Acoustics FAQ

This Acoustics FAQ - with an emphasis on music acoustics - began because questions are often sent to our music acoustics web site. See also the Basics section of the music acoustics web site. If your question isn't answered below, or in the Basics section, send it to

closeup of clarinet keys

General questions in acoustics

  • How can you have negative decibels? Doesn't zero decibels mean no sound?

    Zero decibels is a reference level, not zero sound pressure. So one can have negative decibels. Zero dB is usually set at about the limit of human hearing (in the most sensitive frequency range). The sound level in dB is a measure (on a logarthmic scale) of the ratio of the sound pressure or sound intensity to this reference level. The logarithm of one is zero, so zero dB corresponds to the reference level. Numbers greater than one have positive logarithms, so positive decibels means sound levels greater than that of the reference. Numbers smaller than one have negative logarithms, so negative decibels mean sound levels below the reference level.

    Zero sound pressure or zero sound intensity would actually be minus infinity dB. However, it is impossible to reduce the sound pressure and sound intensity to zero, unless you go to a vacuum, because of the thermal motion of molecules.

    We have a whole web page devoted to the decibel scale, loudness, phons, etc.

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  • What is the relation between loudness and decibels? Is 80 dB twice as loud as 40 dB? How do you translate from decibels to loudness?

    Sound level in dB is a physical quantity and may be measured objectively. Loudness is a perceived quantity and one can only obtain measurements of it by asking people questions about loudness or relative loudness. (Of course different people will give at least slightly different answers.) Relating the two is called psychophysics. Psychophysics experiments show that subjects report a doubling of loudness for each increase in sound level of approximately 10 dB, all else equal. So, roughly speaking, 50 dB is twice as loud as 40 dB, 60 dB is twice as loud as 50 dB, etc.

    nomogram of phon:sone conversion

    Again, see the page devoted to the decibel scale in which the relationships among sound intensity, sound pressure, dB, dBA, sones and phons are explained. We also have a web service with which you can measure your own hearing response curve.

  • Does adding two identical sounds give an increase of 3 dB or 6dB? Why?

    A subtle question: if you look at the page What is a decibel?, you'll see that doubling the sound pressure gives an increase of four in the intensity, so an increase in the sound level of 6 dB, whereas doubling the power increases the intensity by a factor of two, so an increase of 3 dB. There is no paradox here, because to double pressure and to double power you do two different things. Let's see how, starting with a simplified system.

    Let's imagine that we have a completely linear amplifier and speaker. If we double the voltage at the input (like adding two identical voltage signals), the we double the voltage to our linear loudspeaker and thus double the sound pressure it puts out, which gives us four times as much energy, four times as much intensity everywhere in the sound field, and thus a uniform increase in sound level of 6 dB. (The amplifier is also delivering four times as much electrical power.)

    Amplitude as a function of x and y for a Young's Experiment Now imagine that we have two identical, linear amplifiers and speakers, and we'll forget about reflections from walls, floor etc. We put the same signal into the inputs of both. Now each amplifier is producing the same amount of electrical power, so the output sound power is only doubled. On average, the intensity in the sound field is only doubled. Let's consider just a single sine wave (or, if you like, one frequency component of the sound). If your ear is equidistant from the two speakers, then the two pressure waves will add in phase (constructive interference), so the sound pressure at your ear will be twice as great, so you'll a level 6 dB higher when the second amplifier is turned on. However, if you are half a wavelength further from one speaker than from the other, then two two pressure waves will add one half cycle out of phase (destructive interference) and the sound pressure will be zero. So the sound intensity is four times as great in some places, zero in others, and intermediate values in most places. Integrate this over all space and the total power is only doubled.

    This diagram plots the sound pressure (as a function of x and y but not z) from two identical sources separated by three wavelengths. In optics, this would be called Young's experiment.

    What if the sounds are not identical, but have the same level? Then the phase difference will be random. If two signals have pressure amplitude pm and a phase difference θ, then their sum is 2pm(1+cosθ), which of course varies from 0 to 4pm. But the power is proportional to the amplitude squared, and so proportional to 4pm2(1+cosθ)2. The average of (1+cosθ)2 over all θ is ½ so the average of the whole term over all possible phase differences is 2pm2. So again, on average we double the power and so get an increase of 3 dB in sound level.

    Note that the interference effects referred to in our example with two identical amplifiers and loudspeakers are rather difficult to observe unless one really sets up identical sources rather carefully – you won't get them with natural sources such as voices or instruments. You might want to read (also on this page) Why do the sounds of two musical instruments always reinforce, and never cancel out? or How much does one more violin add to the sound level of a section of n violins?

  • What is the loudest sound?

    Atmospheric pressure is about 100 kPa. So, in air at normal pressure, one cannot have a symmetric wave whose pressure amplitude exceeds 100 kPa, because one cannot have a negative pressure in a gas. (One can, however, have negative absolute pressures in liquids.) A peak of 100 kPa in a sine wave corresponds to a pressure amplitude of 70 kPa rms, which in turn corresponds to a sound level of 191 dB (see What is a decibel? for details). You could get a few more dB by making the hypothetical signal a square wave, but it won't be square for long. An asymmetric signal such as a single compressive pulse could, in principle, have a larger amplitude than this. Pulses with amplitudes comparable with 100 kPa quickly distort as they travel. Very close to an explosion, for instance, the sound pressure in the shock wave could exceed 100 kPa. If you know the peak pressure Pp in kPa, you can estimate the sound level as 20 log(35,000,000*Pp/kPa) dB. As to what such a sound might sound like... well, you'd be deaf afterwards. How much damage it does to you (would you explode during the rarefaction?) depends on frequency and duration, but you'll probably be dead, too.

    For sound in liquids and solids, higher intensities and much higher sound pressure levels are possible. For every 10 m of extra depth underwater, you raise the static pressure by one atmosphere. Further, sound levels are usually measured with respect to 20 µPa (see What is a decibel? for details). A wave transmitting the same power in a liquid or a solid has a much higher sound pressure level in dB than one in a gas. (See Acoustic impedance, intensity and power) for details.)

  • Why do cathedrals (and many other large, enclosed spaces) have long reverberation times?

    Clap your hands: soon it is silent. So, where does sound energy go? If there is an open window or door, some goes outside. Any other part of its boundary reflects some of the sound back into the room and absorbs the rest. Acoustic tiles and soft furnishings absorb much more than does a hard, massive surface like a stone wall. The cathedral has good reflecting walls and few open windows or doors – also (ask someone who prays a lot) not enough cushions!

    But there is another important effect. Sound energy is stored in the air, but absorbed at the surface. Usually, a large hall has a much greater ratio of volume to surface area. So,

    The reverberation time measures how long takes to 'fade away'. Usually, it's quantified as the RT60, which is the time for a very brief sound to decay by 60 dB, or to lose 99.999% of its sound energy. (see What is a decibel? for the energy to dB calculation.)

    Wallace Sabine showed that the decay is approximately exponential and calculed the RT60 using what we know as the Sabine equation:

      RT60  =  0.161*Volume/(average absorption coefficient*area*1 metre)
    The RT60 is a fraction of a second in a normal room, but can be several seconds in a cathedral because the big volume of the cathedral stores more sound energy, and the waves don't often encounter a surface, especially an absorbing one. You'll even notice the difference between an empty house and a furnished one: furniture has a lot of surface area, much of it absorbing. It's worth mentioning that, even if the walls were perfectly reflecting, the reverberation time would be finite, because some energy is lost in transmission. SeeWhy is sound absorption in water less than in air? for an explanation.
  • What is acoustic impedance and how is it related to intensity?

    Acoustic impedance Z is the ratio of the acoustic pressure p, measured in Pascals, to the acoustic volume flow, measured in cubic metres per second. Specific acoustic impedance z is the ratio of the acoustic pressure p, measured in Pascals, to the acoustic velocity, measured in metres per second. We introduce and explain these in two separate pages. Acoustic impedance, intensity and power is an introduction in terms of physics and the wave equation. What is acoustic impedance? explains it for musicians and explains how and why the acoustic input impedance is used to characterise wind instruments.

  • Why is sound absorption in water less than in air? According to my text, for a 1 kHz signal in water the loss by medium absorption is about 0.008 dB/100 m. In air, the loss is much greater: about 1.2 dB/100 m.

    Imagine that we could take a very fast picture of certain properties of a sound wave during transmission. The pressure varies from a little above atmospheric, to a little below and back again as we progress along the wave. Now the high pressure regions will be a little hotter than the low pressure regions. The distance between two such regions is half a wavelength: 170 mm for a wave at 1 kHz in air. A small amount of heat will pass from hot to cold by conduction. Only a very small amount, because, after half a cycle (0.5 milliseconds for our example), the temperature gradient has reversed. Although it is small, this non-adiabatic (non-heat conserving) process is responsible for the loss of energy of sound in a gas.

    What happens when we change the frequency? The heat has less distance to travel (shorter half wavelength), but less time to do so (shorter half period). These two effects do not cancel out because the time taken for diffusion (of heat or chemical components) is proportional to the square of the distance. So high frequency sounds lose more energy due to this mechanism than do low. This, incidentally, is one of the reasons why we can tell if a known sound is distant: it has lost more high frequency energy, and this contributes to the 'muffled' sound. (Another contributing effect is that the relative phase of different components is changed.)

    So, let's now dive into the main question. Three different parameters make the loss less in water.

    • First, sounds travels several times faster in water than in air. (Although the density of water is higher by a factor of about 800, the elastic modulus is higher by a factor of about 14,000.) So, for a given frequency, the wavelength is longer and the heat has further to travel.
    • Second, the water does not conduct heat so rapidly as does air. (This may seem odd if you've recently dived into cold water, but the effect in that case is largely due to water requiring more heat for the same temperature change. Not counting the fact that you probably wear more clothes when out of the water.)
    • Third, the temperature of water rises less under a given imposed pressure than does that of air.
    All three effects go in the same direction, and their cumulative effect is substantial, as your text's values suggest.

  • Why do the sounds of two musical instruments always reinforce, and never cancel out? According to the interference of waves, two waves with exactly the same amplitude and frequency can either reinforce (if they are in phase) or cancel (if 180 out of phase). However, when two instruments play the same note, it is always louder, and never softer. Why?

    The answer to this involves several different effects that complicate the sound of musical instruments. To hear the effect of destructive interference, you have first to eliminate each of these effects, and it is rare that they are all eliminated together, which is why you don't normally hear destructive interference in practice. Nevertheless, when two instruments are nearly but not exactly in tune, you do hear the phenomenon of beats (listen to the sound files of beats). This is an example of constructive and destructive interference: the slight difference in frequency causes the phase relationship to change slowly. When the two waves are in phase it sounds loud but, when they are out of phase it is soft.

    Beats between real musical instruments do have variation in loudness, but the loudness doesn't usually go to zero. There are several reasons for this:

    • First, real musical instruments rarely have exactly equal amplitude, so even when they are exactly out of phase, their waves don't cancel. (And they often have vibrato.) Even if they were of equal amplitude, it is unlikely that you would be equally distant from both. Further, this effect is enhanced by the logarithmic response of your ears (see what is a decibel? for details). If two waves cancel to 99%, the resulting sound is 40 dB softer than one instrument alone. Listeners often judge that 40dB softer sounds about 1/16 as loud. So even if you achieve cancellation with this precision (hard to do), the effect is not as spectacular as you might expect.
    • Second, real musical instruments don't play sine waves: =they have several harmonics. Even when the conditions are such that two harmonics cancel, other harmonics may not.
    • Third: unless you are in an anechoic chamber, the points of cancellation are not where you calculate them to be, because of multiple reflections off walls.
    • Fourth: you have two ears. Even if one ear is at the point of cancellation, the other is not.
    • Fifth: your ears are most sensitive in the range 1-4 kHz, which means wavelengths of ~300 mm to ~100 mm. Nearly all instruments radiate fairly strongly in this range, so you must get cancellation in this frequency range to get cancellation. However, because of these short wavelengths the region of cancellation (if there is one, in spite of the complications mentioned above) is very small, and the chance of having even one ear at the point is small.
    • Sixth: instruments and peoples ears move on a scale of at least tens of mm (often more) during performance.

    Despite all of the above, it is possible to set up conditions under which you can experience the interference effects. Simply set up a sine wave source (eg an electronic tuner) in a room. There will be reflections off walls and other objects that cause the amplitude to vary strongly from place to place. Cover one ear, put the other near a wall, and move your head towards and away from it. If you were to drive two identical speakers with the same signal but reversed in phase, and if you did it in an anechoic chamber, then you should get cancellation on the plane of symmetry between the speakers. If you put one ear on this plane, and neglecting the reflections from your body, you'd expect to get pretty good cancellation. (I've never tried this experiment in an anechoic chamber, but I've certainly noticed the effect of reflections from walls.)

    In Berio's Sequenza VII, a sine wave is played throughout on the note B4. It creates an eery ambiance: one doesn't know where it is coming from and it seems to get louder and softer, for several of the reasons discussed above, including the motion of performer and audience.

    When music is recorded with two or more microphones, it occasionally happens that two microphones give signals for one instrument that substantially reduces, by phase cancellation, one or more harmonics. Mixing desks often have a switch that allows the operator to reverse the phase of a channel to reduce this problem.

  • How does active noise cancellation work? And where does the sound energy that gets cancelled end up? More specifically: when I wear those active headphones, the speakers put sound into the headphone and this cancels out the noise from outside. But now there is a bit more sound energy than there was before: the sound energy that was there, plus the energy from the loudspeakers in the headphone. Where does it go?

