I. Classifying Voice-leadings

Musical norms evolve over time. In the eleventh century parallel perfect fifths were tolerated, and perfect fourths were considered more consonant than thirds. In the eighteenth century, parallel perfect fifths were not tolerated, and thirds were considered more consonant than perfect fourths. And in the twentieth century, all manner of traditional prohibitions collapsed, as composers reveled in the sense that everything was permitted. Despite such changes, however, certain musical principles remain relatively constant across styles. One of the most important of these dictates that harmonies should, in general, be connected by efficient voice-leading. That is, notes should be distributed among individual musical voices so that no voice moves very far as harmonies change. Example 1 supplies a few representative passages. In each case, the music exploits the shortest possible path between successive chords, in a sense to be defined below. 1 Efficient voice-leading is not simply a matter of performers' physical comfort, though that is certainly a factor. It also enables listeners to segregate the auditory stimulus into a series of independent musical lines, which is a prerequisite for understanding polyphony. 2 Efficient voice-leading is so ubiquitous that we tend to take it for granted. But upon reflection it is rather remarkable that composers manage so consistently to find the shortest route from chord to chord. There are 57,366,997,447 distinct voice-leadings between two hexachords. 3 There are 288 bijective 4 voice-leadings from a half-diminished seventh chord to the twelve dominant seventh chords. And yet, almost any composer, theorist, or undergraduate harmony student can quickly find the most efficient voice-leading between two hexachords, or a maximally efficient bijective voice-leading from a particular half-diminished seventh chord to any of the twelve dominant seventh chords. The puzzle lies in the mismatch between the large number of possibilities and our apparent ease in sorting through them. 2 We can sharpen our puzzle by noting that students receive little explicit conceptual instruction in identifying efficient voice-leadings. Music teachers tend to provide examples of efficient voice-leading, augmented with relatively uninformative exhortations: we enjoin our students to obey the " law of the shortest way " —that is, to minimise the overall voice-leading distance between successive chords 5 —without telling them precisely what this means or how to do it. And students comply, easily and for the most part without complaint. How can this be? What is it that even the beginning harmony student knows, that allows …