Inharmonicity

# Inharmonicity

Discussion

Encyclopedia
In music, inharmonicity is the degree to which the frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

of overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...

s (known as partials, partial tones, or harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

s) depart from whole
Whole number
Whole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts.Whole numbers may refer to:*natural numbers in sense — the positive integers...

multiples of the fundamental frequency.

Acoustically, a note perceived to have a single distinct pitch in fact contains a variety of additional overtones. Many percussion instruments, such as cymbal
Cymbal
Cymbals are a common percussion instrument. Cymbals consist of thin, normally round plates of various alloys; see cymbal making for a discussion of their manufacture. The greater majority of cymbals are of indefinite pitch, although small disc-shaped cymbals based on ancient designs sound a...

s, tam-tams, and chime
Chime
-Musical instrument or tone:* Chime , an array of large bells, typically housed in a tower and played from a keyboard.** An instrument of this kind with 23 bells or more is known as a carillon...

s, create complex and inharmonic sounds.

However, in stringed instruments such as the piano
Piano
The piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Widely used in classical and jazz music for solo performances, ensemble use, chamber music and accompaniment, the piano is also very popular as an aid to composing and rehearsal...

, violin
Violin
The violin is a string instrument, usually with four strings tuned in perfect fifths. It is the smallest, highest-pitched member of the violin family of string instruments, which includes the viola and cello....

, and guitar
Guitar
The guitar is a plucked string instrument, usually played with fingers or a pick. The guitar consists of a body with a rigid neck to which the strings, generally six in number, are attached. Guitars are traditionally constructed of various woods and strung with animal gut or, more recently, with...

, or in some Indian drums such as tabla
Tabla
The tabla is a popular Indian percussion instrument used in Hindustani classical music and in popular and devotional music of the Indian subcontinent. The instrument consists of a pair of hand drums of contrasting sizes and timbres...

, 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, and stiffer they are), the more inharmonicity they exhibit.

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 exactly harmonic modes of vibration. 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
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

(a tube of mass), which has natural resonances that are not whole number multiples of the fundamental frequency.

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 than 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.

However, when a string is struck or plucked, as with a piano string that is struck by a hammer, a violin string played pizzicato
Pizzicato
Pizzicato is a playing technique that involves plucking the strings of a string instrument. The exact technique varies somewhat depending on the type of stringed instrument....

, 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 wrapped string generally exhibits less inharmonicity than the equivalent solid string, and for that reason wrapped strings are often preferred.

### Sound quality of inharmonicity

In 1943, Schuck and Young were the first scientists to measure the spectral inharmonicity in piano tones. They found that the spectral partials in piano tones are progressively stretched—that is to say, the lowest partials are stretched the least and higher partials are progressively stretched further.

Inharmonicity is not necessarily unpleasant. In 1962, research by Harvey Fletcher and his collaborators indicated that the spectral inharmonicity is important for tones to sound piano-like. They proposed that inharmonicity is responsible for the "warmth" property common to real piano tones. According to their research synthesized piano tones
Synthesizer
A synthesizer is an electronic instrument capable of producing sounds by generating electrical signals of different frequencies. These electrical signals are played through a loudspeaker or set of headphones...

sounded more natural when some inharmonicity was introduced. In general, electronic instruments that duplicate acoustic instruments must duplicate both the inharmonicity and the resulting stretched tuning
Stretched tuning
Stretched tuning is a detail of musical tuning, applied to wire-stringed musical instruments, older, non-digital electric pianos , and some sample-based synthesizers based on these instruments, to accommodate the natural inharmonicity of their vibrating elements...

of the original instruments.

### Inharmonicity leads to stretched tuning

When pianos are tuned by ear by technicians called piano tuners
Registered Piano Technician
A Registered Piano Technician is an individual who has passed a series of technical examinations offered by the Piano Technicians Guild....

, the technician listens for the sound of "beating
Beat (acoustics)
In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....

" when two notes are played together. Piano tuners must deal with the inharmonicity of piano strings, which is present in different amounts in all of the ranges of the instrument, but especially in the bass and high treble registers. The result is that octaves are tuned slightly wider than the harmonic 2:1 ratio. The exact amount octaves are stretched in a piano tuning varies from piano to piano and even from register to register within a single piano—depending on the exact inharmonicity of the strings involved.

Because of the problem of inharmonicity, electronic piano tuning devices used by some piano technicians are not designed to tune according to a simple harmonic series. Rather, the devices use various means to duplicate the stretched octaves and other adjustments a technician makes by ear. The most sophisticated devices include an option to simply record a tuning that a technician has completed by ear; the technician can then duplicate that tuning on the same piano (or others of similar make and model) more easily and quickly.

Piano tuning is a compromise—both in terms of choosing a temperament
Musical temperament
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...

to minimize out-of-tuneness in the intervals and chords that will be played, and in terms of dealing with inharmonicity. For more information, see Piano acoustics
Piano acoustics
Piano acoustics are those physical properties of the piano which affect its acoustics.-String length and mass:The strings of a piano vary in thickness, and therefore in mass per length, with bass strings thicker than treble. A typical range is from 1/30 inch for the highest treble strings to 1/3...

and Piano tuning
Piano tuning
Piano tuning is the act of making minute adjustments to the tensions of the strings of a piano to properly align the intervals between their tones so that the instrument is in tune. The meaning of the term in tune in the context of piano tuning is not simply a particular fixed set of pitches...

.

Another factor that can cause problems is the presence of rust on the strings or dirt in the windings. These factors can slightly raise the frequency of the higher modes, resulting in more inharmonicity.

## Guitar

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" do. 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."
When strobe tuners became available in the 1970s, and then inexpensive electronic tuner
Electronic tuner
The term electronic tuner can refer to a number of different things, depending which discipline you wish to study.In the Discipline of radio frequency electronics an electronic tuner is a device which tunes across a part of the radio frequency spectrum by the application of a voltage or appropriate...

s in the 1980s hit 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.

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

Other stringed instruments such as the violin, viola, cello, and double bass also exhibit inharmonicity when notes are plucked using the pizzicato technique. However, the "inharmonicity disappears when the strings are bowed" because the "bow's stick-slip action is periodic, [so] it drives 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."
Mode locking also occurs in reed instruments such as the clarinet
Clarinet
The clarinet is a musical instrument of woodwind type. The name derives from adding the suffix -et to the Italian word clarino , as the first clarinets had a strident tone similar to that of a trumpet. The instrument has an approximately cylindrical bore, and uses a single reed...

.

• Anharmonicity
Anharmonicity
In classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in simple harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be...

• Electronic tuner
Electronic tuner
The term electronic tuner can refer to a number of different things, depending which discipline you wish to study.In the Discipline of radio frequency electronics an electronic tuner is a device which tunes across a part of the radio frequency spectrum by the application of a voltage or appropriate...

• Pseudo-octave
Pseudo-octave
A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1 , that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency...

• Stretched tuning
Stretched tuning
Stretched tuning is a detail of musical tuning, applied to wire-stringed musical instruments, older, non-digital electric pianos , and some sample-based synthesizers based on these instruments, to accommodate the natural inharmonicity of their vibrating elements...

• String resonance (music)
String resonance (music)
String resonance occurs on string instruments. Strings or parts of strings may resonate at their fundamental or overtone frequencies when other strings are sounded...