    The short answer is that it goes somewhere else: outside the ear enclosures. Let's see how. I'll only talk about low frequencies, because active noise cancellation only works at 'low' frequencies. If your ear enclosure has only one speaker, then it will only work for wavelengths much bigger than the enclosure: let's say wavelengths much bigger than 80 mm, which means frequencies no higher than a few hundred hertz. We'll see why later. I'm also simplifying considerably.

    The first picture below shows an ear enclosure that seals well around the ear, but which lets sound in because it is not completely rigid. It deforms a little (in the sketch, the 'little' has been enormously exaggerated) and so the air inside the enclosure is alternately compressed and expanded, so your ear is exposed to a varying pressure, and you hear a sound. This is the sound that has been transmitted through the enclosure by deforming it. The sound that is not transmitted into the enclosure, is reflected. Here, the reflected sound is less than the incident sound, as indicated by the size of the arrows.

    diagram showing deformable enclosure             diagram showing rigid enclosure              diagram showing deformable enclosure made rigid by acoustic feedback within

    The second picture shows a hypothetical ear enclosurethat is perfectly rigid and that makes a perfect seal around your ear. This doesn't transmit sound into the inside: the air inside is neither compressed nor expanded, so you don't hear a sound. Because no sound is transmitted in, it must all be reflected. This means that the sound level outside the rigid enclosure is slightly higher than that outside the deformable one, because the reflected signal is stronger.

    The third picture shows a simplified schematic of an enclosure with active noise cancellation. A microphone converts the sound pressure into a voltage, passes it to an amplifier and loudspeaker whose combined gain is negative, i.e. it is 180 out of phase with the microphone. This negative gain ensures that the loudspeaker makes a sound signal just strong enough to cancel out the signal at the microphone. So the pressure in the enclosure doesn't vary, and you hear no sound. However, if the pressure inside is not varying, the air inside cannot be undergoing compressions and expansions. So that means that the enclosure is not moving in response to the sound wave from outside. The effect of the speaker and cancellation network is to make the enclosure behave as though it were completely rigid. (Technically we could say that it has changed the acoustic impedance of the enclosure, as measured from outside, infinite, compare to the lower value it had when it was deformable.) If we compare the first and third pictures, we now see the effect of turning on the noise cancellation: it makes the enclosure behave as though it were rigid, and it reflects more sound, so that's where extra sound that didn't come through the enclosure has gone: outside.

    This still leaves the problem: if the second and third diagrams produce the same sound outside, what happened to the sound energy you put in with the loudspeaker? The answer is a little surprising. The work done by a moving object like a loudspeaker is the integral of the force it applies (here pressure times area) over the distance travelled. Neglecting the energy absorbed by the microphone, this is actually zero in the simplified case studied here. The loudspeaker may use up electrical power, but it doesn't transmit any sound power. (In the real world, the speakers do transmit a tiny amount of power, both forwards and backwards. And of course you'll still have to replace your batteries from time to time: amplifiers use up energy just by being switched on.)

    Notice that I have spoken of the pressure inside the enclosure, as though it had a single, value--the same everywhere. This is only true if the wavelength of the sound is much greater than the size of the enclosure, and that explains the limited frequency range of this technique. At high frequencies, the cancellation may work close to the microphone, but not elsewhere.

    Comparing the second and third pictures invites a final question: if a perfectly rigid ear enclosure keeps all the sound out, why bother with active noise cancellation in this application? First, perfectly rigid enclosures that make perfect seals against the head do not exist. However, modestly priced hearing enclosures can often reduce sound by more than 30 dB, and this is better than the only active headphones that I have tried. Further, the hearing protectors work up to higher frequencies. So for serious hearing protection they are recommended. The advantages of the active devices are that they are sometimes more comfortable (eg on hot days) and that you can immediately plug them into a sound channel. For instance, they are widely used by pilots who can then listen to radio or other communications in a noisy cockpit. Hearing protectors require modifications to function as headphones. (However, there may be a market for a comfortable, rigid hearing protector with loudspeakers inside and which are not coloured safety orange.)

    For more information, see Understanding Active Noise Control by Colin Hansen.

  • How does the Doppler effect affect the player and the audience? Since the sound is traveling away from the player, should the player perceive the pitch as lower than somebody in the audience?

    There is no such effect for the player. The Doppler effect arises where there is relative velocity between the observer's ears and the sound source. Even if you have a really weird performance technique, the relative velocity between your ears and your instrument is much less than the speed of sound! If the player or the audience were moving there would be such an effect for the audience, but they'd have to run or bicycle for it to be an important effect. In normal listening circumstances, the sound from the instrument goes towards your ears and the audience's ears at the same speed.

    Incidentally, one might expect a perceptible Doppler effect in the 'bull roarer'*, an instrument used by (among others) some of the native peoples of Australia. This instrument is swung around the head on a cord at high speed. However, other periodic effects in its sound tend to complicate perception of the Doppler effect here.

    * The users don't call the instrument a 'Bull roarer', but the proper names for it are only shared with male initiates of the tribes, so this author does not know them.

  • What are those sounds (such as telephone ring tones) that kids can hear but adults cannot?

    High frequencies. Almost always, adults lose the sensitivity to high frequencies first. It is not well known how much of this is due to age and how much is due to lifetime exposure to noise, because the two are correlated. Children may hear sounds above 20 kHz, but adults rarely can. In middle or old age, the upper frequency limit gradually descends.

    You can find out how well you hear different frequencies -- including your high frequency limit -- on our web service Test your own hearing. In fact, that site may be where some youngsters are downloading their high frequency sounds.

  • Why does helium (balloon gas) make your voice sound strange?

    First, a warning: helium is suffocating and breathing it could be fatal. See Helium inhalation – it's no laughing matter. In order to hear the effect, a single shallow breath is sufficient.

    The frequency of your voice is not much changed by inhaling some helium. The resonances of the vocal tract, and hence usually the formants, are moved to higher frequency. This makes a substantial difference to the sound spectrum of the voice. Its effect is somewhat similar to that of a high pass filter (there are some filter sound file examples on our page about filters). Do either of these change the pitch? Or do they change some other aspect of the voice? Or both? Your answer to this question might depend on whether you listen to speech or to sung notes. We've put up some sound files and an explanation at Helium speech .

  • A horn seems to amplify in both directions: as an old hearing aid, or old gramophone. How?

    Before electronics, a large horn could be used as a hearing aid: the small end was placed at the ear, and the large end pointed towards the sound source. A similar horn was used in a gramophone: the small end was connected to the needle, and sound radiated from the large end. Yes, the horn works in both cases and I can understand why you say that it amplifies in both directions. First, however, let's define amplification.

    An electronic amplifier takes an electrical signal with low power as input, and outputs a signal with high power. The horn does not amplify in this sense: the output signal does not have more power than the input signal. So what does the horn do?

