Music Thoughts: Polymetres and Stuff (Part 2 – Melody and Harmony)

In the last post, we pretty much talked only about rhythm in the context of polymetres (and polyrhythms). And that is pretty much the focus in any introductory text on the topic.

However, the things that happen if we use pitched sounds in repetitive, polymetric patterns are also interesting. So today we’ll look at pitched sounds and see what happens.

General Thoughts

We’ve already discovered that if we run patterns with different length alongside each other and the lengths have a specific property (being coprime, i.e. don’t share the same divisor) they’ll end up in any possible timing relation towards each other.

If we’re using pitched sounds in our patterns, this then means that eventually, any possible combination will appear. So if we have one pattern with three different pitches and length 9, and two more patterns with length 8 and 7 and two different pitches each, then eventually every triad possible with any of those sets of pitches will appear.

So by wisely choosing those sets of pitches, their position in the pattern and the pattern length, we can create specific chord progressions. Or, we can just take some arbitrarily selected pitches, put them in patterns and see what happens.

As the former approach is pretty tricky, we’ll just do the latter and see where we end up.

Example: Keyboard Mindset

The most intuitive way to pick a few notes when given a piano keyboard would be to start at middle C and then play white keys upwards, eventually ending up with a C Major scale. And that’s how we’re going to create our first example.

Take the first three notes (C, D and E) and put them into one pattern, I’ll make it a 7-step pattern and put the notes at steps 1, 3 and 5 (i.e. 2-2-3). For the next two notes (F and G), I’ll use a 6-step pattern and put the notes at steps 1 and 4 (i.e. 3-3 stepping), and finally for A and B use a 5-step pattern and put the notes at steps 1 and 4 (i.e. 3-2).

With that, we have coprime pattern lengths, and a resulting overall pattern length of 7*6*5=210 steps. So if we make the steps quarter notes and use a standard 120bpm i.e. a slow allegro, we end up with a duration of 1:45.

Now which chords (the definition of chord usually being three or more notes) will we get? It’s pretty simple – and combinations from our three patterns {C D E}, {F G] and {A B} will happen eventually but exactly once.

While we kept the tonal part coprime, the same isn’t true for the rhythmic structure. With the symmetry in set 2, the rhythm will repeat after 7*3*5=105 steps.

All of the three-note-combos written out are shown here, though not in the order they will happen:

Chords KB.png

Note that the number of chords is 12 – 3*2*2 i.e. the number of notes per pattern multiplied.

Even though we started with a “C Major mindset”, there is no proper C triad here. And that is no surprise, but rather a result of the choices I’ve made, more specifically the choice to have C and E in the same pattern, which means they can never sound together.

Starting with the standard triads (i.e. two non-degenerate thirds in succession), we get the following chords (in no specific order):

F – G – Em –  Dm – B0

If we add the chords that are of the missing fifth variety (i.e. which are or can be inverted to be a combo of third and pure fifth), we get:

FM7 – Am7

The remaining five chords don’t have an intuitive functional reading in Western harmony – there’s two which can be seen as seventh chords missing the third (A?7 and C?M7 – for our only clear C chord), two sus4 chords (Fsus#4 and Dsus4) and a combo of tritone and fourth (which you might read as an E?9).

But what will the actual succession of notes, intervals and chords be? Rather than trying the tedious task of working that our by hand, I opted to simply use a sequencer software (Cubase) and, by duplicating the patterns on three tracks and then combining the result on another track, get the score. Here’s the beginning of it.

We’re in polymetric territory here. I simply opted to have that formatted in 4/4 because it’s so common – I could’ve gone with 3/4, 7/4 or whatever.

Prog KB.png

Example: Bass Player Mindset

We’ll now use what I’d call a bass player mindset: ascending fourth. Thinking about those odd 7-string bass guitars, they start with F# and then go up in fours up to the C string.

Rather than doing that, to stick with what we have before, I’ll start with B and move to F, this way getting the notes of the C Major scale again.

This time, our sets end up being {B E A}, {D G] and {C F}.

Our chords this time look like this:

PICT_20200213_214142.JPG

As normal triads, we get

C – Dm – B0.

For the third/fifth combos, we have

Am7 – G7 – C?M7 – D?7.

Unfortunately, all that remains are combos of two seconds, which will sound more like miniclusters rather than proper chords. Which is a pity, considering we could do a ii-V-I in C in this one.

I won’t discuss any rhythmic details this time – we can use the same approach as before, or a different one.

Example: Choosing Notes Wisely

We’ve seen so far that we never got the chords for a IV-V-I chord progression. Can we get that?

The subdominant, dominant and tonic contain all seven notes of the diatonic scale. There’s six different ones in the IV and V, and the remaining one (the tonic’s third) is part of the tonic.

There’s several possible solutions to this. Taking the experiences from the previous examples – mainly that we get more third-based triads if we have the pitches in each set next to each other, we’ll go with the following: {D E F} {G A} {B C}.

To both show how this affects the outcome and shorten the full pattern some, we’ll be going with much shorter patterns this time: 5 (with 2-2-1 division), 4 (2-2) and 3 (2-1). And finally, we’ll design our patterns so we start out with C Major right away: {E F D} {G A} {C B}.

Overall pattern lengths are 5*4*3=60 tonally (that’s 15 bars), and a mere 30 rhythmically.

Let’s look at our resulting chords first:

Chords Wisely.png

This one clearly says “C Major”, and it does so because we’ve made it that way.

For standard triads (third/third combo), we get

C – Em – Am – F – G

And for the third/fifth combos, there’s

G7 – Bm7

The remaining chords are a F#11 and three thirdless chords (E?9, C?9 and D?7).

Now there’s potential! We have proper triads on all but one note (the D?7), and consequently, if we accept the D?7 as a D chord, have a descending fifth row from B all the way to F, just like George Russell would’ve liked it.

The resulting progression looks like this (I marked the point where the tonal pattern repeats in bar 16):

Prog Wisely.png

Summary

What we did can be considered already an advanced application of polymetres. While the math for getting specific results is rather boring and cumbersome, I find the approach of aiming for some goals and then see how the rest works out pretty interesting.

If you ask yourself how does it sound? – ask no more: Below you have our Choosing Notes Wisely sets, but with pattern lengths from the first example (7, 8 and 9) and running it a 60 bpm – the resulting playtime is roughly 8:30. In the video, I decided to do this as a loop, though.

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