Estimation of harpsichord inharmonicity and temperament from musical recordings

The inharmonicity of vibrating strings can easily be estimated from recordings of isolated tones. Likewise, the tuning system (temperament) of a keyboard instrument can be ascertained from isolated tones by estimating the fundamental frequencies corresponding to each key of the instrument. This paper addresses a more difficult problem: the automatic estimation of the inharmonicity and temperament of a harpsichord given only a recording of an unknown musical work. An initial conservative transcription is used to generate a list of note candidates, and high-precision frequency estimation techniques and robust statistics are employed to estimate the inharmonicity and fundamental frequency of each note. These estimates are then matched to a set of known keyboard temperaments, allowing for variation in the tuning reference frequency, in order to obtain the temperament used in the recording. Results indicate that it is possible to obtain inharmonicity estimates and to classify keyboard temperament automatically from audio recordings of standard musical works, to the extent of accurately (96%) distinguishing between six different temperaments commonly used in harpsichord recordings. Although there is an interaction between inharmonicity and temperament, this is shown to be minor relative to the tuning accuracy.