    Technically, the horn in each case is an impedance matcher. Let's think of the hearing aid use. Sound in the air (a medium with low density) has relatively small pressure variation, and relatively oscillatory flow of air. Your inner ear is full of liquid (whose density is 800 times greater) and vibrations in this medium require relatively large pressure variation, and relatively oscillatory flow of liquid. In healthy ears, the outer and middle ear serve to convert the wave in the air (low pressure, high flow) to a wave in the liquid (high pressure, low flow). The horn does the same thing: from large end to small, it 'concentrates' the wave so that higher pressure is exerted at the small end. However, the power is the pressure times the flow, so the power can be the same at both ends. (In practice, of course, there are losses.)

    In the case of the gramophone, the vibrating needle vibrates with relatively large force, but it is so small that it doesn't move much air. So it is connected to a diaphragm at the small end of the cone, and exerts a substantial force on it. This creates a wave with high pressure and low flow at the small end of the horn.

    Technically, we say that the horn matches a high acoustic impedance at the small end to a low acoustic impedance at the large end. The electronic analogue of this is the transformer: it takes in high voltage and low current at one end, and delivers low voltage and high current at the other (or vice versa). Neither the horn nor the transformer amplify power.

    Incidentally, the bell on a brass instrument most obviously works as an impedance matcher for sound going outwards. Because it works best for high frequencies, it has important effects for the loudness and timbre of brass instruments. It also works as an impedance matcher for sound going inwards, which sometimes has deleterious effects for the player, as we explain on this page about timpani-horn interactions.

  • Space isn't quite empty. Can it transmit sounds?

    This question is more subtle than it appears. Sound is carried by pressure waves. Imagine the variations in pressure in a sound wave in air: a pressure maximum occurs where the density is highest. Due to molecular collisions, molecules tend to move from high to low pressure. That, plus the momentum of the molecules, produces the wave.

    Pressure, however, is a macroscopic concept. We don't talk of the pressure of a few molecules, we talk about the forces that molecules exert during interactions. To talk of pressure, we need a significant number of molecules. In space, to have a considerable number of molecules/ atoms/ ions, you have to consider large volumes.

    First let's consider just the atoms and molecules in space. The typical distance between them is about a centimetre. So we need a large volume to have a significant number. But it's more subtle than that. A sound wave propagates by intermolecular (or interatomic) collisions. Molecules in air only travel a nanometre or so (their mean free path) before they collide. What is the mean free path (m.f.p.) in space? Just from dimensional considerations, we can guess that it is roughly

      m.f.p. ~ [(atomic cross-sectional area)(number density)]−1

    (There's also a numerical factor in there, of order 1, but we shan't need it for this approximate calculation.)

    So, taking the atomic area as ~ 10−20 m2, and number density (using the separation value quoted above) ~ 106 molecules m−3, we get a m.f.p. of about 1014 m or 0.01 light years. Collisions are rare. So, to talk of sound waves, we'd need to consider wavelengths of longer order than this. Very low frequencies.

    In a plasma, things are more complicated, because the ions interact by electric and magnetic fields. (Strictly, I don't know whether a pressure wave in a plasma should be called a sound wave, but let's not quibble.) However, while the medium was a plasma, the radiation and matter were strongly coupled, so the temperature of the radiation was about the same as that of the matter.

    The speed of sound is roughly proportional to the square root of the ratio of the temperature to the atomic or molecular mass. In the early universe, temperatures were very high and, early enough, the speed of sound was comparable with the speed of light!

    As the universe itself expanded, the wavelengths of the pressure variations of the early universe have been expanded by a similar factor – and complicated by gravity. So the sound waves have now ultralow frequencies. And are difficult to measure. However, the temperature variations in that early plasma, when the universe was only a few hundred thousand years old, those variations are the subject of intense research effort. They are called the cosmic microwave background (CMB). The wavelength of these (electromagnetic) waves has also expanded, which is why it is now microwaves rather than light.

    These waves are of interest to cosmologists, including Mark Whittle of the University of Virginia. He came to our lab to talk about it and one of the postdocs here, Alex Tarnopolsky, made some sound files for him, transposed them up into the audible range. Mark now uses it in his seminars and calls it the "primal scream of the infant universe".

    You can hear the first million years (transposed upwards by 50 octaves) on his website, where he has a rather nice popular account of these acoustics aspects of cosmology.

General questions in music acoustics

  • When we were playing two recorders, we heard a low buzzing sound, sometimes making a chord with our notes. What is that?

    It is called the Tartini tone. It is most often noticeable when the two notes being played are sustained, about equally loud, not too low in pitch (say treble clef and above) and moderately loud. I've put a section about it on a page called Interference beats and Tartini tones where you can hear sound files made with pure sine waves that show the effect quite well. There is also a page called Tartini tones and temperament: a musician's introduction.

    If you listen for the Tartini tone, you can use it to tune your chords: the tuning you will arrive at is Just Intonation, and not Equal Temperament. In sustained chords, the former usually sounds better. (A multimedia introduction is at Tartini tones, consonance and temperament.)

    The effect may be due to nonlinear effects (heterodyning) in the ear, or perhaps due to higher processing in the auditory cortex, or (more likely, I think) both of these.

    sound spectrum showing two notes and their Tartini tone The strong Tartini tone that we sometimes hear is usually generated in the ear. In the case of the two recorders (above), there is no vibration in the air at the difference frequency. However, when two notes are played simultaneously on the same instrument -- as played here on a violin by John McLennan -- it is possible to produce a difference tone via small non-linearities in the instrument. This is explained in more detail in Interference beats and Tartini tones and in Tartini tones and temperament: a musician's introduction.

  • What are heterodyne and comination tones? How do they relate to beats and difference tones? And where do they come from?

    Nonlinear systems, in response to two signals with frequencies f and g, produce heterodyne or combination signals, with frequencies mf +/- ng, where m and n are integers. To me, the two terms have the same meaning, although heterodyne is used more frequently in radio and TV.

    One of the heterodyne terms, that with frequency f-g, is called the difference tone in acoustics and the Tartini tone in music. (I'm choosing f > g.) If (f-g) is rather smaller than f or g, then the phenomenon is called interference beats or just beats: the combined signal is an oscillation with frequency (f+g)/2, amplitude modulated by the beat frequency (f-g). When the beat frequency is less than a dozen Hz or so, it can be clearly heard as a variation in amplitude, and is commonly used to tune instruments. (Beats at mf-ng are also thus used.) See Interference beats and Tartini tones for examples and sound files.

    Where do such terms come from?
    Non-linear systems inherently produce them. I shan't go through the algebra here (though it's not difficult) but one can expand a non linear response using a Taylor series. The series includes higher order terms, which become more important as the signals get larger. Simple trigonometric identities substituted in the product terms give the heterodyne terms.

    In electronics, the nonlinear response of diodes was traditionally used for modulation and demodulation (producing heterodyne terms).

