Advanced Mathematics 3: Polyamory

The notation used was originally made by Urs Liska over at Lilypondblog.org.

In music, as in life, you can have relationships with whoever you want. In both cases, however, cultural and popular acceptance of this freedom is only in its early stages. Many countries still forbid homosexual relationships, just as many musical genres frown upon polynumeric rhythms. Today, we’ll take a look at polyrhythm, polymetre, polytempo, and non-dyadic – also mistakenly known as irrational – time signatures.

If you haven’t already, please read our previous post on microrhythms.

Polyrhythm

What is a polyrhythm? Simply enough, it’s two or more rhythms played at the same time. But, what is a rhythm? In this case, a rhythm constitutes a tuplet feel. For the sake of explanation, let’s say a tuplet is what happens inside the time of one beat. Therefore, 1-tuplet rhythm will be equal to a quarter note; 2-tuplet (or just tuplet) consists of two eighth notes; 3-tuplet (or triplet) would be three notes that each last one third of a beat, 4-tuplet would be four sixteenth notes, 5-tuplet (or quintuplet) five fifths of a beat, and so on.

However, this is only part of the truth. The other side is often hidden, either for convenience or for conciseness, but it’s important to point it out here. A tuplet needs always be a ratio. What’s missing from the above examples is a second number so that the tuplet can be written like this: \(A:B\) (read “A to B”). \(A\) denotes the number of notes that are to be played in the span of \(B\) notes. One of the most common polyrhythms out there is \(3:2\), where, if this tuplet feel is applied to eighth notes, you need to play three notes during the time it would take to play two normal eighth notes. It’s important to point out that these three notes should be of equal length.

A 3:2 polyrhythm. 3 notes are played by the treble staff while the bass staff plays 2 notes.

You can use any number you want as \(A\) and \(B\), even decimal numbers or fractions if you wish, but keep in mind that someone – or something – needs to play it. If you can make a computer play it, then good, but if you’re playing with humans, always ask for their consent before writing a riff in \(\frac{17}{3}:\frac{11}{7}\).

Other relatively common tuplets are \(5:4\), also known as quintuplets, \(5:3\), which are quintuplets in triadic time, such as \(\frac{12}{8}\), \(\frac{6}{8}\), or \(\frac{3}{4}\), where five notes are played in the span of three, \(7:5\), which is one of Efrain Toro’s harmonic rhythm examples, and \(4:3\), which, as Adam Neely pointed out here, is used widely in pop music.

You can also have more than two rhythms playing at the same time. One of the most simplest examples of this is with the \(5:3:2\) polyrhythm. It seems more complex than it really is. Basically, 5 notes are played in the span of 2 while 3 notes are played in the span of 2. Harmonically speaking, this equals to a simple major chord in just intonation (more on that later).

A 5:3:2 polyrhythm where the treble staff plays one voice in 5-tuplets and one in 3-tuplets, and the bass staff plays 2 tuplets.

Harmonic Rhythm

I will briefly talk about harmonic rhythms here, since it’s tied to the idea of polyrhythms. Rhythm is harmony. Simple as that. Except, they both occur at different orders of magnitude. For example, a \(3:2\) polyrhythm might happen at anywhere between 30 bpm to 300 bpm (and even more). To start hearing rhythm as a pitch, however, you’d need to play steady sixty-fourths at at least 100 bpm, and that’ll give you a note that’s on the lower threshold of human hearing, at 26.7 Hz. Middle C (261.63 Hz), is the same as playing steady sixty-fourth notes at 981.11 beats per minute. Let that sink in for a minute.

Therefore, an interval is just like a super-sped up simple polyrhythm. That’s in fact, how just intonation works. If you want to mimic an equal-temperament interval, however, you’ll need a much more complex polyrhythm. For example, a \(3:2\) polyrhythm, sped up, will give a just perfect fifth interval, whereas an equal-tempered perfect fifth, slowed down, will give a polyrhythm of about \(1499:1000\)… quite different, but also almost identical.

