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.