The deviation of a set of frequencies from an exact harmonic series. The most common musical application of the term is in discussion of the natural mode frequencies (or partials) of a stretched string. For a uniform, completely flexible string, the mode frequencies (or resonance frequencies) are members of the harmonic series represented by the following formula, in which n is the harmonic number (counting the fundamental as the first harmonic) and F is the fundamental frequency:
frequency of nth harmonic = nF
The stiffness always present in a real string increases the frequency of each mode by an amount which depends on the mode number. The resulting inharmonic series of mode frequencies is given, to a good approximation, by the following formula:
frequency of nth mode = nF(1 + bn2)
The inharmonicity coefficient b can be calculated if the properties of the string are known. For a string of fixed radius and sounding length, the inharmonicity diminishes as the tension is increased.
For the relatively supple strings normally used on bowed and plucked instruments, the inharmonicity coefficient is very small, and such strings can be treated for most musical purposes as having exactly harmonic natural mode frequencies. The inharmonicity of piano strings, on the other hand, is far from negligible. A typical bass string on an upright piano may have an inharmonicity coefficient b of 0·0002; in this case the stiffness will have an insignificant effect on the frequency of the first string mode, but will increase the frequency of the twentieth mode by nearly 8% (a sharpening of more than a semitone).
Since the frequency spectrum of the sound radiated by a plucked or hammered string consists of the inharmonic natural mode frequencies of the string, the degree of inharmonicity can have a significant effect on the perceived timbre of the sound. String inharmonicity is also generally considered to be one of the causes of the octave stretching found on well-tuned pianos.
O.H. Shuck and R.W. Young: ‘Observations of the Vibrations of Piano Strings’, JASA, xv (1943), 1–11
D.W. Martin and W.D. Ward: ‘Subjective Evaluation of Musical Scale Temperament in Pianos’, JASA, xxxiii (1961), 582–5
H. Fletcher: ‘Normal Vibration Frequencies of a Stiff Piano String’, JASA, xxxvi (1964), 203–9
C. Carp: ‘The Inharmonicity of Strung Keyboard Instruments’, Acustica, lx (1986), 295–9
P. Barbieri: ‘The Inharmonicity of Musical String Instruments (1543–1993) with an Unpublished Memoir by J.-B. Mercadier (1784)’, Studi musicali, xxvii (1998), 383–419
N.H. Fletcher and T.D. Rossing: The Physics of Musical Instruments (New York, 2/1998)
MURRAY CAMPBELL