A geometry of music: harmony and counterpoint in the extended common practice

Abstract

Click to increase image sizeClick to decrease image size Notes The expression ‘extended common practice’, used by the author, is slightly but usefully misleading, since the technical sense refers to a musicological convention (roughly classical and romantic music), while the literal sense is all-inclusive but ill-defined. More scientific exposition of some of this material has appeared previously (see p. 116, footnote 1). They are actually singularities on a differentiable manifold, lying on a border. ‘My goal is not to provide a complete theory of all music, but rather to discuss a few general features whose musical importance is for the most part unquestioned’ (p. 10). Aware that this foundation, while sociologically very strong, may seem debatable to the more systemic-minded, Tymoczko is very modest when stating the psychoacoustic grounds of his system: ‘In this sense, it seems that at least some of the five features are necessary for tonality, at least as I personally understand it’ (Ibid.) Although in this domain, there is a precise, checkable definition of genericity (for instance what is true on a Baire dense open subset of the phase space). This work, first presented at the John Clough Memorial Conference in Chicago in 2005, later appeared in 6 Callender, C., Quinn, I. and Tymoczko, D. 2008. Generalized voice-leading spaces. Science, 320: 346–348. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar].