Algebra of Harmony: Transformations of Just Consonances (original) (raw)

2017

Abstract

The paper focuses on mathematical aspects of harmonies in extended just intonation and their relations. The first part lays down a theoretical framework for the investigation of structural features of such harmonies. Among other aspects, it addresses symmetry, inversion, and multiplication of harmonies. The second part explores transformational relations among harmonies of the same type, while the approach is intrinsically dualistic. Riemann-Klumpenhouwer’s concepts of Schritts and Wechsels are generalized for ‘harmony spaces’ in extended just intonation. This enables a deeper analysis of harmonic ‘neighborhoods.’ Finally, a graphical representation of the complete neighborhood of a harmony, called ‘neighborhood network,’ is presented along with several simpler and more complex examples.

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