  The musical realm is a promising area in which to expect to find nontrivial
topological structures. This paper describes several kinds of metrics on
musical data, and explores the implications of these metrics in two ways: via
techniques of classical topology where the metric space of all-possible musical
data can be described explicitly, and via modern data-driven ideas of
persistent homology which calculates the Betti-number bar-codes of individual
musical works. Both analyses are able to recover three well known topological
structures in music: the circle of notes (octave-reduced scalar structures),
the circle of fifths, and the rhythmic repetition of timelines. Applications to
a variety of musical works (for example, folk music in the form of standard
MIDI files) are presented, and the bar codes show many interesting features.
Examples show that individual pieces may span the complete space (in which case
the classical and the data-driven analyses agree), or they may span only part
of the space.
