The deep connections between the art of music and the science of maths, from Bach to Bacharach

Paul Desmond’s Take Five, recorded by The Dave Brubeck Quartet, was the biggest-selling jazz song of all time. The late Burt Bacharach loved experimenting with time signatures: “Do You Know the Way to San Jose?” is in cut common time, but the phrase length is five bars instead of the usual four. This hyper-metre evokes surprise and has an uplifting effect.

We can add the two notes and plot a graph or combine them in a diagram called a Lissajous curve or harmonograph. A perfect fifth, shown as blue curves in the diagram, produces a simple pattern; its aesthetic appeal is in harmony with the sound of the chord. Since the numbers in the ratio are small, the combination is harmonious.

The waveform and harmonograph get more complex as larger number ratios occur: the combination of C and C♯ has a shrill sound, and the Lissajous curve is correspondingly intricate . All is not lost: a sound that is not of itself pleasing often occurs in music only to be relieved or resolved by a more concordant sound.

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