Inharmonicity

In music, inharmonicity is the degree to which the frequencies of overtones (also known as partials or partial tones) depart from whole multiples of the fundamental frequency (harmonic series).

Acoustically, a note perceived to have a single distinct pitch in fact contains a variety of additional overtones. Many percussion instruments, such as cymbals, tam-tams, and chimes, create complex and inharmonic sounds.

Music harmony and intonation depends strongly on the harmonicity of tones. An ideal, homogeneous, infinitesimally thin or infinitely flexible string or column of air has exact harmonic modes of vibration.^[1] In any real musical instrument, the resonant body that produces the music tone—typically a string, wire, or column of air—deviates from this ideal and has some small or large amount of inharmonicity. For instance, a very thick string behaves less as an ideal string and more like a cylinder (a tube of mass), which has natural resonances that are not whole number multiples of the fundamental frequency.

However, in stringed instruments such as the violin, and guitar, or in some Indian drums such as tabla,^[2] the overtones are close to—or in some cases, quite exactly—whole number multiples of the fundamental frequency. Any departure from this ideal harmonic series is known as inharmonicity. The less elastic the strings are (that is, the shorter, thicker, smaller tension or stiffer they are), the more inharmonicity they exhibit.

When a string is bowed or tone in a wind instrument initiated by vibrating reed or lips, a phenomenon called mode-locking counteracts the natural inharmonicity of the string or air column and causes the overtones to lock precisely onto integer multiples of the fundamental pitch, even though these are slightly different from the natural resonance points of the instrument. For this reason, a single tone played by a bowed string instrument, brass instrument, or reed instrument does not necessarily exhibit inharmonicity.^[1]

However, when a string is struck or plucked, as with a piano string that is struck by its hammer, a violin string played pizzicato, or a guitar string that is plucked by a finger or plectrum, the string will exhibit inharmonicity. The inharmonicity of a string depends on its physical characteristics, such as tension, stiffness, and length. For instance, a stiff string under low tension (such as those found in the bass notes of small upright pianos) exhibits a high degree of inharmonicity, while a thinner string under higher tension (such as a treble string in a piano) or a more flexible string (such as a gut or nylon string used on a guitar or harp) will exhibit less inharmonicity. A wound string generally exhibits less inharmonicity than the equivalent solid string, and for that reason wound strings are often preferred.

The physical origin of this inharmonicity is the dispersion of waves in a stiff string. In an ideal flexible string, the wave speed is constant as a function of frequency. Looking at the resonant frequency of a string with two fixed ends, this means that the frequency of the harmonics increases linearly with the mode number. The added dispersion due to the stiffness, which is most prevalent in the thick bass strings, means that as the frequency increases, so too does the wave speed in the string. The result is that modes of the stiff string are no longer perfectly harmonic.