    In music, nonlinearities occur in several places. The basilar membrane in the inner ear is nonlinear, and it is plausible that Tartini tones are generated here. I have read, however, that Tartini tones can be heard when one note is input to one ear and another to the other, via headphones. I've not experienced this myself but, if it happens, it suggests that they are produced in the neural processing. (Neural processing is highly nonlinear.)

    They may also be produced in a nonlinear source. For instance, when they are produced by double stopping on a string instrument, one would expect the two notes to interact through the hair of the bow and through the bridge. The bow-string interaction is highly nonlinear, so this can produce heterodyne terms. This is shown in the example given above. When wind players produce multiphonics, they also produce heterodyne signals. To take one particular example that we have studied in detail: when a didjeridu player vocalises at a pitch different from that of the drone he is playing, strong heterodyne terms are produced.

    Finally, it is possible to produce heterodyne terms in a microphone. No microphone is perfectly linear and so sufficiently strong signals could, in principle, produced heterodyne terms in the microphone itself.

    Tartini tones sound unusual because one doesn't have a sense of direction associated with them. One can imagine that they are being produced 'inside one's own head', which may in fact be true. However, it's worth noting that, even in a slightly reverberant space, the same may be said of a pure tone. Listen to the sound files on Interference beats and Tartini tones and see if you agree.

  • What is the frequency, and how does it relate to pitch? Where do harmonics fit in? Does the frequency of the lowest harmonic determine the pitch?

    The frequency f is the number of vibrations per unit time. For example, when a tuning fork sounds the note A4, its tines vibrate 440 times per second. Its frequency f is 440 cycles per second, which is usually written as 440 Hertz or 440 Hz. (The unit of frequency is named for Heinrich Hertz, a pioneer of electromagnetic radiation.) The pitch of a musical note is determined by the frequency of its sound wave.

    Pitch differences depend on the ratio of frequencies. If the frequency is doubled, the pitch rises by an octave, independent of starting frequency. If it increases by a factor of 3/2, the pitch rises by a fifth. (It follows that pitch is proportional to the logarithm of the frequency.)

    The duration of each cycle of a vibration is called its period, T. If there are f cycles in one second, then each cycle must last 1/f seconds. In other words, the frequency is the reciprocal of period, or f = 1/T. So, for the note A4 at 440 Hz, the period is 2.27 milliseconds.

    To relate pitch to frequency and back, see notes, frequencies and MIDI numbers. We give an introduction to pitch, loudness and timbre in Quantifying Sound.

    Musical tones usually comprise vibrations that are periodic. Such tones may be considered as the sum of pure tones from the harmonic series. So a note with frequency f usually contains also components with frequencies 2f, 3f, 4f etc. These components are called harmonics and the component with frequency f, the highest common factor, is called the fundamental. This is discussed in more detail in What is a sound spectrum?

    We usually hear a pitch corresponding to the fundamental. However, we hear that pitch even if the fundamental is absent -- called the missing fundamental. For periodic tones, our sense of pitch is determined by the spacing of harmoncs in the region of several hundred Hz. So, for example, a small loudspeaker is very inefficient at 40 Hz. Consequently, when the sound of a bass playing its lowest note (E1 at 40 Hz) is played on such a speaker, the radiated sound has strong high harmonics, but almost no fundamental. Nevertheless, the pitch we hear is E1. The radiated sound might include the tenth, eleventh, twelfth etc harmonics at 400, 440, 480 Hz etc, and the spacing between these is 40 Hz.

  • What is all the fuss about temperament? Is it just an academic problem for musicologists? And what does it have to do with the circle of fourths?

    Litres of ink have been spilt on this topic, and I expect that there are many web sites. But it is by no means a purely academic problem, so let me give a quick introduction. For example, suppose a violist tunes up nice pure fifths C3, G3, D4, A4, removing interference beats as he does so. These will give intervals with frequency ratios of 3:2. The violinist tunes in unison to the last three of these notes, ie G3, D4, A4, but then adds E5. So the ratio of the frequency of the violinist's E string to the violist's C is (3/2)*(3/2)*(3/2)*(3/2) = 5.063. If they played these open strings together it would sound (to many people, anyway) uncomfortably out of tune. The fifth harmonic of C3 at 5.000 times the fundamental clashes and produces beats with the open E5 at 5.063 times that fundamental. Of course in practice, while the violist must play the C3 as an open string, the violinist will rarely play E5 on open string, so in musical context the violinist solves it by adjusting the position on the fingerboard and/or adding some vibrato (or by glaring at the violist!).

    Wind and bowed string instruments can rapidly change the pitch by changing embouchure or finger position respectively and so, for sustained chords, can tune to eliminate beats or to achieve some other effect. A harpsichordist or a pipe organist does not have the option of changing the pitch (quickly) according to harmonic context. For these instruments, one must make a compromise between getting nice fifths and nice thirds. Not all organists, and very few harpsichordists, think that equal temperament is a sufficiently good compromise for music containing thirds because it favours the fifths far too much at the expense of the thirds. Just intonation is no good if you go beyond the home key, and even within that key the chord on the second note of the scale is very ugly.

    Mean tone temperament gets the thirds right, and this spreads the dissonance over four perfect fifths. (e.g. tune C3 - E5 in the ratio 5:1 and make the fifths equal to the fourth root of 5, which is 1.495, which is close to the just fifth of 1.500. This is acceptable in itself, but it becomes much worse if you modulate into different keys.

    A harpischordist might get away with mean tone thus: put all pieces in the sharp keys in the first half of the concert and then retune during interval to play the pieces with flats in the second half.

    Many people opine that the best compromises are the so-called Well-temperaments like those of Vallotti & Young and those of Werckmeister, which spread the dissonance over more fifths. Some musicologists think that Bach wrote the Preludes and Fugues to demonstrate one of the Well-temperaments, perhaps Werckmeister.

    See Tartini tones and temperament: a musician's introduction and the multimedia introduction at Tartini tones, consonance and temperament.

  • Why don't pianists and guitarists bother with temperament?

    Wind and bowed string instruments have nearly *exactly* periodic sounds, and thus their partials are almost exactly harmonic. For these instruments, equal tempered thirds in sustained chords don't sound great.

    Some guitarists possibly do use equal temperament. Others can adjust the tuning according to the key that they are to play in, and can do so quickly. They may use several different temperaments without noticing it. Some good players clearly do adjust the tuning by pushing the finger that stops one string parallel to the string, so as to increase or to decrease its tension.

    Pianos have strong transients, which mean that they don't have periodic sounds. Further, they have three strings for most notes, and these are tuned slightly differently (to assist sustain) which gives chorus effects that disguise the problem. There is the further complication that the partials of thick steel strings are sharper than harmonics and thus their sounds are not exactly harmonic. Try it on a guitar using the very high harmonics, particularly with solid steel strings.

    Pianists get used to equal temperament, and may even prefer it to others, but string quartets sometimes have trouble playing quintets with piano.

