The Integrated Scale Desirability Function: A Musical Scale Consonance Measure Based on Perception Data

Richard J. Krantz and Jack Douthett
Bridges London: Mathematics, Music, Art, Architecture, Culture (2006)
Pages 183–190


Based on previous work by Krantz and Douthett [1, 2, 3] and Sethares [7, 8], briefly reviewed in Section 2, a universal measure of the equal-tempered musical scale consonance is developed in Section 3. Preliminary results applying this, so-called, Integrated Scale Desirability Function for scales made up of complex tones show that higher frequency components of complex tones are necessary to explain the emergence of our usual 12-tone equal-tempered musical scale. These results are discussed in Section 4. The formalism is also applied to the well-known Bohlen-Pierce scale. Here, too, it appears that higher frequency components are necessary for the emergence of the historically important 13-note to the “tritave” Bohlen-Pierce scale. These results are also discussed in Section 4. Subsequently, in Subsection 4.4, we show that complex interval spectra can be “sculpted” so that non-standard equal-tempered musical scales turn out to be consonant as measured by the Integrated Scale Desirability Function.