Is there a magical connection? (There is some mathematical music at the end of this to listen to!)
I remember having this discussion at home as a child on many occasions. My father was a successful musician with an interest in Maths. Lancelot Hogben’s “Mathematics for the Million” was one of his favourite books. I was a budding mathematician with an interest in Music. It is remarkable how many Mathematicians turn out to be excellent musicians, and vice versa.
My father saw a mathematical form in the music he wrote. One piano piece was called “Upside Downside”. It was written as a leaving present to the minor prep school down the road in Surrey that I left at the age of 9, called Downside School. The clever bit was that you could turn the score upside down, rotating the page through 180˚ and play it from that viewpoint, and it was exactly the same. For him the challenge of making this work was a mathematical puzzle. He did a similar thing when he and some other composers were asked to write short pieces as birthday presents for Prince Charles (his 21st? Not sure.). Charles played the cello and the clarinet. My Dad wrote a cello part that became the clarinet part when you turned it upside down. He loved the mathematical aspects of Bach’s fugues as much as Schoenberg’s serial music.
There seem to be various aspects to the link between Music and Maths that make sense to me. One is this mathematical pattern forming. In tonal music the harmonies are numerical patterns and the repetitions, inversions and key changes are mathematical transformations. Serial music is even closer to an algorithm with its insistence on strict rules of which notes can be used. Even probability gets called on. Mozart created a parlour game where you created pieces from his fragments with the roll of dice, and this has been echoed and extended into the 20th century by Cage and Stockhausen.
There is a more philosophical point as well. Bertrand Russell described Maths as “The subject where we do not know what we are talking about nor whether what we are saying is true”. He meant that it is an abstract creation and that the objects of mathematics do not actually exist in the real world. There are no perfect circles or straight lines except in our imagination. Where will you find a number Five? You can see symbols that indicate it, you see sets or lengths or areas that have it as a property, but where is the Five itself? What colour is it? What does it feel like to touch? What does it look like? Not the numerical symbol for five, of which there are many, but an actual Five. These mathematical objects do not betray any of the properties that real objects hold.
Music has a similar sort of abstraction. If I look at the score of a piece of music, it is precisely that. It is of the music but it is not the music itself which hides uneasily behind that score. In the same way that we might have a performance of the music, or a recording of the music. Each gives a property of a representation of something that we cannot grasp in our hands. This is different from the visual arts where a piece of art is exactly what it is. A painting, sculpture or film is the artwork. It is not a proxy for the hidden abstract form. I think that this connection of a shared abstraction is a powerful link between Music and Mathematics. It is why musicians like my father have no difficulty understanding concepts such as “The Beauty of Mathematics” that GH Hardy alludes to when he says “I am interested in mathematics only as a creative art”
So to some music. My pieces based on the Logistic Mapping and the Lorenz Butterfly map are already on these pages. I thought I would say a bit about the process behind them. I am doing this because I have been invited to do so in a Maths course we offer by my friend and colleague Russell Scott. He teaches Mathematical Genie as part of our Elements Programme at Island School. In Genie they cover all the really interesting bits of Maths that most syllabi leave off. He offers starters, but the students choose their research area and explore it, whether it be Game Theory, Puzzles, Large Scale Euclidean Modelling, Fractal Geometry or wherever the muse takes them. Elements courses are mostly like this. There are 56 of them at the moment, all with a high degree of student ownership of content and process. The classes are all mixed ages and the courses are not constrained by anyone except the teacher and the students.
The Music Process
Music, at least digitally produced music, is just a set of numbers. Note values, note lengths times etc are all numbers. So I use a mathematical tool to generate the numbers and then assign the variables to them. The most recent pieces have been created using the Fibonacci process. The programme creates sets of notes in chords or individually by repeatedly asking for the next number in the Fibonacci sequence. Doing that actually can sound quite dreadful, so decisions need to be made. What instruments? What range of values can I use for the notes? What range for the times between notes?
Inspired by Brian Eno’s “Music for Airports” I wanted to make something that had that ambient feel. This is it. All note values, the sizes of the chords, the times notes appear or change are generated by the next Fibonacci number. These numbers are recycled modulo a very large number, as otherwise they will overflow my computer! The modular base has to be large enough for the series not to repeat in the time of the music. Or does it? I need to think about that? Anyway, you can create any number of different pieces by choosing different starting numbers. 1 and 1 give the normal series, but 1 and 3 give a different series hence a different piece.
This next piece is basically the same but less ambient as it has a time function that is quadratic. It speeds up as the time variable is the ordinate of a parabola, giving a value that rises to a peak in the middle of the music and falls again to the end.
Is there an aesthetic element to writing these pieces of music? I would say definitely yes. There is a constructual similarity to the process of writing 12 tome music, although this is very different from that. The algorithm provides only a set of numbers and several decisions need to be made in order to turn it into a piece of music that I am satisfied with.
My theory is that I can mix my mathematical rules and my aesthetic values to adjust and change the piece until I like it, but with the mathematical rule at the core. The Logistic and Butterfly pieces, found on this page, are done in similar fashion, as is the Reich inspired minimalist piece Ford Circles.