    So what of harpsichords? They have thinner, softer strings than do pianos and so their partials are more nearly harmonic. Also, harpischord players are more likely to have studied different temperaments, and are more likely to be playing with wind and string players who are conscious of the issues.

    See Tartini tones and temperament: a musician's introduction and the multimedia introduction at Tartini tones, consonance and temperament. (By the way, I've recently written an orchestral overture called Circle of Fourths, which uses the circle as a chord, and might be considered as an experiment in orchestral temperament.)

  • Whay do orchestras tune to A? And why to the obooe? String players tune their open strings. Violins, violas, cellos, basses all have an A string. (So do guitars, mandolins and most others.) Further, A is the highest string on viola and cello and second highest on violin. (Brass bands usually tune to Bb, so that the trombones can tune their instrument with the slide all the way in.)

    Why tye oboe? First, the oboe has a strong, readily identified timbre, especially in its bottom octave. Second, the oboe requires very little air, because the hole between the two halves of the double reed is very small. Consequently, a player can play a note for a long time relatively easily, if required.

  • What is stretch tuning?

    The frequency ratio of octaves on a piano are found to be slightly greater than 2:1, especially for the very high and very low notes, and especially for small pianos. The reason for this is that the octaves are usually tuned to eliminate interference beats between the fundamental of the upper note and the second partial (or resonance frequency) of the lower note. The second resonance of a struck string usually has a frequency that is slightly higher than twice that of the first, because of the finite bending stiffness of the string.

    Playing an instrument that only sounds one note at a time, or singing, many musicians stretch octaves. However, playing chords with other instruments, they usually play 2:1 octaves.

  • What causes broadening or finite width of harmonic peaks in a Fourier transform?

    If a signal is perfectly periodic (i.e. repeats exactly after a period T) and infinitely long, we might expect its Fourier transform to have infinitey narrow peaks at frequency f  =  1/T and the other harmonics 2f, 3f etc. When we sample such a signal and use a program to calculate the Fourier transform, we obtain peaks of finite width. (If we sample a signal with vibrato we might see even broader peaks, but let's restrict this to a strictly periodic signal.) In fact, we could get very narrow peaks if we did one of two things.

    First, we could use a very long signal and a very long sample window for the transform. This would give narrow peaks, going to zero width as the signal and window length approached infinity.

    Second, we could use a window whose length was exactly an integral number of periods nT. (Of course, without having performed a Fourier transform on a very long sample we don't know exactly what T is...)

    In practice, we rarely do either of these: even if the software used for the transform may offers a choice of sample length, its maximum value may not be long enough to produce very narrow peaks. In general, the width of the peaks is of the order of 1/t, where t is the length of the sample window – except in the case where the window has a length nT.

    This is a specific example of the impossibility of measuring time or frequency with infinite precision. I've devoted a whole page to this: it's called Heisenberg's uncertainty principle and the musician's uncertainty principle.

  • Why are so many physicists and mathematicians (and engineers and ...) good at music?

    I am often asked this, and one can make a few observations about the similarities between the two. For instance, in physics or maths we start with a relatively small number of definitions and laws and with these, we attempt to explain almost everything in the universe. We build elaborate and detailed patterns in a heirarchy of structures, starting with quite simple elements. In music, we start with a relatively small number of pitches and durations. Again, we build elaborate and detailed patterns in a heirarchy of structures, starting with quite simple elements. The physicist and the musician recognise these heirarchies and underlying structures, and find the elegance and beauty in them. I've elaborated on this and other ideas in a paper called The creation and analysis of information in music.

    However, on a more pragmatic level, I think that it is helpful to look at it from the other direction. A good musician knows about practice: that an hour's solid work yields only modest advancement and that regular practice is necessary. S/he is capable of abstraction at several levels. S/he is capable of processing information rapidly and precisely.

    So, take someone who is good at abstractions, capable of processing information rapidly and precisely and has the temperament to work in order to progress. Is it surprising that some musicians have aptitude for physics? All that one would need to add to this list, I suspect, is curiosity and wonder about the world. It's not surprising that many intelligent, creative people are good at physics, maths and engineering. It's also not surprising that they are good at music.

  • Why does a loud note still sound different from a soft note, even when you turn the volume down?

    In many musical instruments, there is an oscillator (such as a reed, or the player's lips) that behaves in a non-linear way. For small vibrations, however, the behaviour is nearly linear. So louder playing means more non-linear behaviour, and more non-linearity means more higher harmonics. This is explained in more detail in, for example, how a reed works in a clarinet. There are sound files and spectrograms illustrating the effect in What is a sound spectrum?, whence this illustrtion:

    spectra and spectrogram of a crescendo

  • Why does rubbing your finger around the rim of a wine glass make a note?

    Like a percussion instrument, the glass will vibrate with a range of frequencies when you tap it (lightly). However, when you rub your finger around the rim, you are continuously putting in energy and so you produce a sustained note.

    It works like this: over a time of a millisecond or so, the fingertip 'sticks' briefly to the glass, then 'slips' a little and, if the conditions are just right, the glass will vibrate so that the finger 'sticks' again, one period of vibration later. The mechanism is rather like that of the violin bow on the string: see Bows and strings. Wine in the glass impedes the vibration of the part that it occupies, so you can tune the note by drinking some of the wine. Glasses of different sizes also have different pitches. Some glasses ring for longer when you tap them: there is less internal loss of energy in the glass. These seem to be easier to play with the fingertip, too.

    Benjamin Franklin invented an instrument called the 'glass armonica', consisting of a set of glass bowls, of varying sizes, on a spinning axle. The player touched a bowl to make a note. One of Mozart's last compositions (K617) was for glass armonica, flute, oboe, viola and cello.

Questions related to string instruments

  • What is the difference between artificial and natural harmonics?

    This contrast is made, to my knowledge, only by string players. Natural harmonics are those that are played on an open string, whereas 'artificial harmonics' are the harmonics of a stopped string. The latter are of course more difficult to play, as you need one finger to stop the string and another to touch it at the desired node. See Strings, standing waves and harmonics and Standing waves.

  • How important are the materials from which string instruments are made?

    For string instruments, in which vibrations of the material of the instrument is what radiates sound, the mechanical properties of the materials are of great importance. The stiffness, density, anisotropy and losses in the wood (or other material) are all important to the response and performance of the instrument. A complete answer would be very long. Briefly: the bridge is not quite stationary: it must move a little so that it takes a small quantity of the energy out of the string in each vibration cycle. This is used to vibrate the body and, particularly for low frequencies, a substantial area of the body must move in order to transmit energy effectively into the air. For a loud instrument and a good sustain of a plucked note, relatively little of this energy should be lost in the body. Further, the mechano-acoustical properties of the body should help make the sound interesting. The spectra of different notes should have overall shapes that are different (but not too different) so that the instrument has some character. For bowed string instruments especially, it is important that the body has properties that vary rapidly with frequency, so that a vibrato induces substantial changes in the spectrum. This is quite important to the characteristic warm sound of a bowed string vibrato. (See also How important are the materials from which wind instruments are made?)

