  After our book---Mathematics and Music: Composition, Perception, and
Performance, 2nd edition, CRC Press, 2020---was published, we found a striking
example of the importance of the Tonnetz for analyzing the harmonic structure
of The Beatles' song, "In My Life." Our Tonnetz analysis will illustrate the
highly structured geometric logic underlying the numerous chord progressions in
the song. Spectrograms provide a way for us to visualize chordal harmonics and
their connection with voice leading. We shall also describe the interesting
harmonic rhythms of the song's chord progressions. A lot of this harmonic
rhythm lends itself well to a geometric description.
