Inharmonicity is a well-known property of stiff strings such as those used in the modern piano. Effects on piano tuning (e.g., the stretched octave) have been suggested but have never been fully investigated. We have measured the inharmonicities of the strings of a medium-sized grand piano. The measured inharmonicities are in excellent correspondence with the predictions by formula from the physical properties of the strings. Strings with higher frequencies usually have higher inharmonicity than strings with lower frequencies; this cancels out part of the effect of inharmonicity on beat frequency.
Short musical fragments consisting of a melody part and a synchronous bass part were mistuned in various ways and in various degrees. Mistuning was applied to the harmonic intervals between simultaneous tones in melody and bass (harmonic mistuning), which caused at the same time a mistuning of the melodic intervals between successive tones in the melody part (melodic mistuning of melody) and/or the bass part (melodic mistuning of bass part). The fragments were presented to musically trained subjects for judgments of the perceived quality of intonation. Results showed that the melodic mistuning of the melody parts had the largest disturbing effects on the perceived quality of intonation, followed closely by the harmonic mistuning. Melodic mistuning of the bass was less influential. It could be reasoned that the deviating interval size was probably of more importance in the perception of harmonic mistuning than the presence of beats.
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.