Gordon's Progress

I've been thinking about rhythm. Here's something for a music theorist with an interest in percussion, or a percussionist with an interest in music theory...

Let's start with a definition of polyrhythm from http://www.xs4all.nl/~marcz/Polyrhythm.html, which is a fairly simple introduction to the subject.

[i]Poly rhythm is the systematic exploitation of different rhythms performed simultaneously.[/i]

The page's author, Marc, then goes on to refine this definition by ruling out trivial cases. Silly fellow.

Let's start with the most trivial. Two drummers each playing the same steady beat. 1:1 - They are playing in [i]unison[/i].

Next most trivial: One drummer's beat is twice as fast as the other. 2:1 - [i]Octave[/i].

First non-trivial. One drummer plays three beats in the same time the other plays two. 3:2 - [i]Perfect Fifth[/i].

Next non-trivial. One drummer plays four beats in the same time the other plays three. 4:3 - [i]Perfect Fourth[/i].

Ideas like "perfect fourth" are not normally associated with drum beats. Perhaps they should be.

Because: [i]A tempo of 120 BPM is equivalent to a tone of 0.5 Hz.[/i]

Drum beats are very low frequency notes, so...

Beats that play nicely together follow the same rules as notes that play nicely together.

(There is a difference - phase does not matter with audio frequency notes. It matters with low frequency beats.)

At the moment this is just a thought. I'm not seeing any references on the 'net to people who have made the same observation and developed it further. Which either means (1) It is a silly thought because I have missed something obvious or (2) I am looking in the wrong places. (There is also the rather unlikely (3) - this is an original and non-silly thought.)

So I am left with either of two questions - what is the obvious thing I missed, or - where should I be looking?

Great post, Gordon.

In the examples in the link, the clock time between the first beats of each measure is the same -- that is, regardless of the rhythm, they all "line up" on beat one.

The definition of "polyrhythm" is a little hard to describe mathematically because there is always a "lowest common denominator" issue.

For instance, in the example of two beats against three, this could appear as a simple (non-poly) rhythm in 6/8 wherein the duple division lands on beats 1 and 4 while the triple division lands on beats 1, 3, and 5.

Since any combination of numbers can yield a common pulse (say 35 pulses to represent 5 against 7) the notion of "beats that cannot fit into each other" cannot be obtained mathematically.

However, the author's point concerns the notion that multiple percussion parts can either contribute to a common (non poly) rhythm or emerge as independent rythmns that sound simultaneously, depending on how they are written.

So, what is one to do? How does one achieve a poly-rhythmic texture?

The placement of accented beats is an important factor in polyrhythm, too. To have two drums beating together in eigth notes, according to the definition, is NOT a polyrhythm. However, if one drummer accents every 4th beat and the other accents every 3rd beat, then the effect of independent rhythm emerges.

Steve Reich's "Drumming" is a famous exposition of rhythmic "phasing" -- that is when two repeating drum parts are not equal in length and as they iterate they move in and out of "phase" with each other. "Drumming" is a remarkable work and I highly recommend it as a study of rhythm and as highly enjoyable music.

Another group that is very adept at poly-rhythm is "Tool" -- on the intros to many of their songs one wonders how all those beats are going to come together! In fact, it was some of those "Tool" poly-rhythms that inspired the opening of my "PCT 125" track that appears on the Electronic Bible Chapter III collection.

Why PCT 125? Well, the time signature is 5/4 -- 25% more than common (4/4) time. I had to come up with a title. :)

do you know conlon nancarrow?
he composed pieces (studies) that use the idea of comparing rhythmical ratios to musical intervals.

Hi Theremind, not until today. I found him at http://artofthestates.org - a jolly useful resource.

Hi Kev. Listening to Drumming Part 1 of streamed from artofthestates as I type. Which is a bit frustrating - if I could download a file I could speed it up (I guess) about three or four hundred-fold to hear the drum beats as tones and find out if it sounds like it's been through a phaser.

Your explanation of what that page was getting at helped. One can consider a complex drum pattern as a sum of simple beats occurring at different frequencies, as one can interpret a complex waveform as a sum of sine waves. If the various component sine waves are all in simple harmonic relationship then the complex waveform sounds as a single tone, but if the relationship is less simple they will sound as two or more tones with some degree of consonance between them. This is equivalent to the separation into distinct rhythms of a polyrhythm.

Accented beats, if I understand correctly, are beats played a little louder and tend to happen rhythmically, for instance every fourth beat. This is the same, pretty much, as a second identical drum being played at one quarter of the frequency of the first drum, like a fundamental and overtone.

