Musical instruments from a physicist's point of view

A musical instrument, from a physicist's point of view, is a system that generates sound waves and radiates them in the environment.

It can be broken down into the following elements:

1. a vibrating element (more appropriately a wave guide), which is the first source of oscillations in the elastic medium part of the instrument (strings, plates, membranes, air, etc.);
2. a resonator that selects some vibration frequencies and gives the sound waves a distinctive shape. For wind instruments, the bore is tuneable, for example by means of holes, while for string instruments, intonation occurs by acting directly on the vibrating element (the strings), while the resonator is the instrument's body;
3. one or more impedance adapters enhance the transmission of mechanical energy through the various vibrating parts of the instrument and, finally, from the instrument to the surrounding air;

The three elements indicated are functional and do not necessarily correspond to distinct structural elements in each instrument. Sometimes, they are grouped together in one single element and other times the final impedance adapter is missing (or not necessary) or several internal adapters are needed besides the final one that couples the instrument with the air.

instrument	vibrating element	medium	resonator	impedance adapters
violin	strings	strings, top, back, air	body	bridge, sound post
flute	air	air	bore with holes	not needed
oboe and clarinet	reed	air	bore with holes	various sections of the bore and bell
trumpet	lips	air	bore with pistons	bell
timpani	membrane and air	elastic membrane	kettle	not needed
triangle	triangle	steel bar	none	none

Our division does not show the pleasant characteristics that we attribute to the sound generated by a musical instrument. However, in its generality, it points out the chain of phenomena that lead to the formation of sound and highlights the acoustic particularities of each instrument. The physical description of these particularities is very general and involves many of the principles that we have developed in these pages.

Role of the vibrating element

Each musical instrument generates mechanical vibrations thanks to an element that can oscillate around an equilibrium position when it is excited by the player.

The initial vibrating element can be:

• a string (of metal, catgut or synthetic material) for string intruments, which can be plucked, struck, bowed, etc.;
• a reed (single or double, made of cane, metal or synthetic material) for wind instruments with reeds;
• the player's lips for brass wind instruments;
• direct air, as for the flute or pipe organ;
• a membrane (of catgut or synthetic material) for membrane percussion instruments;
• a wood or metal bar or plate, along with more exotic elements found in idiophone percussion instruments;
• the vocal chords of the human voice.

Instruments must be able to execute many notes, which means that each of them must have a selection mechanism for the oscillation frequency produced. This mechanism varies greatly, for example:

• for voice emission, the note frequency produced is the same as the vibration frequency of the vocal chords, which is tuned by using specific muscles to vary tension.
• Something similar occurs in brass instruments, where the performer can regulate both the tension of their lips and the pressure emitted by their breath.
• For stringed instruments, frequency is indirectly selected by varying the length of vibrating strings. For the piano, there is a wide range of pre-tuned strings, while for bowed string instruments, there are fewer strings and the performer varies the length of each with their fingers. This gives rise to the problem of intonation, which must be maintained throughout the entire performance (while tuning of free strings takes place before performing).
• For the guitar, intonation occurs as with the bowed strings by shortening the strings with the fingers, however, the presence of preset bars on the fingerboard guides the performer to play only fixed notes. There is also a guitar without frets, known as the "Hawaiian".
• For wind instruments with reeds, the oscillation source (the vibration produced by the reed) varies very little in frequency (see the acoustic examples for the oboe and clarinet) and selection occurs directly in the instrument's resonating body by varying the effective length, i.e. by opening and closing the holes. Since the vibration of an air column in a bore is the same as that on a string, intonation can be compared to that of the strings; however, it is not identical because the bore holes are located at fixed distances by the manufacturer similarly to guitar bars.
• The story is even more complex for percussion instruments because they include "non-tuneable" instruments, which do not produce sounds of definite pitch.

Therefore, the sound frequency produced depends on both the geometrical and physical characteristics of the vibrating element.

Role of the resonator

The resonator of a musical instrument is usually its body, which is made up of one or more cavities. Some examples are the mouth and nose of the human body, the body of string instruments and the bores of wind instruments. Many percussion instruments (such as the marimba or timpani) also have hollow bodies.

The work of the resonator is to damp or enhance the vibration of the initial oscillator at certain frequencies. As more thoroughly explained on the page about resonance, the resonator can absorb energy from an external source (in this case, the vibrating element of an instrument) particularly efficiently only in some bands of frequencies corresponding to its eigenfrequencies. In the case of musical instruments, the resonator always absorbs energy from the primary vibrating element and transmits it to the air surrounding the instrument. Therefore, it transforms mechanical energy from the initial vibration into radiating sound energy, also often functioning as a sound "transmitting antenna" (see how antennae work).

