Overtone

An overtone, even partial tone or partial tone [n 1] is a sound that resonates with a multiple of the frequency of a fundamental tone. The composition of the overtones makes the characteristic of the sound or timbre of most instruments.

• 3.1 The harmonic series 3.1.1 as sheet music sample
• 3.1.2 as a table

• 3.2.1 inharmonicity
• 3.2.2 Noise shares
• 3.2.3 blur

• 4.1 harmonics of the human voice
• 4.2 overtones of different instruments
• 4.3 Effect of harmonics: brilliance and dullness

• 6.1 The organ and its register
• 6.2 Residualtöne

Basics

Examining the sounds generated by acoustic instruments or vocals, so these are never simple sine tones [n 2], but they are composed of a root, which is perceived as pitch, and mitklingenden overtones together:

Listen? / I

? Listen / i hear: the fourth overtone of cis '''' alone / i?

In the figure above, the large sine wave on the left represents the root note; pictured right next to superimpose overtones in the form narrower sine waves, the big wave. In aerophones such as the flute and the violin Chordophonen as are the frequencies of the overtones approximately integer multiples of the fundamental frequency. This means that a fundamental frequency of 440 Hz harmonics of the frequencies can be mixed with approximately 880 or 4400 Hz, but not, for example, 550 Hz in the musical acoustics, this refers to the harmonics. Wherein other sound sources ( for example, in tubes, rods, plates or bubble ) and harmonics appear, which frequencies have non-integral ratios of the fundamental frequency. The human listeners find it here all the more difficult to recognize a musical pitch.

While partials are components of an overall sound that is produced by excitation of all or several of the possible natural oscillations of a vibratory body are stimulated at the conceptually related natural tones of wind instruments by so-called overblowing individual harmonics, which are then perceived as the root. The same applies to the Harmonics in stringed instruments.

Depending on the sound source, the composition of the overtones one, so that in addition to noise and their factors in the temporal profile of the signal is very specific, especially the harmonic content for the characteristic tone of musical instruments as well as human and animal voices responsible.

Harmonic

A composite of harmonic partials sound event is referred to as sound in acoustics as sound and music, and is listed as a note.

Harmonics are always in relationship to the fundamental frequency. How exactly this relationship is described, depends on the chosen mathematical model. The choice of the fundamental frequency is objectively difficult to determine and is determined in terms of music primarily from the perceived or quoted root. In the analysis or synthesis of sound events can be chosen differently from acoustic or measurement point of view, the fundamental frequency. Fundamental and harmonics must therefore always be understood in context. In many cases, however, reaches a simple descriptive model that assumes the frequencies of the harmonics as integer multiples of a perceived as a sound fundamental frequency.

For example, pitch a ' and the first four harmonics

This table shows the root note a 'and its first three harmonics with their respective order n and their frequencies. The nth harmonic has generally the frequency n · f

The root is the first harmonic, an octave above the 2nd harmonic, which is the first overtone. [N 1]

The simple harmonic model - harmonic series

This model is known since ancient times and represents a " first approximation " dar. for musical practice, such as the over-blowing of wind instruments, playing Flageoletttönen on stringed instruments, the overtone or the organ registry, this model is sufficient. In the physical examination and the electronic simulation of sounds, however, this model has its limitations.

The human ear takes periodic oscillations as true tones (in the sense of musical tones ), where the period of oscillation determines the perceived pitch. An analysis of the amplitude spectrum of an audio signal of approximately periodic oscillations, eg by means of the short-time Fourier transform, so it is composed of

• A tone corresponding to the period of oscillation,
• And frequency, the integral multiple of the fundamental frequency corresponding to the harmonic overtones.

The overtone series

The following are the first 16 related to the key of C partials are shown as examples. This restriction is arbitrary for the sake of clarity. Theoretically, the partial tone series is continued upward with ever decreasing distances to infinity.

As sheet music sample

Here the partials in the table above notes are displayed. It should be noted that because of the continuously decreasing upward pitch intervals an exact reproduction in musical notation (at least in the higher range of the partial tone series ) come close (and eventually not more) is possible. Also, not all agree overtones with the pitches of common tuning systems. In the following example, note the overtones with the sounds of equal temperament tuning are compared. The deviations up or down are indicated in cents.

