Podcast transcript
Mathematics and music
Dr Marcel Jackson
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Transcript
- Matt Smith
Hello. This is me, Matt Smith, welcoming you, the listener, to this particular La Trobe University podcast. Today we’ll be hearing from Dr Marcel Jackson. He’s a mathematician from La Trobe University. He’ll be talking about the connection between music and mathematics, and how they’ve got a lot more in common then you’d normally think.
- Marcel Jackson
People often feel that music has a sort of a mathematical connection. Certainly a lot of people I think involved in mathematics will have some kind of musical ability, or they’ll play an instrument, or something like that. I was interested in trying to find out what aspects of rhythm in particular had some kind of mathematical basis. That seemed to be a more obviously mathematical property, since it’s sort of a sequence of ons and offs, if you like. Of course, that’s a simplistic version of what rhythm might be, but nevertheless, it starts to sound a little bit mathematical. In fact there is quite a bit of literature out there on this kind of approach, but yes, so rhythm seemed to me to be something that would be well analysed mathematically, and indeed, this has been something that has been done by a number of people.
- Matt Smith
Can you put it down to a mathematical formula at all? What would music be?
- Marcel Jackson
I think it would be too much of a simplification to reduce everything to a mathematical formula. And I don’t think a mathematical approach to analysing music would ever really tell you what makes something good, or what makes something bad, but nevertheless there may be certain kinds of properties that are reflected in mathematical structure, if you like. So something in particular that had been looked at quite a bit in looking at rhythm is an issue about the equal spacing of beats in a rhythm. If you have some underlying time span of say, sixteen beats for example, and you wanted to distribute some smaller number of emphases within these sixteen beats, that’s easy if, for example, you wanted to distribute four emphases within the sixteen beats, because four goes into sixteen four times, so you could just evenly space them at every fourth beat. But if you wanted to place, for example, five emphases amongst the sixteen beats, then clearly you can’t do this evenly because five doesn’t go into sixteen. And so in this situation you have to have some degree of unevenness, but there may be an interesting mathematical problem here, as to what does it really mean to space these beats as evenly as possible. Can you quantify that concept?
- Matt Smith
Yes, because if you space something four evenly in sixteen beats, you would technically just get four beats. Repeated consistently you get that kind of symmetry there.
- Marcel Jackson
Yeah, that’s correct. So, you can see here that it’s a number theory. I mean, maybe fairly elementary number theory but nevertheless there are some mathematical properties coming into play. So with a situation like five into sixteen, because five doesn’t go into sixteen, you’re forced to have some kind of unevenness. So people have produced a fairly elegant kind of classification, if you like, of what it means to be most evenly spaced, and here it is convenience to abstract a tiny bit for your sixteen beats, so since a rhythm presumably would normally be thought of as a finite pattern of beats that we repeat over and over again. In a musical context, of course, a rhythm isn’t usually just repeated indefinitely. It might change after a while, but the very use of the word ‘change’ there suggests that we did have an idea about what rhythm meant – it was the unchanging bit, and when we decided to deviate from that pattern, then we were now changing the rhythm and moving to some other rhythm. So from a mathematical perspective though, if you wanted to analyse a particular rhythm then, you’re really thinking of it as some kind of pattern that is repeated, and a standard way to visualise this is essentially just to think of it as something that cycles around. A standard way to visualise a rhythm is on say, sixteen fundamental beats, would be some kind of necklace configuration. So we imagine a necklace of beads, and there’s sixteen beads on our necklace, and we’re essentially just going to select some of those beads for the emphases. The pulses, maybe, of this rhythm. In the instance where we were selecting four of the beads, or five of the beats, to be given some emphasis, this would be a bit like selecting four or five of those beads on our necklace, to be maybe coloured a different colour. And we’re going to consider two rhythms the same, if they correspond to the same necklace, up to rotation.
If we put a necklace on our heads and rotate it around a little bit, it’s still the same necklace. One arguably could say that if you took the necklace off and turned it around and put it back on your head, then you might sort of have a different pattern then. That would correspond to playing a rhythm in reverse. Anyway, so there’s a lot of geometry and mathematics associated with patterns on a ring like this. This is
- the kind of thing that’s used. Matt Smith
How about the cycles in scales? When you’re playing on the piano.
- Marcel Jackson
We were talking about rhythm just before, but in a way, the idea of equal spacing sort of works quite nicely when one looks at a piano keyboard. So even if you don’t play the piano, everyone’s sort of familiar with the fact that it has a slightly unusual pattern of black keys, nestled in amongst the white keys. And actually it’s a really good instance of the kind of thing I'm talking about there because while it’s true that the piano keyboard is not infinite, in that it doesn’t have infinitely many keys – it’s only about an arm span – it clearly sort of repeats this pattern in octaves. I think actually this is a nice instance of something that sort of looks a little bit mathematical. And so one could look at the possibility of trying to construct scales on the notes of the keyboard, and really, if one looks within an octave on a piano keyboard, there’s effectively twelve different semi-tones, so if you include all the white notes and the black notes within an octave, then there’s twelve different instances, and so if you wanted to construct some kind of equal spacing scale, where you were to select amongst those twelve different semi-tones, so these are the black and white keys, if you were to select seven of the twelve, but try and make them as equally spaced as possible. Interestingly this really does correspond to, in effect, just choosing the white keys on the keyboard.
So if you just look at the white keys, and if you had a keyboard in front of you, you’d see this, that some of the white keys are only a semi-tone apart, like from a B to C, or from an E to an F, whereas others are a whole tone apart, so again there’s some unevenness there in a scale that uses only the white keys. Now of course there are quite a few different basic scales that use only the white keys, so there’s basically seven different modes, as they might be called. Nevertheless they all have this slight unevenness to the spread, and it’s really just because of the fact that seven doesn’t go into twelve. But it is sort of fun that the most equal way to distribute seven, if you like, special beads from amongst twelve, can be thought of as exactly the white keys.
