Theory of Helmholtz-Beat Frequencies

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Abstract
In musical tuning, deviations from the simple frequency ratios of pure consonant intervals are often necessary. These deviations are called temperings. They result in beats in the sounding interval. Rules are developed according to which the beat frequencies can be determined, both exactly and by way of easy integer approximations. Beat frequencies of consonant intervals are most easily expressed as relative beat frequencies, the quotient of the beat frequency and the lower fundamental frequency of the interval. The relative beat frequency is a constant for a certain interval in a certain tuning, whereas the absolute beat frequencies vary with fundamental frequencies. Also described are the relationships between the beat frequencies of the three intervals that make up a consonant triad. Numerical data are given for five model tunings: Pythagorean, equal, Silbermann, meantone, and Salinas.

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