The analogy is not perfect - two sine waves at the same frequency but 180 degrees out of phase will cancel each other out, but two simple rhythms at the same frequency but 180 degrees out of phase sound like a rhythm of twice the frequency. Hence my wondering about the effect of speeding up Drumming.

Syncopation is also a phase thing - saying the beats all line up at beat one is the same as saying there is no syncopation - and it doesn't have an equivalent in harmonics - leastwise I can't think of one.

Gordon, I may be way off base here but I don't think syncopation (or lack thereof) really affects where the 1 beat lines up.

Syncopation might tend to disguise where certain beats fall or muddy the waters a bit when you are conting the beats but the one should still be the same place with or without syncopation.

Your musings are very interesting and there seem to be some definite paralells between rythm and tones.

Kind of makes me think of granular synthesis.

I wrote [i]A tempo of 120 BPM is equivalent to a tone of 0.5 Hz.[/i]

Er, no it doesn't. BPM means beats per minute, Hz means per second. 120 BPM = 2 Hz. Silly me.

Er, maybe I don't understand syncopation right. It's about messing with the accents, shifting them away from the first beat of a bar? For example, a back beat - (from wikipedia) "is a term applied to the beats 2 and 4 in a 4/4 bar." If I shifted the accent on the 2nd and 4th beat to the 1st and 3rd it wouldn't be a back beat.

Yes, Granular Synthesis. Genius! The percussionist describes very low frequency notes, marking out various harmonic elements of the note by making rhythmic percussive sounds.

My wanderings have moved from rhythm to metre, and thence to tempo and from there to a most evocative word, Pulse. The pulse of a song. The theoretical ultra low frequency note or notes that form the overall structure of a piece, around which the drummer synthesises his rhythms, which are implied in accented notes and which determine changes in tempo.

Some changes in tempo can be jarring, others easy transitions - and it's about tempos that are in simple ratio to one another - a melodic consideration. And when it comes to polyrhythms - two or more tempos running simultaneously, harmonic considerations apply. And timing considerations, of course. By way of example - Eight bar blues.

So, structure of music - melody, harmony, rhythm - first unsounded and at infrasonic levels to make a framework, inside which sit the audible frequency harmonies and melodies and rhythms and within each individual sound harmonics and so forth.

My, I nearly invented the rhythmicon there, didn't I.

Anyway, my impatience to show off my new pedal got the better of me. There's a new piece on my mySpace profile (http://myspace.com/beatfrequencyuk) - [i]Butterflies of Vertigo[/i].

This is a quick improvisation, putting the PS-5 and expression pedal through its paces with a delay unit attached to each of the PS-5s outputs.

I was initially going to call it Improvisation 1, or similar, but that's a bit dull. So I set off for a Beckett quote I had in mind, from one of his later novellas, Worstward Ho! or Ill Seen, Ill Said - or maybe Company - something about the howls of the damned - this mad laughter, these wild cries - but on the way I got distracted by a much better quote that I had not seen before, or had forgotten...

[i]I shall state silences more competently than ever a better man spangled the butterflies of vertigo.[/i]

I'm not stating silences competently, yet, but I think the piece spangles the butterflies of vertigo fairly well. :-)

And in breaking news from [i]Duh[/i] City, here's an obvious relationship between pitch and tempo.

If you play a loop at half speed, it's an octave lower. 1:2

The Line 6 loop modeller does that. It also does reverse, which is also interesting, seeing as, as long as you're using the volume for expression, and not Kurstinesque envelope shaping, a theremin sounds the same forwards and backwards.

But that's all the Line 6 does. It sure would be neat if it could slow down and speed up by other ratios that correspond to musical intervals. One thing that makes it extra neat is that a variable speed loop could be controlled by the same pedalboard arrangement as described above for selecting chords for a multi-voiced theremin, but this time using it to play the tempo of the piece. See previous musings re rhythm.

And of course you could have more than one loop on the go at the same time, all playing at different but related tempos. That could be fun!

For more fun with the Line 6 Delay Modeller, listen to what Bela Emerson (http://www.myspace.com/belaemerson) does with her Cello running through a couple...

Very nice.

It appears that the Electro-Harmonics 2880 (http://tinyurl.com/g2459) does just what I described. (See the instructions pdf (http://www.ehx.com/instructions/2880_INSTRUCTIONS.pdf).)

Well, almost - the tempo can be adjusted in increments of a semitone, which I guess means equal temperament rather than just intonation. Does this, I wonder, mean the loops will slowly drift out of sync?