A just intonation major chord (root, third, and fifth) would be represented by the harmonic ratios \(1:1\) (root), \(5:4\) (major third), and \(3:2\) (perfect fifth). To make them fit inside one polyrhythmic ratio, we have to make a small mathematical trick, and we get the \(6:5:4\) polyrhythm. That’s a very simple chord, and we already get barely-playable polyrhythms. That’s a testament to how efficiently our brain processes and interprets pitches being played together. Just for curiosty, a minor seventh, flat fifth chord (in just intonation), would give the polyrhythm of \(9:7:6:5\), which is surprisingly simple and beautiful.

Therefore, while harmonic rhythm isn’t doomed to play only simple intervallic ratios, it has a steep difficulty curve when trying to mimic chord proportions with rhythm.

Non-Dyadic Time Signatures

I want to quickly introduce the concept of non-dyadic time signatures, which some call “irrational time signatures”. Irrational time signatures are part of the non-dyadic realm, but calling all non-dyadic time signatures “irrational” is wrong and misleading. So… what does “non-dyadic” mean, anyway?

Well, it simply means that the denominator of the time signature – the lower number in \(\frac{4}{4}\), \(\frac{7}{8}\), or \(\frac{15}{16}\) – is not a power of 2. Good, but what does this imply?

The denominator of the time signature tells you what is the value of the beat in the measure. For example, \(\frac{3}{4}\) means that there are to be three quarter notes in the measure, while \(\frac{11}{8}\) denotes a bar that has eleven eighth notes. So, if the denominator is 1, it represents whole notes, 2 means half notes, 4 is for quarters, 8 is for eighths, and so on. But what if it’s not a power of 2? If your bar is in \(\frac{3}{3}\), that means that there are three third notes. And what is a third note, you ask? Well, it’s simply three half notes in triplet feel, so that they each last one beat plus one third of a beat.

Why not simply use \(\frac{4}{4}\) and triplet half notes, then? Well, the \(\frac{4}{4}\) time signature implies a beat on every quarter note, and you might not want that. If you use quarter notes in \(\frac{3}{4}\) instead, then, perhaps you’ll need to change the tempo of your piece for one measure only, and perhaps even for one instrument only, which, again, you might not want.

That being said, non-dyadic time signatures are almost only ever really useful when used in the middle of a normal, dyadic, composition. It needs the comparison to stand on its own, because otherwise it’s almost always better to use a time signature in powers of 2.

However, you can do more things with a non-dyadic time signature than with a simple tuplet notation. Tuplets are required to be full. You wouldn’t write a triplet of notes with only two notes, or just one! You could write two notes and a rest, for example, but the triplet is filled by the rest, and is not left open. I have one actual tale of a compositional struggle that I faced, years ago.

I was writing a song, and then needed to switch to triplets, but wanted to include a motif of eleven notes. I don’t remember quite exactly the details, but I know it was too complex for me to simply change the tempo of the piece for that little passage. Maybe I was just too lazy to re-write the part. At the end, the eleven triplet notes only fill up three beats and two thirds. At the time, what I did to reconcile this is add one beat on the fourth repetition. It worked pretty well in the end, but I might not have to alter my original vision if I had known about non-dyadic time signatures; I would’ve simply wrote that part in \(\frac{11}{12}\)!

Here’s a random example I made using one \(\frac{4}{4}\) bar, followed by a \(\frac{5}{6}\) measure, which holds five triplet quarter notes, then back to a normal \(\frac{4}{4}\) again and finishing up with four quintuplet eighth notes in a \(\frac{2}{5}\) bar.


Polymetre

While polyrhythm is the layering of different rhythms playing on the same number of beats, polymetre is the layering of different numbers of bet playing in the same rhythm. Simply put, it is playing in multiple time signatures at once.