Whether the effect of selection (narrow band) or that of coupling (broad band) will be dominant depends on the type of instrument.

1. String instruments, for example, need a sophisticated mechanism of impedance matching to the surrounding air because the strings alone cannot efficiently transfer their vibrations to the air. Moreover, the body is not able to produce the fundamental of sounds, which have already been selected by the player's fingers that shorten the vibrating portion of the string. Therefore, for string instruments, the body must be resonating efficiently in a large interval of frequencies. The geometry of the strings has been optimised over the centuries by the experience and the study of master violinmakers in such a way that their resonance bands are not too narrow and do not highlight certain notes to the detriment of all the others.
   • For example, in the case of the violin, the resonator is the body with its curved shape that responds to natural frequencies at 600 Hz and 1,000 Hz (called wood resonances), as well as having other resonances closer to the range from 2,000 to 4,000 Hz. The resonance in this range is also called Helmholtz resonance, due to the air that enters and exits the body through the F-shaped holes. It has a frequency of 300 Hz.

Note: In some cases we use to say that the body of an instrument amplifies the sound of the string. It is clear that in this case the word "amplification" does not refer to the total energy. The energy coming out of the instrument, is always less than, or equal to the energy of the source, according to the laws of Thermodynamics. It is meant instead that a part of the input mechanical energy is converted by the resonator into radiating sound energy. For instance, a plucked string vibrates, but does not radiates sound effectively, while the same string, coupled to a suitable resonator, is able to transform a large fraction of its mechanical energy into sound energy. The concept of amplification has a very different meaning when applied, for instance to HiFi. Those kind of amplifiers use an input audio signal to control an external source of power (coming from the electrical network) to get an output signal having acoustical power larger than the controller signal.

1. On the contrary, for wind instruments, we have seen that the source oscillation cannot always determine the pitch of the sound final on its own. There are various examples:
   • for the woodwinds, such as the flute, oboe and clarinet, the work of the bore is partly similar to the strings of the string instruments. The bore contains an air column set in vibration by the reed or the performer's breath. The response of the bore is the very narrow resonance peaks of the frequencies determined by its length. Therefore, the bore receives the incoming (complex) vibration from the reed and selects from it only frequencies that are compatible with the oscillation modes of the air column. These modes are then determined by the boundary conditions at the ends of the bore and, therefore, can be modified by the holes made in them.
   • for the brasses, such as the trumpet, the bore has no holes and the only notes compatible with the oscillation of the air column are, therefore, the sharp resonances of the same (corresponding to the natural harmonics of the bore). Of course, it is possible to obtain other resonances by varying the length of the bore using specific valve mechanisms.

Role of the impedance adapter

The sound generated by a musical instrument is carried by the air. However, the musician never directly excites the air outside the instrument but rather a part of the instrument itself, over which they have control. It is the job of the instrument to transfer sound energy (i.e. transmit it) to the outside air as advantageously as possible so that it can be heard at the maximum distance with the minimum expenditure of energy.

