Click to read my article Mathematics, Physics and A Hard Day's Night, which appears in the October 2004 issue of the refereed CMS (Canadian Mathematical Society) Notes. The article provides the first scientific evidence for the intrumentation, notes and voicing of the famous opening chord of The Beatles' movie of the same name.

A mathematical process called Fourier Transforms can be used to decompose a chord or note into the frequencies that make it up. The diagram below shows a picture of the sound wave for a sample of the opening chord.

The list of frequencies and their amplitudes (the latter corresponding roughly to volumes) can be visualized by a spectrogram, with a vertical bar at each frequency whose height is the amplitude of the frequency. Below is a spectrogram of the 48 frequencies of highest amplitude.