While at the same stage mood except the root note and its octaves No sound exactly matches the partial tone series, the differences in just intonation are much rarer. In the following example, note the deviant out of tune harmonics are characterized and compared with those given in brackets sounds of just intonation.

As a table

The colors used in the table are based on the music -color synesthesia.

Table footnotes

From the last row of the table shows that it is possible (see just intonation ) of the overtone derive all the intervals of the diatonic scale. In particular: Halftone ( frequency ratio of 16/15 ), large and small whole tone (9/ 8 and 10 /9), minor third ( 6/5), major third ( 5/4), fourth ( 4/3), fifth (3 / 2) and octave ( 2/1).

Limitations of the simple model

Many instruments or vowels of the human voice is an essential part of the sound of periodic oscillations, which can be described with the simplified model representation of the fundamental and harmonic overtones to a good approximation, as for example in vibrating strings of stringed instruments ( Chordophonen ) or vibrating air column of wind instruments ( aerophones ). However, this contact in reality more or less significant deviations from the theoretical integrality of the overtones.

Inharmonicity

Deviations from the harmonic ratios of the partials are individually depending on the type of instrument. These known under the term inharmonicity deviations are, for example, the piano mainly due to the high string tension due and make the mood a so-called stretching required. The high string tension caused at least for the bass strings very thick strings. Thicker and shorter strings have a higher proportion of inharmonicity. The more detailed analysis of such harmonics is much more complex and requires more complex models to describe than the analysis and description of very harmonic tones. (See also the audio signal )

Noise components

In addition, non- periodic oscillations occur which have a rather broad band frequency spectrum and can not be described by the fundamental and harmonic overtones, such as stop sounds stringed instruments, wind instruments and organ pipes blowing noise and consonant with the human voice. The analysis of these sound components requires advanced electronic measurement technology and mathematical models whose solutions are only possible with powerful computers.

Fuzziness

Mathematically, only sinusoidal oscillations are if they continue and persist indefinitely. Vibrations are in practice only quasi-periodic or nearly periodic. The sine function extends to infinity on both sides and cutting the duration leads mathematically to something else, a time-limited wave. Werner Heisenberg shapes for the term ' Wellikel ', or ' wave - particle ' to draw attention to the unimaginable this term content. This short-lived oscillations do not sharply demarcated from each other overtones, but a smeared band. In psychoacoustic consequences arise when cutting long-running continuous, static sine tones or broadband Sinustongemischen artifacts.

For short -running operations of this kind - as they occur in all instruments where not always energy will be submitted, ie to the plucked and percussion instruments (including the piano ) - the basic requirement of the continuous signal is still not even given in approximation.

In the culture of engineering you went most of the situation that operations are long-lasting and slowly varying ( in the modulation of a radio station that is the case ). Only then will yield the Fourier transform and the resulting implicit in the article following terms make sense. Only in recent decades have come to realize prevailed that in rapidly changing and short -running operations, the wavelet transform needs to be applied, what terms such as frequency must be re- interpreted. For fundamental tracking are now a variety of different methods in use.

Music involves much such operations. In this respect, criticism is also from this point of view to practice on traditional ideas. Too much our ideas are shaped by the electronics in many areas completely sufficient today spreading models. That one of the complex relationships already before Hermann von Helmholtz, a mathematical theory to explain the timbre by harmonics in the doctrine of the Sensations of Tone as a Physiological Basis for the Theory of Music ( 1863) published, was aware shows an excerpt: The music and musical instruments: in their relation to the laws of acoustics, Friedrich Georg Karl Zamminer, 1855, page 176 " All sounding body, which their substance, their shape, their may be elastic and voltage condition, except the vibrations in all the earth, which give the fundamental, yet infinitely many Abtheilungsarten and just as many overtones capable. The vibrational states of which they are able to accept, are so varied, the less easy it is their shape. Only cylindrical and prismatic columns of air and similar to this vibrating rods of small diameter have such a simple harmonic series as the stretched strings; far richer already is the amount of harmonics in bodies which, like plates and stretched skins, spread in a flat or curved surface, the most varied of any extended in every sense of solid masses and airspace. "

Overtones and timbre

Harmonics of the human voice

In the human voice vibrates exactly as in most sound-producing physical systems, a complex overtone spectrum. In the special vocal technique of overtone singing can bring these high frequencies to Dominate.