1. The wind instruments start out with an advantage because the air within them is part of the instruments themselves. When the air inside is set in vibration, the bore selects a particular frequency and its integer multiples, after that it is enough to find a way to carry this vibration, which already occurs in air, outside. Now, we will look at visible shape differences amongst the wind instruments.
   • The woodwinds have a narrow bore with holes. The flute is perfectly cylindrical, the oboe and clarinet are not, however the only difference is a small bell at one end. This is because for the woodwinds, contrarily to what we may think, most of the sound is transmitted from the holes. Therefore, the holes of these instruments are their impedance adapter.
   • However, for the brasses, we have all seen that they terminate with a conical bell, the function of which must be clear by now. Since the brasses do not have holes, all the sound energy must be transmitted from the end of the bore opposite the player. The bell connects the narrow bore of the instrument to the external environment by gradually enlarging and matching its impedance to that of the open air. If there were no bell, sound waves would reflect almost completely at the open end holding most of the sound energy within the instrument. For more on this topic, see the pages on reflection, impedance matching, antennae and interference and the questions and answers section.
2. Percussion instruments with definite pitch are the ones that most need resonators to both add a feeling of definite pitch to the sound produced and to make it last longer. On the contrary, instruments with indefinite pitch almost never need an impedance adapter for sound radiation. The reasons are many:
   • Firstly, these instruments often emit a lot of sound energy in impulsive waves similar to those coming from explosions. This energy is very concentrated in space and time and tends to spread effectively over distance thanks to air dispersion.
   • Secondly, pitch can be indefinite for various reasons; either because the sound has an impulsive source or because it produces non-harmonic overtones. In both cases, it does not make sense to filter it because a significant part of the emitted energy would be wasted in the filter.
   • Finally, these instruments are often very large (such as the tam-tam or drums) and their coupling with air occurs through large elastic surfaces and is already automatically "matched" when the membrane oscillates in such a way to produce a radiating dipole. For more on this, see the page on antennae and interference.
3. The case of the strings is more difficult. As we already illustrated in the previous paragraphs, the vibrating element has narrow band resonances and functions as both an excitation source and a frequency selector, while the resonance body functions mainly as an impedance adapter and, therefore, has the function of increasing the transmission efficiency of sound in the air. The strings also have various internal adapters, which are used like intermediate steps to match the impedance of the metal strings with that of the wood surfaces. Two examples are the bridge and sound post of the violin.
   • The bridge receives the vibration from the strings and its feet lever the top plate of the instrument's body transforming the one-dimensional transversal oscillation of the strings into the two-dimensional oscillations of the top.
   • The sound post is a small wood cylinder that couples the top of the instrument with the back allowing the oscillations of the two surfaces to become volume oscillations in the air contained in the body.
4. Finally, for some instruments, the energy released by the oscillator is such that the need for matching between the oscillator and the body of the instrument is less important. In several cases, tricks are used to "slow" the energy transfer from the strings to the body and, therefore, increase the duration of the oscillation and sound. These phenomena occur for instance in the piano, in which:
   • the "percussion" of the hammers on the strings generates an oscillation with a lot of energy
   • the sound board, having impedance that is much different than that of the strings, absorbs energy slowly. Most of the energy is reflected by the bridges, where the strings are fixed to the board making their oscillation last for a long time.
   • another slowing mechanism of the process of energy transfer from the strings to the sound board is obtained, in the case of double and triple strings, by tuning them to slightly different frequencies. This causes the strings to not all vibrate in phase perturbing the regularity of oscillation that makes the transmission mechanism more efficient. Obviously, this can lead to the objection that the strings of the same note are out of tune! As explained on the page about critical bands, a slight difference of intonation does not produce "unpleasant" sound effects but rather causes the formation of slow beats, which are a peculiar trait of the timbre of the sound of the piano.

About the source-filter model for reed instruments

In this video, in which a rudimental wind instrument is built with only a whoopee cushion and several plastic tubes, we can see the ideas illustrated in the preceding paragraphs put into practice.

© 2006 The University of Salford, with the kind permission of Prof. Trevor J. Cox, https://hub.salford.ac.uk/sirc-acoustics. Translation and subtitles by Carlo A. Rozzi.

In-depth study and links

• To see the phenomena of reflection in sound tubes and resonance in cavities at work, we suggest you visit our 2D Wave Applet. On the 2D Wave Applet Guides page, you will find instructions on how to carry out the guided experiments.

• If you wish to further study the dependence of the speed of mechanical waves on the physical properties of elements in oscillation, visit this page.

• On this page, the description of the phenomena is purposely kept in general terms. The way these phenomena actually occur in reality depends on the structural details of each instrument, the skill of the player, etc. If you wish to see some examples, visit the pages on the Oboe, Clarinet, Trumpet, Violin, Piano, Timpani and Guitar.

• In the sections String instruments, Wind instruments and Percussion instruments, you will find more information on various excitation techniques (and, therefore, the distribution of energy through the various partials) of the oscillator of a musical instrument.

• If you wish to know how to calculate the resonance frequencies of a regular-shaped cavity, visit the page on stationary waves in 3D. You will discover that for resonant bodies of musical instruments, regular-shaped cavities should be avoided. Irregular shapes create resonance frequencies that are more homogeneously distributed along the frequency spectrum even when there are more accentuated resonance ranges (remember the example of the violin).

• If you wish to see a review of the eigenfrequencies selected by various vibrating elements, visits these sections:
  • Eigenfrequencies
    • Eigenfrequencies of bars
    • Eigenfrequencies of bores
    • Eigenfrequencies of strings
    • Eigenfrequencies of circular membranes
    • Eigenfrequencies of rectangular membranes
    • Eigenfrequencies of circular plates

Bibliography and references

• Neville H. Fletcher, Thomas D. Rossing, The Physics of Musical Instruments, Springer, second edition, 1998
• Thomas D. Rossing, Science of Percussion Instruments, World Scientific, Series in Popular Science - Vol. 3, Singapore, 2000
• Arthur H. Benade, Fundamentals of Musical Acoustics, Oxford University Press, 1976, second edition Dover, 1990
• Arthur H. Benade, Horns, Strings, and Harmony, Dover, 1992
• Harry F. Olson, Music, Physics and Engineering, Dover, 1967

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