  • Why do you get odd tunings if you tune by harmonics?

    If your instrument has four strings, tuned in fifths or fourths (violin family and bass) then it is likely that you will tune by harmonics, and you won't get very noticeably "odd" tunings. This question was asked by a guitarist.

    Some background, and a method for tuning by harmonics, are given in Strings, standing waves and harmonics. But here is a specific guitar question. "I tune by first setting my E strings to a standard pitch, then using harmonics to match this string to adjacent strings. On the lowest E string, when one I hit the 4th fret harmonic to get G# (the third of E), it is slightly flattened. On the 7th fret harmonic to get B (the fifth), it is pretty much on an even division of E. When the chord is played after tuning this way, there are no beats, and this is what sounds in tune. Tuning the whole guitar this way yields some flat and some sharp strings, that all sound good together. I understand that this is the essence of temperement, well vs even or just. My question is, why is the 4th fret harmonic flat and the 5th and 7th frets on, and why does this eliminate beats? Is there a mathematical explanation that can be easily transmitted here?"

    Your guitar fretboard is designed to produce pitches that approximate equal temperament - i.e. each of twelve semitones has the same frequency ratio (in that sense they are equal - our sense of pitch is close to logarithmic). An octave is 2:1 so that makes each semitone the twelfth root of two, 21/12, which is about 1.059. (Americans: semitone translates as halfstep.)

    The ratio between the third and second harmonics of an exactly periodic sound is 3:2 which we call a perfect fifth in just intonation, or a pure fifth. (More on the "exactly" later, and see Sound spectrum for more explanation.)

    The third harmonic, touched at the seventh fret: Seven equal-tempered semitones is 27/12 = 1.498, which is quite close to 3/2 (= 1.500). So an equal tempered fifth, plus an octave, almost equals the third harmonic, and so produces only very slow beats at ordinary pitches. Further, to play a fifth on the seventh fret, you've reduced your string length by about 1/3, so this is where you touch the string to get the third harmonic.

    Your fourth fret observation: stopping the string here ought to give an equal tempered major third (four semitones). Touching the string here will give the fifth harmonic, which is two octaves and a major third above the fundamental. Four equal-tempered semitones is 24/12 = 21/3 = 1.260, which is not very close to a 5/4, which is a major third in just intonation. Here is the root of most of the problems which require temperament, which is described in another FAQ called temperament. See also Tartini tones and temperament: a musician's introduction and the multimedia introduction at Tartini tones, consonance and temperament.

  • What is the "secret of Stradivarius"?

    Why this question is difficult. Although much has been written on this, it is a subject that attracts a lot of speculation and few facts.

    The astonishing thing about Stradivarius is that virtually all of the instruments attributed to him are judged to be excellent, and most of them comparable in quality with the very best modern instruments. To a small extent, some argue that this may be a tautology: his instruments have so long been treated as the optimum, that is almost impossible to do better: an instrument sounding brighter than all (modified) Strads would be judged to be too bright; one sounding mellower would be judged too mellow.

    To start, there are some complicating factors: One of the most obvious reasons why these instruments sound so good is that they are almost always played by superb violinists. It is also possible that much of the comment about the quality of Stradivarius' instruments may be complicated by the fact that the people who own or play them have a strong financial incentive to maintain their market price. Consequently, it is usually usually difficult to get them to agree to double blind tests. Non-owners who have the chance to play one very rarely do so blind, and often have expectations that may colour their opinions and/or their assessments. So one must be cautious. However, on a FAQ in music acoustics, we cannot get away from this question, so here is my contribution.

    How do they compare? Note that I wrote 'comparable' above and not (necessarily) 'better'. In a recent study conducted at the 8th International Violin Competition of Indianapolis, 2 famous Strads, one Guanari and three excellent modern violins were played and compared by 21 experienced players in double blind conditions. One of the modern violins was rated best, and one of the Strads last. (The lead author, Claudia Fritz, was a PhD student in this lab several years ago.)

    Differences among strads. However, it is important to note that Stradivarius' violins are judged to differ subtantially in character from one to another, just as modern instruments do. This is in part because wood samples differ substantially in mechanical properites, even if taken from the same tree. Makers can compensate for these differences to some extent.

    Strads don't sound like the violins Stradivarius made. The word 'modified' in the preceding paragraph is important. Few, if any, of Stradivarius' instruments today sound anything like the instruments he made, so no-one knows what a violin made by Stradivarius sounds like. Over the intervening years, virtually all have been subjected to the following changes:

    • disassembly,
    • a bass bar has been fitted, the plate and bar retuned, and the sound post has been replaced with a thicker one,
    • the necks and fingerboards have been removed, discarded and replaced with longer, heavier necks and fingerboards at a greater angle to the body,
    • new, taller bridges, with different shapes and acoustical properties, have been fitted,
    • they have then been strung with more massive strings (steel instead of gut) at a much higher tension and tuned about a semitone higher,
    • quite a few of them have been repaired in ways that would be expected to make substantial changes to the sound.
    To have an idea of the sound that Stradivarius would have heard from his instruments, go to a concert played by a group specialising in authentic performance. Some will play on old instruments, or more commonly on reproductions of old instruments, using gut strings and usually pitched about a semitone lower than the modern instrument. Many of them are in face imitations of violins by Stradivarius. The contrast with the modern instruments is striking. So the difference between how a Stradivarius violin sounded as he made it, and how it sounds now is great. John McLennan did his PhD in this lab, studying these changes. His thesis and related research is on this link.

    That said, the highly modified Stradivarius violins are still judged to sound good, so is there a secret?

    Is it in the varnish? Some people talk of secrets in the varnish. Most makers and players agree that violins sound better 'in the white', i.e. before varnish is applied, than after. So one of the tricks in varnish (certainly not a secret) is to use only only enough to protect the wood, and not enough to change the sound much. For those who like the romantic idea of the "secret of the Stradivarius", it's very attractive, however, to imagine some secret ingredient, and the varnish is a convenient place to imagine putting it.

    Is it in the wood? Was the wood special, or did it become special due to environmental or artificial treatments. Again, for those who like the romantic idea of the "secret", this would be a convenient place to imagine putting it. We list some reported additives below.

    So how did he do it? How is it that Stradivarius made a lot of consistently good quality instruments? Although some 'secrets' are listed in the next paragraph, it seems that nobody knows for sure. So here are some observations and speculations of mine. First, he had good materials. Modern demand is high and modern makers compete for limited stocks of the best wood. (The demand by the aviation industry for spruce in the first half of the twentieth century did not help: a lot of potential violins were shot down in the first world war.) Second, he had good training: he was an apprentice of Nicolo Amati. The third and most important point is this: he was a virtuoso maker. Take someone really gifted with all the right talents, give him superb training, provide him with a large supply of excellent materials, then put him in a town where good instruments are bought for good prices, in competition with other good makers. Result: Excellent instruments.

