McCartney: high D3 played on the D-string, 12th fret on Hofner 500/1 electric bass George Martin: D2-G2-D3 played on a Steinway Grand Piano Ringo Starr: subtle snare drum and ride cymbal This gives the notes: G-B-D-F-A-C (the B is a harmonic). One of the interesting things about this chord (as described by Pedler) is how McCartney's high bass note reverberates inside the soundbox of Lennon's acoustic guitar and begins to be picked up on Lennon's microphone or pick-up during the sounding of the chord. This gives the chord its special "wavy" and unstable quality. Pedler describes the effect as a "virtual pull-off". In 2004, Jason Brown, a mathematics professor at Dalhousie University, published a report titled "Mathematics, Physics and 'A Hard Day's Night'", in which he analysed the properties of the song's opening chord using Fourier transforms. He concluded that Martin's piano contribution provided the important element in the chord beside Harrison's playing. In November the following year, Wired published an article on Brown's use of Celemony's Melodyne Editor with Direct Note Access technology to further analyse the chord. Brown's findings were partly challenged in 2012 by another mathematician, Kevin Houston from the University of Leeds. Houston, who also used a Fourier transform, attributed a greater importance in Lennon's contribution on acoustic guitar, rather than the piano notes played by Martin. Harrison played a repeated guitar arpeggio, outlining the notes of the opening chord, thereby ending the song in a circular fashion. Martin said that the ending was his idea: "Again, that's film writing. I was stressing to them the importance of making the song fit, not actually finishing it but dangling on so that you're into the next mood." The song contains 12 other chords. Music The song is composed in the key of G major and in a time signature. The verse features the VII or major subtonic chord that was a part of the opening chord as an ornament or embellishment below the tonic. Transposed down a perfect fifth, the modal frame of the song though pentatonic features a ladder of thirds axially centred on G with a ceiling note of B and floor note of E (the low C being a passing tone). According to Middleton, the song, "at first glance major-key-with-modal-touches", reveals through its "Line of Latent Mode" "a deep kinship with typical blues melodic structures: it is centred on three of the notes of