow this standard; Carter does), tuplet notes will subdivide just like normal ones. This means a quintuplet quarter note split into 4 will yield 4 quintuplet sixteenth notes, just like its non-tupleted cousin will split into 4 normal sixteenths. Each normal quarter note in the bar could be divided into 5 quintuplet sixteenths. Either way we get there, 5×4 or 4×5 results in 20 quintuplet sixteenths, four of which add up to a quintuplet quarter. Knowing this, we can re-notate the rhythm as follows.

The quintuplets renotated

First the pulse, subdivided by accents, then written with ties

Now instead of a mysterious 5 floating somewhere in the bar we have attacks in relationship to the quarter note pulse, easily realized with a facility in subdivision. The purely mechanical accuracy of this method is, in many cases, only a first step; ideally, a tuplet like this should be realized without the syncopation accents implied by the re-notation. When you know exactly where each note lands in relation to the regular pulse of the piece, you can perform the tuplet smoothly, knowing proper points of reference throughout.

Any interaction of pulses can be rationalized in this manner – a more complicated example yielding a similar method: 5 for 3 over 3/32 in Aaron Cassidy’s I, purples, spat blood, laugh of beautiful lips. We can use our knowledge of the standards of tuplet notation to find how the pulses interact. First, we can imagine each iteration of the the 5 over 6 to have dots, which would make the first note of the quintuplet 9 pulses long, and the second note 6 pulses long. This 15 part division of the bar is in fact the LCM, allowing us to easily place the 32nd note every 5 pulses (15/3) and the 32nd note of the quintuplet on the 10th iteration of this pulse, leaving us with attacks on iterations 1, 6, 10 and 11 of a 3/32 measure hypothetically subdivided by 15 128th note quintuplets!

The same method can be used in the abstract to learn a polyrhythm. What does 5 over 3 in general sound like? Find the LCM (15), write out the digits 0 – 14 on a piece of paper, and circle every multiple of 3 and put a square around every multiple of 5. Following the numbers as a regular pulse, tap both hands at 0, the left at circled numbers and the right at numbers with a square. Voila, 5 over 3, mathematically done! Practice feeling each number as the ‘main pulse’, and each as the ‘cross rhythm’.

Two ways of feeling a 5:3 polyrhythm

Two ways of feeling a 5:3 polyrhythm

Generally, feeling the higher prime as a pulse will be easier, as it will feel subdivided by a lower prime. Take the above example: the 5/4 measure requires accurate placement of only triplets, while the 3/4 measure would require quintuplet level accuracy. An ability to shift between the two feels can help with accurate performances of polyrhythms and tuplets.

Dec 10


Composer Karim Haddad’s Motets present a complex rhythmic language based on the medieval concept of prolation taken to a modern extreme. The notation is beautiful and evocative, but tremendously difficult to realize as a performer.

The original, complex notation for Haddad's motets

Conceptually lucid, impossible to perform (currently)

Realizing this (thanks Karim!), the composer has also presented a ‘quantized’ version of the score, translating the many layers of tuplets into their duration in milliseconds, then mapping this information onto the closest approximation afforded by 2-8 subdivisions per quarter note. The result can be extremely variegated series of tuplet subdivisions of a regular beat, requiring accurate placement of say, the final septuplet of one beat, and the final triplet of the following beat, as in the below example.

The human-readable quantized notation for Haddad's motets

Eminently readable and aurally identical

This notation, while immensely more familiar and readable, poses a unique challenge. It requires of the musicians the ability to switch subdivisions in some cases, every beat. This is not something that musicians are normally trained for; we are usually very good with duple and triple subdivisions of the beat and can fake our way through fives, and switching between them involves some grinding of gears. I wanted to train myself to instantly hear these higher-prime subdivisions in the same immediate way we hear duple and triple divisions.

I began by practicing each division on its own, putting on the metronome and simply subdividing, for example, an even five, ensuring that each attack was of equal length and unaccented. Of course 5s and 7s end up in groupings of 2 and 3, but I found it most useful to keep the subdivisions flexible, later improvising groupings with the metronome, exploring aspects of the subdivision. Then I would move between several subdivisions, alternating fives and fours, feeling the even pulse of each, and how they relate. After I felt more comfortable in each subdivision, I wanted a way to practice long strands of changing subdivisions like I knew I would find in the Motets. Using a favorite tool of practice and composition, random.org, I created a string of hundreds of random numbers between 2 and 8, presented in 4 columns. I put my trusty Dr. Beat on 4/4 and dove in, reading each number as a subdivision. It’s tedious work, but even a few sessions found me more confident in dealing with simple subdivisions of the beat. In a way, there’s no difference between triplets, sixteenth notes, and septuplets; they’re all periodic subdivisions, evenly filling the space between two pulses. There’s no special secret to learning them, only the familiarity that we have from years of duple/triple based music that has led us into rhythmic complacency and fear of the higher primes!

Nov 10

Elementary Training

Contemporary music is full of sundry rhythmic challenges. Before we even get there though, let’s make sure we are completely fluent in the rhythmic language of common practice music. My favorite workout for basic rhythmic exercises comes in Paul Hindemith’s perennial favorite Elementary Training for Musicians. If the title seems condescending, wait until you read what he has to say about singers!

“As for singers, nobody denies that most of them are launched on their careers not because they show any extraordinary musical talents, but because they happen to have good voices. On account of this advantage a singer is usally excused from any but the most primitive musical knowledge — knowledge such as could be acquired by any normal mind in a few weeks of intelligent effort.”

Ouch, Paul, ouch. The text takes you step by step from reading a simple series of vertical lines as regular pulses through the furthest traditional notational difficulties of Hindemith’s time. He even has a remarkable prescient turn in a page where he describes the derivation of what have been come to be known as “irrational meters”, with denominators other than powers of 2. But my favorite feature of the text is the way he forces a physical incorporation of the rhythmic concepts at hand with what he calls coordinated action. This consists of speaking the given rhythm while conducting with one hand, tapping it with the left hand while conducting with the right, tapping it with the foot while conducting, and every possible combination of limbs and rhythmic interactions.

A rhythmic exercise from "Elementary Training"

Try tapping, singing, conducting, etc. as prescribed

While a given exercise may be simple with only your dominant limbs in play, a simple redistribution of the material across your body can force a radical re-learning of the rhythmic concept at hand. The literal embodiment of rhythm in a deep and conscious way (not just toe-tapping) has a transformative effect.