Music Thoughts: Polymetres and Stuff (Part 1 – Introduction)

One of the fun things you can do with a bunch of pattern sequencers is having them run at different pattern lengths and see what happens. I’m planning on a series of posts on the topic of different phrase lengths, polymetres, polyrhythms and things in that context.

This post is trying to explain some of the basics. More hands-on details may come later.

Definitions and such

I’ll try to bring the concepts of metre, phrase length and rhythm into context – as a basis to make them poly.

Generally, the term poly can be described as meaning more than one at the same time – a description that will be clear to all the synth folks among you.

Let’s start with the phrase length for now, because that is something with little amount for disambiguity. In our pattern sequencer mindset, if you have two sequencers running from the same pulse (the same clock), but both playing a pattern with different length, then you’re in poly-phrase-length territory (although that term is not one I’ve heard that often), at least most of the time.

Note that the actual phrase length can be shorter (but never longer) than the length of the sequencer pattern. As a simple example, if you have a standard techno/disco/house/whatever bass drum without any accents, then although you might run that in a 16-step pattern, the actual phrase length is four (because it then repeats a further three times).

The topic of the metre is more tricky. It is usually defined as a (smallest) repeating structure of beats and accents, and has in standard notation a close correlation to the time signature, e.g. 4/4 or 6/8.

It already gets somewhat tricky here, as you can describe a piece with changing metre (e.g. a traditional Zwiefacher) either as a repeating sequence of metres (here: (3+3+2+2)/4) that then repeats throughout the piece, or could also call it a 10/4 metre.

Another thing, and that is something relating to standard (Western-style) notation, is that even if you have a polymetre, you often do not notate it as such, but rather notate one of the metres. And there’s a good reason for that – imagine you had a piano piece where the right hand was suddenly split into two staffs, one in 4/4 and one in 3/4. That would be impractical to read (also imagine calling out bar numbers for ensemble pieces).

1280px-Son_clave_3_side_and_2_side-B.png
Son clave – CC-BY-SA 3.0 Dr clave

Rhythm, maybe the most intuitive of the terms, is usually also described as something like a recurring series of timed events (i.e. pretty similar to what we’ve discussed so far). Interestingly, this time for use of the poly attribute one usually also considers things that aren’t written but implied – for example by the metre. A prime example is the son clave, which even though it doesn’t have parallel structures by itself, is set upon an underlying 4/4 (or 2/4) metre and such becomes a polyrhythm, as this results in a 3-over-4 rhythm (in its first half).

Introductory Example: 3 against 4

A really easy example is something with 3 against 4. To implement that, you need e.g. two hardware sequencers with adjustable pattern length and a maximum pattern length of 4 or more. In this example, I’m using a Korg SQ-1 in two-track mode. Track A is set to three steps (and triggers a CY-522’s hihat), Track B is set to four steps (and triggers a DFAM).

PICT_20200206_111142
The beauty of the modular world: these things are really easy to set up!

We will now consider the pulse on the sequencer, i.e. the length of each step, to be a quarter note. With a trigger on each track’s first step (and only on that), the result can be notated like this:

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While deceptively simple, you can use this example (and setup) to try out anything we’re discussing today. How does the transcription look in 4/4? Experiment with different phrase lengths and more than one trigger per phrase. Is this thing polymetric 3 against 4, or could you also consider it polyrhythmic with a single meter at 12/4? Play around!

How do those things shift around?

In our example we’ve seen that the two separate tracks shift around against each other, from coinciding to various spaced situations back to coincidence. Is there a general rule?

There is, and for it, we first have a look at the least common multiple (lcm).

If we have two phrases of length a and b going against each other, the overall pattern of both playing will repeat after lcm(a, b) steps.

Without going into the details, we take both numbers a and b, do an integer factorization, throw out all the factors that both share once, and multiply the rest. Sounds strange?

There is a reason why it sounds strange. The problem of the integer factorization is one where the complexity is not known.

In our example, we have a = 3 (with only one factor, 3), and b = 4 (equaling 2*2). So the lcm (and the resulting length before it repeats) is 3*2*2=12 (=a*b in this case).

As a counterexample, if our first sequence was six steps long, we’d have a = 6 = 3*2 and b = 4 = 2*2. As both share the “2” factor, the result wouldn’t be 6*4=24, but again 2*2*3=12.