The different sounds of vowels comes about through their specific Obertonaufbau. Due to the individual size and shape of the mouth and throat, some frequencies are amplified by resonance, others subdued. The frequency ranges which are each reinforced, is also known as formants.

Harmonics of different instruments

The specific sound of an instrument is derived from the following parameters:

• Which harmonics are present at all
• How in relation to each other according to these harmonics
• As the volume and frequency of the individual harmonics changes, while the sound is heard
• What added noise ( impact noise, blowing noise ... )

The following instruments have a particularly characteristic Teiltonaufbau:

• String instruments have a very rich partial tone spectrum
• Clarinets emphasize the volume of the odd partials
• When bassoon the fundamental is much weaker than the first overtones
• Bells often emphasize the thirds very strong and the Obertonzusammensetzung is complex
• Tuning forks almost only produce the root

For instruments with simple Obertonzusammensetzungen are the frequencies of the overtones approximately integer multiples of the frequency of the fundamental. These include the chordophones ( stringed instruments ) and the aerophones with a vibrating column of air. This is of course only an idealized assumption; so there is a inharmonicity in real (not infinitely thin ) strings. Especially the very small deviations from the ideal harmonic make the sound of a single instrument distinctive and alive.

For most woodwind instruments which is very close to the idealized assumption for many stringed instruments does this quite well. In piano, however, the integer frequency ratio is only approximately satisfied. In particular, the very high harmonics are quite far apart from the frequencies with integer ratios to the root. The higher one ascends the ladder of the overtones, the more soft the frequencies of the harmonic exactly. It has even been found that the piano 's timbre is related very much with this deviation from the exact harmonic overtones. For example, to imitations of a piano listen to piano not particularly similar, if this deviation of the harmonic series is not taken into account in the artificial production of sound.

The natural frequencies and their harmonic overtones depend on the tone generator, and are determined by the dimensions and nature of the body. There are tools in which let the Obertonzusammensetzungen relatively easy to describe, and others that require very complex description models. For instruments with complex Obertonzusammensetzungen many frequencies of the harmonics to each other in complicated non-integer ratios. The overtones of the Membranophones with round diaphragm have the natural frequencies of a Bessel 's differential equation. In idiophones may differ depending on the form of the ensemble give very different overtone series - with the mallet as they are the natural frequencies of the bending vibration of a beam.

Man-made from pure tones overtone spectra are called synthetic sounds (see Sound Synthesis, synthesizers ). A pure sawtooth wave is characterized in that it contains all of its harmonics to the fundamental, which is why she prefers began in the days of analog electronic musical instruments as output swing.

Effect of harmonics: brilliance and dullness

The amount of harmonics in the overall spectrum and the resulting timbre by words such as brilliance, sharpness, purity, dullness, inter alia, are described.

In general, sound sounds more brilliant (violin), sharper (trumpet ) or colored (oboe, bassoon), the more overtones they have, and the purer and clearer (flute ) or pale or dull (deep clarinet, Indoor organ stops ), depending less they have.

Pure tones without overtones which are sine tones, can virtually never be generated. Approximation, they can be produced by mechanical means only with very low sound levels ( tuning fork or cavity resonators, very gently picked up). Electronically generating approximately pure sinusoidal tones is possible. They sound dull, wide and flowing at a lower frequency, certain organ stop coming to the close. At higher frequencies, the difference to the sound with overtones is lower because these harmonics are outside the audible range. An example of the situation for medium frequencies is the 1,000 -hertz tone of the television test pattern, but the speakers adds his own harmonic spectrum again by its distortions. Since the total energy occurs only in a narrow frequency range, high-output sine waves can be very unpleasant. Ever sine tones are a touchstone for each speaker, as the risk of electrical and mechanical overload is very high on the one hand, on the other hand noticeable distortion products with audible levels immediately and mechanical design problems to be disclosed with sometimes buzzing or hissing resonances.

