Frequency ratios of spectral components of musical sounds

A recently developed high resolution frequency tracker @J. C. Brown and M. S. Puckette, ‘‘A high resolution fundamental frequency determination based on phase changes of the fourier transform,’’ J. Acoust. Soc. Am. 94, 662–667 ~1993!# has made it possible to measure the ratios of the frequencies of the upper harmonics of a sound with respect to its fundamental frequency with high accuracy. Calculations were carried out on digitized sounds produced by a clarinet, alto flute, voice, piano, violin, viola, and cello. The sounds produced by the stringed instruments included examples played pizzicato and bowed both with and without vibrato. Measured ratios were exactly equal to integers for all instruments except the piano and strings played pizzicato. Anomalous behavior was observed for the fundamental frequency for vibrato sounds played by stringed instruments with the frequency deviation exceeding the extrema of the other harmonics divided by their harmonic number by about 1% on average. Piano inharmonicity was proportional to harmonic number squared in agreement with Fletcher @J. Acoust. Soc. Am. 36, 203–209 ~1964!#. The major limitation on this calculation was found to be instrumental fluctuations. © 1996 Acoustical Society of America.