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Visualized using a spreadsheet – both for 3 against 4 (top) and 6 against 4 (bottom), it takes 12 steps until both sequences realign.

We’ve seen that the two tracks before shifted to any possible relative distance based on the pulse. This only happens if the sequence lengths have no common divisor. So if we again have the track B from above and track A with six steps (and again only a trigger on the first step), we’ll never reach a situation where the hihat sound exactly one step after (or before) the bass drum.

The longest length for given sequencers

Let’s stick with the Korg SQ-1 in a two-track configuration as we did before. Each track has a maximum length of eight steps. What is the longest overall phrase length until both tracks realign?

For only two tracks, it’s usually pretty simple: set one to the maximum length (here: 8) and one to one step less (here: 7). With that, the overall phrase repeats after 7*8=56 steps (which, thinking in sixteenth notes as before, would be 14 quarter notes).

For more than two tracks, we’re back to using the least common multiple: if we had a third track and were aiming for that maximum, setting it to still one lower at 6, we’d only get an overall phrase length of 168, whereas if we set the third track to 5, we’d get 7*8*5=280 (which would be 70 quarter notes, quite a lot of variation to get out of only eight sequencer steps).

What if I only have one sequencer?

We’ve seen in our example that the overall phrase length is 12 steps. This means that if we only have one sequencer at hand, we could also set it to a pattern length of twelve and repeatedly program our two patterns from above so they fill the twelve-step pattern. Really anything that can play two distinct sounds at any given time and allows for a pattern length of 12 would do – such as a polyphonic Volca or Pocket Operator.

Examples and Stuff

In no specific order, a few examples on the topics we’ve been discussing.

Afro Blue (Polyrhythm, and depending on the version Polymetre)

The most famous version in jazz circles is probably the one by John Coltrane. The track does have a mild case of polyrhythm in the main theme, as the 3/4 (played as quarter notes by the bass in most versions) is set against dottet quarters in bar 3 of the head.

Untitled.png
Much simplified transcription of Coltrane’s interpretation

In the Coltrane version, all of the ensemble plays pretty much jazz – meaning the artists don’t let themselves be pigeonholed into a specific rhythm or metre (and thus making the discussion somewhat obsolete).

The version I picked from composer Mongo Santamaria is different from his album version. It features the same piano montuna as the album version (which, with its last note of the two-bar-phrase, nicely complements bass and theme in its first eight bars).

The interesting part here is the theme (played by the horns). For the first eight bars (a four-bar phrase repeated), it’s quarter notes. For the second eight bars, we have a tricky thing:

Untitled.png
Shift between 3/4 (yellow), 2/4 (green) and 4/4 (red) in the melody.

In bar ten, the two quarters followed by a whole (i.e. four notes, tied to the next bar), contrasts with the first eight bars and is a two-beat measure followed by a four-beat measure before going back to 3/4 – changing meters. The bar divisions in the melody are here marked in colour.

If you’ve got some spare time, why don’t you do a transcription of Santamaria’s intro to this one? Which metres do you hear?

Franz Schubert Impromptu No.1 (Polyrhythm)

This is a nice one, independently of today’s topic. Bulding from a relatively simple theme (albeit with some spice in the form of dotted notes), Schubert continues to make things more interesting as we progress.

On page two of this score, he already brings in a ternary bassline (eight triplets) to contrast with the dotted quarters and eights. In bar 54, the juxtaposition of eights in the melody and eight triplets in the bass gives us the first clear “odd rhythm” impression (interestingly, the straight eights feel off here).

In the part starting about here, we have a combination of quarters, dotted quarters, straight, dotted and triplet eights, which prepares for what in comtemporary music would be called a drop – Schubert’s version of the “big snare drum roll”.

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Pretty much Polyrhythm by Franz Schubert (D. 899)

Outlook

This should do for a brief introduction, and also as an idea what you can play with. For some further writing, a few of my ideas (which I may or may not follow up upon) are:

  • Proper musical use of the concept,
  • Applications in popular (and not-so-popular) music,
  • How to use it tonally (melodies, chords),
  • Hands-on workflows for typical gear.

If there’s anything else you’d like to be covered – let me know!

2 thoughts on “Music Thoughts: Polymetres and Stuff (Part 1 – Introduction)

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