# Local Consonance and the Relationship

Between Timbre and Scale

### by William A. Sethares

The principle of local consonance is based on an explicit parameterization of Plomp and Levelt's consonance curves. It explains the relationship between the spectrum of a sound (its timbre) and a tuning (or scale) in which the timbre will appear most consonant. This relationship is defined in terms of the local minima of a family of dissonance curves. For certain timbres with simple spectral configurations, dissonance curves can be completely characterized, and bounds are provided on the number and location of points of local consonance. Computational techniques are presented which answer two complementary questions: Given a timbre, what scale should it be played in? Given a desired scale, how can appropriate timbres be chosen? Several concrete examples are given, including finding scales for nonharmonic timbres (the natural resonances of a uniform beam, ``stretched'' and ``compressed'' timbres, FM timbres with noninteger carrier-to-modulation ratios), and finding timbres for arbitrary scales.

This paper first appeared in the September 1993 issue of the Journal of the Acoustical Society of America and can be downloaded here, and a somewhat simpler version (one without mathematics) appeared in the December 1993 issue of *Experimental Musical Instruments* and is now available online, and there are computer programs to help you draw dissonance curves. These ideas foreshadow many of the key ideas in *Tuning Timbre Spectrum Scale*, and are actualized in the CDs *XENTONALITY*. and Exomusicology.

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