This happens a lot in progressive metal, more than we think, and that is mostly due to the incapacity of some of the most popular music notation softwares to deal with it. What you’ll often see is a motif, a theme, a riff that doesn’t quite fit in the measure it’s in. For example, let’s take the ending riff of Meshuggah’s song “This Spiteful Snake”, since I worked with it recently for my microrhythm experiment. It’s the same as the opening riff, except it has a polymetre with the lead guitar. Here’s how it is written in normal tablatures:

Coda, or outro, of “This Spiteful Snake”, by Meshuggah.

As you can see, the upper staff, which is the rhythm guitar, play a riff that repeats itself, but not in the same time as the second staff, the lead guitar. The main riff is a repetition of two measures, \(\frac{6}{8}\) and \(\frac{7}{8}\), four times, and then a fifth, incomplete time, where the second measure has one eighth removed, resulting in a \(\frac{6}{8}\) bar, before cycling back completely. The lead melody has a simple eight-bar \(\frac{4}{4}\) structure. Here it is in polymetric notation, thanks to Lilypond (I’m new to this software, so excuse all imperfections with this image):

Coda, or outro, of “This Spiteful Snake”, by Meshuggah, in polymetric notation.
(Click the image for a full-size view.)

In addition, you can see that this polymetre also includes polyrhythm, thanks to the lead guitar playing triplets. Indeed, the two are not mutually exclusive. However, the basic idea behind polymetre is that two motives that don’t have the same time signature can be played together, whatever might be happening inside the measures.


Polytempo

In one sentence, polytempo is just another way of writing polyrhythm. Usually, a piece will use polytempo in order to alleviate the notations. This is because, once again, the two conflicting tempi will display a certain ratio between them. For example, a \(3:2\) polyrhythm would be the same as writing a polytempo piece in 180 and 120 bpm, since the ratio between these two tempi is 3 to 2. There are other changes happening, of course, like the choice of time signatures. If you have a piece in \(\frac{4}{4}\), 120 bpm, where one voice is playing eighth notes while the other is playing triplet eighths, then the polytempo equivalent of it will leave the first voice unchanged while the second will play straight eighth notes in \(\frac{6}{4}\) and at 180 bpm.

A 3:2 polytempo measure.

Of course, that is pretty much the most basic and uninteresting use of polytempo you could probably think of. More interesting examples can be found on Jute Gyte’s albums, for example on Perdurance. “At the Limit of Fertile Land” features a \(8:7\) tempo ratio near the end of it, while the climax of “I Am in Athens and Pericles Is Young” features four tempi together in a \(7:6:5:4\) ratio.

“At the Limit of Fertile Land”

The \(8:7\) ratio relates to the just intonation septimal supermajor second, which is a wider major second than can be found on a regular piano. It can also be a flatter minor seventh, depending on which tempo you decide to follow. That might explain why there is a certain pull towards a resolution found in it. Sevenths and seconds usually exert a strong pull melodically and harmonically towards the tonal centre.

“I Am in Athens and Pericles Is Young”

The \(4:5:6:7\) ratio is similar to a just intonation major chord with a harmonic seventh. That might explain why the polytempic passage doesn’t pull towards a resolution, it already has a pretty low level of dissonance, and it is in stable interaction with all of its tempo components.

As you see, the discussion on harmonic rhythm, started earlier, seeps into other realms of rhythmic experimentation, and is not bound to polyrhythms. Polytempo, and, to a lesser extent, polymetre, can also put this theory to work, albeit on slower time scales in general. While harmony takes place in an instant, polyrhythms play out in the span of beats, while polymetre and polytempo take a few measures to mentally analyze their ratios.


Conclusion

I hope this has been fun for you, because it has been for me. It also brought me some unexpected surprise in the form of Lilypond, a thoroughly programmable music notation software, with which I was able to make (amateurishly) some of the images in this post. I’m sure I will use this a lot in the future!

Stay tuned, because I still have one or two rhythmic discussion topics up my sleeves, namely the swing feels and the Carnatic music rhythmic theory.

On May 13 2018, this entry was posted.
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