    The various 'secrets of Stradivarius'. It seems that, whenever someone claims to have discovered the 'secret of Stradivarius', the media become excited about it. It has been suggested that we should maintain a list of these 'secrets of Stradivarius'. Here are a few, to which readers may wish to add by emailing us:

    • Chemical treatment of the wood - un-named components. Nagyvary et al, 2006.
    • Chemical treatment of the wood - urine and dung. Nagyvary et al, 2000.
    • Chemical treatment of the wood - Borax. Nagyvary et al.
    • The 'little ice age' made the wood grow more slowly. Grissino-Mayer and Burckle, 2004.
    • The 'little ice age' allowed fungus to grow on the wood. Michael Rhonheimer and Francis Schwarze, 2009.
    • Stradivarius used the 'golden ratio' in the geometrical design. Anon., but widely mentioned, even by some violin makers.
    • "Experience. Stradivarius made his best violins between 1710 and 1720 at age 60 (died at age 93). His contemporaries didn't live to be 60 years old, so lacked experience." (Contributed by Jan Woning.) This hypotheis could be tested by comparing instruments made in his forties or earlier with instruments made in his sixties or later.
    • Shape and density. Sirr and Waddle, 2011. Measuring the shape and density of the wood using Xray scans could allow accurate replication of shape and density. But, given the variability in wood properties, would this produce the same acoustic properties? BBC report here.
    • Imperfections in the geometry. Tiny modifications including the geometry of the f-holes can be decteced by synchrotron radiation. Zanini 2012.
    So, a range of theories. But it is worth reading this recent study by Claudia Fritz.

  • How important to violin quality is ageing, exposure to climatic variation and the amount of playing?

    We have begun a long term study on this and related questions.

  • How can you work out the appropriate size for a sound hole when designing an instrument?

    Usually, the air resonance of a string instrument is set somewhere near the tuning of the second lowest string (a bit lower for guitars, a bit higher for bowed strings). I have included a discussion of the calculation on Helmholtz resonance, which gives a simple equation, and some warnings about the approximations used.

  • How can a resonating chamber amplify sounds? Where does the extra power (seemingly) come from?

    Let's compare a string on immoveable mountings (an unplugged electric guitar approaches this) with a string on an acoustic guitar. In the former, the bridge (almost) doesn't move, so no work is done by the string. The string itself is inefficient at moving air because it is thin and slips through the air easily, making almost no sound. So nearly all the energy of the pluck remains in the string, where it is gradually lost by internal friction.

    In contrast, the string on the acoustic guitar moves the belly of the instrument slightly. Even though the motion is slight, the belly is large enough to move air substantially and make a sound. So the string converts some of its energy to sound in the air. Consequently, its vibration decreases more rapidly than does that of a similar string on an electric guitar. (Internal losses in the string are still very important, however.)

    So there is no extra energy: the energy for the sound comes from the string. Which raises an obvious question: if there is no amplification, how does such a little vibrat make such a lot of sound? The answer is that our ears are rather sensitive (see our page on decibels and hearing). Consequently, even a small energy (even less than a millijoule) over several seconds makes a reasonably loud sound.

  • How much does one more violin add to the sound level of a section of n violins?

    Let's make a few simplifying assumptions: that each violin radiates the same power P (not usually the case in amateur orchestras!), that the listener is equally distant from all of them (hard to arrange) and that there is no simple phase relationship among the sounds from the different violins (this one is safe).

    Consider n violins, each with power P, that produce total power nP. Say that the intensity, at our listener's ear, due to one violin is I1. Thanks to our simplifying assumptions, the intensity due to n violins is then In = nI1. At this stage, you may need to look at our section on decibels, sound level and loudness. The sound level L1 (in decibels) for one violin, Ln for n violins and Ln+1 for n violins are all given by the definition

      L1 = 10 log(I1/I0) ,       Ln = 10 log(In/I0)       and       Ln+1 = 10 log(In+1/I0)

    where I0 is an arbitrary, finite reference intensity (and is not the sound level due to zero violins!). Now to the question: the increase in sound level when you go from n to n+1 violins is

      ΔL = 10 log(In+1/I0) − 10 log(In/I0)

    Because of our simplifying assumptions, the intensity is proportional to the number of violins. The site on decibels etc shows you how to handle logs, and explains why log(a) - log(b) = log(a/b). This allows us to write:

      ΔL = 10 log[(In+1/I0)/(In/I0)] = 10 log(In+1/In) = 10 log((n+1)/n) = 10 log(1+1/n).

    Reaching for your calculator, you'll see that adding the second violin adds 3 dB to the sound level produced by the first, the third adds 2 dB to the level produced by the front desk, the fourth adds 1 dB, and so on, and adding the 15th violin gives you an extra 0.3 dB. The decibel page gives you sound file examples of how much changes of 3 dB, 1 dB and 0.3 dB in level sound like.

    Of course, there is much more to it than that: multiple instruments give chorus effects that make the sound more complicated and give it a different quality. But if your orchestra has 15 firsts, the biggest difference will be your empty chair on stage.

  • Why does a thicker string sound less bright (have weaker high harmonics) than a thin string?

    All else equal, a thick string doesn't bend as easily as a thin one: it is harder to produce a sharp corner in a thick string. So, when you pluck or bow a thin string, you create a shape that has sharper corners. When you look at the harmonics needed to make up this shape (see What is a sound spectrum? for background), you'll see that more and stronger higher harmonics are required to make a sharp corner. So bowing or plucking a 'hard to bend' or stiffer string puts in fewer high harmonics.

    Further, a substantial fraction of the energy you put into a string is not converted into sound, but is lost in bending and unbending the string. So the stiffer string usually loses its high harmonics more quickly.

  • Finally, at the same pitch, a thicker string is usually shorter than a thin one: to play E4 on the top string of a guitar, you use the whole length. To play it on the B string, you need 3/4 or the length. To play it on the lowest string, you need only 1/4 or the length. So, all else equal, the high harmonics require sharper corners on the lower strings.

    I've referred to stiffer or 'hard to bend' strings rather than just thick strings. It is possible to make a thick string (or, more importantly, one with a high mass per unit length) that bends relatively easily by using a thin core and winding wire around it to increase its mass per unit length (and therefore stiffness). The low strings on guitars, pianos and usually violins are wound strings. This allows them to have stronger higher harmonics, and also improves their harmonicity. However, if you make the core too thin, the string is easy to break.

    For some basics about string vibrations, see Strings.

  • What causes sympathetic vibrations in a string? How is it different from the coupling between string and soundboard?
  • Sympathetic vibrations: You pluck the A string of a guitar, then damp it. You notice that the (high) E string is now vibrating, although you didn't touch it. Or you play an A on the G string of your violin, and