# Inharmonicity

Harmonicity , a series of partial tones

Harmonicity , inharmonicity or partial detuning refers to a phenomenon of vibrating strings . But it is also a general characteristic of acoustic signals of monophonic , polyphonic or complex nature.

Harmonicity describes how precisely the frequencies of the harmonics of a sound are multiples of the fundamental frequency . If the harmonicity is high, the acoustic signal sounds very pure and is described as "static"; if the harmonicity is lower, the sound contains more beat and is described as "lively". ${\ displaystyle a _ {\ mathrm {H}}}$${\ displaystyle f _ {\ mathrm {H}}}$${\ displaystyle \ mathrm {H}}$${\ displaystyle h \ cdot f_ {0}}$

Closely related to this is the inharmonicity , which can be calculated directly from the harmonicity. Harmonicity and inharmonicity provide good information about the internal structure of the harmonics - this applies to sounds in general. This feature can be used well for the identification of musical instruments. Intoxication is closely related to this. ${\ displaystyle a _ {\ mathrm {IH}}}$

## Physical origin

### Vibrating string

If you make a string vibrate, the movement of the string is very complex: it vibrates both in its entire length and in sections, namely in its half length, in its third length, in its quarter length, etc. These sections vibrate faster than the whole string, twice as fast, three times as fast, four times as fast, etc. The oscillation of its entire length produces the fundamental or 1st partial , the respective sections the overtones , i.e. the 2nd partial, the 3rd partial, the 4th partial etc.

The frequencies of the partials behave like the whole numbers, i.e. 1: 2: 3: 4 ...

In practice, however, one encounters the phenomenon that the overtones vibrate faster and thus sound higher than they should theoretically. This phenomenon, called inharmonicity or partial detuning, became known at the end of the 19th century.

The inharmonicity depends on the diameter, length, frequency and modulus of elasticity of the string material. Its value is proportional to the square of the diameter of the string, inversely proportional to the 4th power of the change in length and inversely proportional to the square of the change in frequency.

In practice this means that the inharmonicity value increases, the shorter, thicker and weaker the string, the "stiffer" it is. A high value for the inharmonicity means that the string itself sounds “wrong” or at least “bad”. Piano strings show significant inharmonicity values, especially in the bass and treble . The fact that the deepest bass notes of a low piano - with relatively short but thick bass strings - sound audibly “worse” even to the layman than on a long concert grand piano is essentially due to the inharmonicity.

### Vibrating bodies in general

The acoustic resonance analysis of vibrating bodies shows that each body generates its own vibration pattern, which in the simplest case is mainly dominated by a fundamental vibration or natural vibration . Bodies can be excited to different natural vibrations, the relationship of the vibrations to one another being determined individually by the nature of the vibrating body. Normally, these natural vibrations are not in an integral relationship to one another. In this case, the sound pattern usually consists of a mainly perceptible oscillation of the basic oscillation and many other oscillations that can also be viewed as overtones , although these do not have to be in a harmonic relationship with the basic tone. However, each natural oscillation has its own individual overtone spectrum, which leads to a complex sound image.

## Musical meaning

To a certain extent, however, the inharmonicity is also responsible for the liveliness of the piano sound. A favorable course of the inharmonicity can be achieved by the appropriate construction or the optimal calculation of the scale length (length, diameter and tension of the individual strings). The piano tuner must be able to assess the inharmonicity of a piano aurally to such an extent that the instrument sounds good despite these “wrong partials”. From what has been said so far, it also follows that a one hundred percent exact matching of two pianos that are not structurally identical is not possible.

Simple electronically generated tones have no inharmonicity and therefore sound unnatural. This hearing impression can be significantly improved by additionally generated inharmonic frequencies.

## Mode coupling

On string instruments , the plucked strings in pizzicato show strong inharmonicity. This disappears with bowed strings, because the bow movement and the non-linear stick-slip effect induce a mode coupling on the string , the string movement is practically exactly periodic. There is also mode coupling with reed instruments such as the clarinet. To describe the overtone spectrum, a simple model with overtones in exact integer ratios is sufficient .

## Mathematical description

A delimited time range of harmonics . Amplitude of the -th of harmonics . is the total number of filter bands and the index on a single band of frequencies obtained by short-time Fourier transform and recognized as correlated samples. However, even the extraction of these tapes requires extremely complex processes. ${\ displaystyle H}$${\ displaystyle m}$${\ displaystyle h}$${\ displaystyle H}$${\ displaystyle A_ {m} (h)}$${\ displaystyle K}$${\ displaystyle k}$

Harmonicity:

${\ displaystyle a _ {\ mathrm {H}} (m) = 1 - {\ frac {2} {f_ {0}}} \ left ({\ frac {{\ sum _ {k = 0} ^ {K} } {\ vert f_ {k} -k \ cdot f_ {0} \ vert \ cdot A_ {m} ^ {2} (k)}} {{\ sum _ {k = 0} ^ {K}} A_ { m} ^ {2} (k)}} \ right) \,}$

Inharmonicity:

${\ displaystyle a _ {\ mathrm {IH}} (m) = 1-a _ {\ mathrm {H}} (m) \,}$

The area covered by harmonicity is less than 0.1.

${\ displaystyle f_ {a} -k \ cdot f_ {0} \ leq f_ {a} \,}$