Inharmonicity

Harmonicity, inharmonicity or Teiltonverstimmung refers to a phenomenon of vibrating strings. But it's generally a feature of acoustic signals monophonic, polyphonic or complex nature.

Harmonicity describes exactly how the frequencies of the harmonics of a sound at the multiples of the fundamental frequency. If the harmonicity large, the acoustic signal sounds very pure and static, with less harmonicity of sound contains more beat and vitality.

Closely related to this is the inharmonicity, which can be calculated directly from the Harmoniziät. Harmonicity and inharmonicity give a good information about the internal structure of the harmonics - this generally applies to sounds. For the identification of musical instruments, this feature can be used well. Closely related to this is the noise-likeness.

Physical development

A vibrating string

Putting a string to vibrate, so it comes to a very complex movement of the string: She swings both in its entire length as well as in sections and that in its half- length, in their third length, in their quarter- length, etc These sections oscillate faster than the whole string and twice as fast, three times as fast as 4 times quickly, etc. The vibration of their entire length generates the root or first partial tone, the respective portions of the harmonics, ie the second partials, the 3rd partial, 4 partials, etc.

The frequencies of the partials thus behave like the integers, ie 1: 2: 3: 4 ...

In practice, however, one encounters the phenomenon that the overtones vibrate faster and thus sound higher than they should in theory. This phenomenon, which is called inharmonicity or Teiltonverstimmung was announced at the end of the 19th century.

Inharmonicity is dependent on the diameter, length, frequency, and the modulus of elasticity of the string material. Its value is proportional to the square of the diameter of the string, is inversely proportional to the fourth power of the length change and inversely proportional to the square of the frequency change.

In practice, this means that the Inharmonizitätswert rises, the shorter, thicker, and less tense the string is, the more " stiff " it is. A high value for the inharmonicity means that the string is in itself "wrong" or at least " bad " sounds. Piano strings have especially in the bass and treble on significant Inharmonizitätswerte. The fact that the deepest bass tones from a low Pianos - with relatively short, but thick bass strings - audible for the layman "worse" sound than that of a long concert grand piano, finds its reason very much in the inharmonicity.

Vibrating body in general

Analysis of the acoustic resonance vibrating bodies shows that each body produces its own vibration pattern is dominated mainly by a fundamental or natural frequency, in the simplest case. Body can be excited to oscillations different properties, wherein the ratio of the vibration is determined with each other individually on the nature of the vibrating body. Normally, these natural vibrations to each other in any whole number ratio. The sound pattern is in this case usually consists of a mainly perceptible vibration of the fundamental frequency and many more vibrations, which can also be considered as harmonics, although they need not be in harmonious relation with the root. Each natural frequency has, however, again for an individual harmonic spectrum, which leads to a complex sound image.

Musical meaning

To a certain extent, the inharmonicity but also responsible for the vitality of the piano sound. A favorable course of inharmonicity can be achieved by an appropriate design and the optimal computational determination of the scale length ( length, diameter and tension of each string ). The piano tuner must be able to assess the inharmonicity of a piano aurally so far that despite these "false harmonics " the instrument finally sounds good. From what has been said, there is also that one hundred percent accurate combination of two voices not of identical pianos is not possible.

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

Mode coupling

For stringed instruments pizzicato, the plucked strings on strong inharmonicity. They disappear in coated strings, because the arcuate movement and the non-linear stick-slip inducing a mode coupling on the string, the string motion is practically exactly periodic. Mode coupling is also available in reed instruments such as the clarinet. For a description of the overtone spectrum therefore is usually sufficient to each other, a simple model with overtones in exactly the integral ratios.

Mathematical Description

A demarcated time range of harmonics. Amplitude of the th harmonic. is the total number of filter bands and the index on a single band of frequencies that are obtained by short time Fourier transform, and identified as correlated samples. Already, however, the recovery of these bands requires extremely complex process.

Harmonicity:

Inharmonicity:

The area which is covered by harmonicity is less than 0.1.