Poverty and Polyphony: A Connection between Economics and Music

Rachel W. Hall and Dmitri Tymoczko
Bridges Donostia: Mathematics, Music, Art, Architecture, Culture (2007)
Pages 259–268


In polyphonic music, simultaneous melodic lines, or “voices,” create a sequence of vertical harmonies or “chords.” The melodic voices determine mappings, or voice leadings, from the notes of one chord to those of the next. While we have clear intuitions that some voice leadings are smaller than others, our intuitions are not robust enough to determine a precise “metric” of voice-leading size. Tymoczko (2006) proposed that any voice-leading metric should be consistent with two principles: (1) small voice leadings move their voices by short distances, and (2) small voice leadings move their voices along non-intersecting paths. We show that the partial order imposed on voice leadings by these constraints is equivalent to the submajorization partial order, originating in 1905 with the economist Lorenz. We further show how to use submajorization to compare distances between “chord types.” Finally, we highlight surprising connections between the results discussed in this paper and problems in welfare economics.