Beauty and Tragedy In the Mathematics of Music Some things are
too good to be true. Others are too good not to be true. The
elegant correspondence between musical consonance and simple
mathematical ratios is too good not to be true. The infinitely
creative system of musical harmony—based on a small set of
canonical intervals and chordsarises out of this correspondence.
And yet the system has a flaw—a rip or tear in the musico-
mathematical universe. The most consonant musical interval is
the octave, so much so that two tones an octave apart bear the
same letter name, for example, A. The next most consonant
interval is the fifth (for example, A to E). If you pluck the A
string on a guitar and then pluck it again while placing your
finger halfway up the fingerboard (allowing only half the string
to vibrate and doubling the frequency), the pitch goes up by an
octave to A. If instead you place your finger one-third the way
up the fingerboard (allowing only the remaining two-thirds of
the string to vibrate and incr