In a multi-way speakers ( electroacoustics ) is primarily responsible for the brilliance of the tweeter, so the sound brightness and tone playback.

Higher harmonics are of mechanical musical instruments usually quieter (level weaker) than lower.

• To a higher frequency in the mechanical sound generators are only stimulated much weaker than lower (for example, takes on a vibrating string, the vibration amplitude of the harmonics with increasing frequency from ).
• On the other hand, higher frequencies are attenuated more in the air. Therefore, in a sound over large areas the brilliance of the play is usually relatively poor.

Audibility of harmonics

Usually overtones are not perceived individually, but give the sound of a note. In certain cases or under special conditions but they can also individually be heard or made ​​audible.

• The vocal technique of overtone singing makes the overtones clearly perceptible. Examples are the Mongolian throat singing and Tuvinian peoples. Even in Western music has been around since the late sixties a revival of Obertonkultur.
• Also in the instrumental area you can make overtones clearly audible. Typical tools for this purpose are, for example, the didgeridoo, Fujara or singing bowls.
• For stringed instruments, sounds can be generated in the pitch of harmonics by harmonic - play (see Flageolettton ). The string with the fretting hand is only slightly affected rather than press on the fretboard.
• On the piano, one can make audible overtones in two ways: By depressing the keys of a chord from the harmonic series gently without touching the hammer, the string, and then strikes the key-note in the bass short and strong. The overtones now generate a response to the undamped strings of depressed holding the keys that you can hear clearly.
• By the manner described in silence presses down a button in the bass range, and then strikes one or more sounds from the corresponding overtone short and strong. By the undamped resonant bass string is made to vibrate at the frequencies of these overtones. The ailing tones sound more like an echo, although the associated strings were damped.

Applications

The organ and its register

Particularly important is the harmonic overtone series at the organ. Through various organ stops, each individual harmonic overtones, with few exceptions produce ( aliquots ), can be generated tones using a simple kind Additive Synthesis. In pipe organs only "on" or "off" the register is possible. The harmonic overtones most used are octaves ( 2nd, 4th, 8th, 16th, ... partial tone ), fifths ( 3, 6, 12, ... partial tone ) and major thirds ( 5th, 10th, Parti ... Alton ), in modern organs, the minor seventh ( 7th, 14th, ... partial tone ) and the major ninth (9th, 18th, ... partial tone ).

One of them inspired sound synthesis takes place at the Hammond organ. Here, the proportions of the partials can additionally be varied by slider.

Residualtöne

The human auditory cortex is able to perceive a ( even partially ) sounding overtones, the fundamental frequency, even if they will not sound. This " added " root is also called Residualton.

Music theory and didactics

The existence of harmonics was used for a scientific explanation and justification of tonal systems of music for a long time, on the understanding generally of the simple model of integer frequency or string length ratios.

• The first standing in connection with overtones theory is attributed to Pythagoras, this was around 2500 years ago.
• For didactic purposes ( teaching the accompaniment, figured bass, harmony and melody and composition theory ) is probably the first Johann Bernhard Logier ( 1777-1846 ) has made the overtone series advantage. His teaching " harmonious mitklingenden " of the tones was always controversial in his lifetime; his didactic works with high reflected their simple, build on each other basic rules, however, may be considered as the beginning of modern, still valid today music theory. ( See above all: JB Logier, system of music science and the practical Composition with epitome of what is usually meant by the expression General -Bass, Berlin 1827, p 11. Circle of fifths, basso. Pp. 15ff, from p. 53 begins the teaching of overtones. )
• One of the last attempts to justify a theoretical system of the overtone series and other acoustic phenomena ( eg, combination of tones) can be found in Paul Hindemith's instruction in music theory. Also Hindemith's system is very controversial in the scientific world. Real tones or sounds are today limited mathematically ascertainable, therefore, each system will eventually reach its limits. An aesthetic system is therefore to legitimize difficult science.

Undertone

Reflects the harmonic overtone series one, created the theoretical to their symmetrical harmonious undertone, which is caused by frequency division, downward supplements. In nature undertones are extremely rare; they sometimes occur with bells and gongs. It is not certain whether it is in fact a tones undertone. In practice, they are created when Trautonium, when subharchord and the undertone.

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