- Matt Smith
So it’s a bit too presumptuous to say that mathematicians can work out a formula for a famous song, or a song that everybody’s going to like if this formula is followed. But have you found that there are certain genres that lend themselves better mathematically, or even certain songs?
- Marcel Jackson
Well, yes, world music seems to be a place that has led to a lot of the mathematics in this area, because in a lot of African rhythms and also rhythms from South Eastern Europe and India and places like that – these areas have very complicated rhythmic structures to their music. And a lot of musicologists I think have tried to look at what are the common themes to these, because often rhythms that are used in one place, from a Western classical music perspective, might seem extremely exotic, might nevertheless be used in a totally different area, leads one to ask the question as to why this might be the case.
This is part of the reason that people started to look at the notion of equal spacing, or trying to space beats as equally as possible, because this seems to be one of the common themes to these unusual rhythms. Once you have a classification of what it means to distribute the pulses within some time span as evenly as possible, and you start to list possible rhythms that might have this property, it turns out that essentially all of them do really occur in world music. All of the simple ones, obviously, if you were distributing 87 beats into 600 or something like that, there’s not many rhythms that repeat on a scale of 600, but for smaller numbers, it seems to be some sort of mathematical explanation for these common rediscovery of complicated looking rhythms.
- Matt Smith
Is there a specific song that you can take me through that the audience might be familiar with?
- Marcel Jackson
With strangely enough, there are some in popular music that have some, at least vaguely exotic structure, people might be familiar with. Some good instances here might be the song by the Stranglers, Golden Brown, of maybe 1979 perhaps. This is a sort of a well-known piece of music but starts out with a repeated passage and repeats it about four times, and there it might be grouped perhaps in a combination of six-eight, followed by seven-eight, but at least at this abstracted level, we just are looking at ons and offs if you like. It really groups as a group of three-threes followed by a group of four. This is an instance of distributing as evenly as possible, four beats from within thirteen. Now thirteen is an unusual time span, for Western music at least, but I think this would be a familiar example.
There’s a lot of other well-known ones too in South American music, such as the bossanova tends to have lots of similar things like this. So bossanova tends to have an underlying rhythm that is quite similar to the one we had there in Golden Brown, but that would be the equal distribution of five beats in sixteen. Six in a more natural time span, because sixteen is a more commonly occurring time span of something to repeat, but again five doesn’t go into sixteen, so that the five beats are distributed in a different pattern.
- Matt Smith
Where else are these concepts of maths and rhythm and music maybe applied in normal life?
- Marcel Jackson
There is an interesting connection with, of all things, spallation neutronics source accelerators, which is some kind of acceleration chamber idea. One of the places where the issue of distributing beats as equally as possible actually arose in this context. So there, there’s huge machines that have some kind of cyclical behaviour, something might be running at ... it might be 600 hertz and you have to select some of the peaks in this to send down some kind of signal, or produce some kind of pulse. And for technical reasons, these need to be equally spaced, partly because there’s huge surges in voltage and things like this, and apparently it puts strain on the equipment to have these too close together. So interestingly, one of the algorithms for producing what turns out to be the most equally spaced distribution of k-beats from amongst n, to start to speak a bit mathematically, has come exactly from this. There has been a nice and quite simple algorithm for producing exactly this and it was in the context of these accelerated chambers.
- Matt Smith
So where do you see as a direction that you’d like to take this kind of research?
- Marcel Jackson
My interests in it really came through a student who had a scholarship with the Australian Mathematical Sciences Institute. He was a bit of a percussionist, and also a very keen and budding mathematician. It seemed like a fun little thing to look at. And one of the things we were looking at was the idea of extending these repeating rhythms to a notion of an infinite rhythm, because there are a lot of patterns that are investigated in mathematics, sometimes because mathematicians like to look at things like patterns, but also because infinite patterns really do arise in a lot of different contexts. Dynamical systems is an area where these arise for example, and the patterns that tend to emerge as kind of natural if you like, in these mathematical contexts, turn out to have a lot of properties that are in common with what rhythm should have. So a nice way to think about the motivation for having a repeating finite pattern which is what we would maybe have called a rhythm, is the idea that you need to be able to identify that rhythm. So at least you are listening to a piece of music under which the rhythm sits. As the music goes on, you need to be able to identify that it’s still somehow the rhythm that you’re hearing relates to what you heard before.
So there you have a property that little patches of combinations of beats that you hear at one point, you keep on hearing, over and over again, at some kind of vaguely regular interval. Now in the case of a rhythm that’s just repeated exactly, so some finite pattern that’s repeated over and over again, of course, you really do hear these little patterns at regular intervals, but there are a lot of non-repeating patterns, so these are now infinite patterns. You could think of them as like an infinite tapping out of beats, that kind of thing, that goes on forever though, but they nevertheless have the property that each little patch, if you like, that you can hear, will nevertheless appear at bounded intervals, so it may not appear exactly every five beats, but maybe every block of ten beats might contain that pattern somewhere within it. These turn out to be the kinds of properties that arise naturally in the mathematical contexts, so in fact minimal dyna symbolic dynamical systems produce exactly this property for example.
- Matt Smith
That was Dr Marcel Jackson, and you can also listen to a public lecture that he did which was recorded and put on iTunes U. That’s all the time we have today for the La Trobe University podcast. If you have any questions, comments or feedback about this podcast or any other, then send us an e-mail at podcast@latrobe.edu.au.
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