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).
Not to be confused with Enharmonicity or Anharmonicity.
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.
Guitar[edit]
While piano tuning is normally done by trained technicians, guitars such as acoustic guitars, electric guitars, and electric bass guitars are usually tuned by the guitarist themselves. When a guitarist tunes a guitar by ear, they have to take both temperament and string inharmonicity into account. The inharmonicity in guitar strings can "cause stopped notes to stop sharp, meaning they will sound sharper both in terms of pitch and beating, than they "should". This is distinct from any temperament issue." Even if a guitar is built so that there are no "fret or neck angle errors, inharmonicity can make the simple approach of tuning open strings to notes stopped on the fifth or fourth frets" unreliable. Inharmonicity also demands that
some of the "octaves may need to be compromised minutely."
[6]
When strobe tuners became available in the 1970s, and then inexpensive electronic tuners in the 1980s reached the mass market, it did not spell the end of tuning problems for guitarists. Even if an electronic tuner indicates that the guitar is "perfectly" in tune, some chords may not sound in tune when they are strummed, either due to string inharmonicity from worn or dirty strings, a misplaced fret, a mis-adjusted bridge, or other problems. Due to the range of factors in play, getting a guitar to sound in tune is an exercise in compromise. "Worn or dirty strings are also inharmonic and harder to tune", a problem that can be partially resolved by cleaning strings.[1]
Some performers choose to focus the tuning towards the key of the piece, so that the tonic and dominant chords will have a clear, resonant sound. However, since this compromise may lead to muddy-sounding chords in sections of a piece that stray from the main key (e.g., a bridge section that modulates a semitone down), some performers choose to make a broader compromise, and "split the difference" so that all chords will sound acceptable.
Mode-locking[edit]
Other stringed instruments such as the violin, viola, cello, and double bass also exhibit inharmonicity when notes are plucked using the pizzicato technique. However, this inharmonicity disappears when the strings are bowed, because the bow's stick-slip action is periodic,[7] driving all of the resonances of the string at exactly harmonic ratios even if it has to drive them slightly off their natural frequency. As a result, the operating mode of a bowed string playing a steady note is a compromise among the tunings of all of the (slightly inharmonic) string resonances, which is due to the strong non-linearity of the stick-slip action.[1] Mode locking also occurs in the human voice and in reed instruments such as the clarinet.[7]