Playing music sharpens the brain. It’s proven. I’m a musician, but I’ve also spent a number of years studying mathematics and physics. That is unlikely to have made me a better musician or composer, but playing music from an early age has, quite possibly, made me better at math. Today, I like to let both disciplines talk to each other, and use mathematical ideas in my composing. They help me find sounds and rhythms that I might never have made otherwise. I want to make music that hits me viscerally, but in surprising, unobvious ways.
I want to show you one example involving Fibonacci numbers. Fibonacci was a 13th-century Italian mathematician who brought the Indian-Arabic number system to Europe. He also wrote about this set of numbers that now bears his name. I became intrigued by these numbers some years ago, and have used them to structure much of my work ever since.
The Fibonacci sequence begins: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and continues from there. Each number in the sequence is the sum of the previous two numbers, and it continues ad infinitum. If you look at the ratios of two successive Fibonacci numbers, and keep going up the sequence, you get: 1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619, 1.618 ... As you go up the sequence, this ratio gets closer and closer to a famous irrational number called the “golden ratio”: 1.6180339887.
That ratio has been observed frequently in dimensional proportions across many different contexts — in architecture from the Pyramids of Giza and the Parthenon, to constructions by Le Corbusier and Mies van der Rohe; images by artists from Da Vinci and Albrecht Durer to Juan Gris, Mondrian and Dali; and rhythmic durations and pitch ratios in works by composers from Bartok and Debussy to John Coltrane and Steve Coleman. (Coleman introduced me to this whole idea.)
What interests me about Fibonacci numbers is their scaling property. Because the ratios get successively closer to the golden ratio, the ratio 5:3 is not the same as, but “similar” to the ratio 8:5, which is “similar” to the ratio 13:8, or 144:89, or 6,765:4,181. But what do I mean by as vague a term as “similar”? This is a question I explore musically with my trio’s version of Mystic Brew, a 70s soul-jazz classic by Ronnie Foster.
RHYTHM
The harmonic rhythm in Foster’s original is asymmetric in a Fibonacci way: a short chord and then a long chord, three beats plus five beats, totalling eight beats. It’s standard four-four time, with one added feature: if you were to step to the beat, you’d hear a chord when you take your first step, and then another chord while your knee is aloft between the second and third steps. This is a rhythm that you hear in all kinds of places — think of the opening chords of Michael Jackson’s Billie Jean.
In our version of Mystic Brew, we work with that asymmetry and move it through Fibonacci-like transformations. We perform an asymmetric “stretch” that maintains the same “golden” balance over the entire measure. But we don’t transform simply by multiplying, as you might when shifting from duple to triple metre, or when doubling the quantities of a recipe, say. Rather, we try to preserve an “impression” of the original — the short-and-long-ness of it — to see if we can this way achieve that feeling of similarity.
Suppose you had a round pie and eight guests; you know how to divide that pie into eight equal pieces, and you know exactly what that pie would look like with three pieces missing. Now, suppose five more friends unexpectedly show up. You have the same pie and 13 guests. How do you divide a circle into 13 by eye? A decent short cut would be to imagine it divided into eight with three pieces missing, and cut that shape. Then, divide the smaller section you’ve just cut into five equal pieces, and the larger section into eight. Your result would be close enough.
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