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| 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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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.
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.
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 corresponds to a pressure amplitude of 70 kPa rms, which in turn corresponds to a sound level of 191 dB. 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.
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.
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:
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.
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.
The second picture shows a hypothetical ear enclosure that 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 with negative gain (ie gain with a phase shift of 180°), which goes then to a loudspeaker. The circuit is such 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.
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.
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.
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.
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. The wave is propagated 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
(There's also a small numerical factor in there, 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 suppose that that is a sound wave.
The speed of sound is roughly proportional to the square root of the ratio of the temperature. In the early universe, temperatures were very high and, early enough, the speed of sound was comparable with the speed of light!
The cosmic microwave background (CMB) has expandeds as the universe itself expanded. The wavelengths of the sounds of the early universe have been expanded by a similar factor. So the sound waves have now ultralow frequencies.
These waves are of interest to cosmologists: Mark Whittle of the University of Virginia has done some research on it. He came to our lab to talk about it and one of the postdocs here, Alex Tarnopolsky, made some sound files for him, transposed 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 some aspects of cosmology. Download it here.
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. If you listen for it, 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.
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.
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. |
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.
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 fundamentall. 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.
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.
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.
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.
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.
For more about temperament, see the tutorial and application written by Simon Caplette, one of our students.
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.
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.
I am often asked this, and one can make a few observations. 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.
Let's now 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.
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:
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.
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.
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?)
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 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 temperment, 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.
Why this question is difficult. Much has been written on this. It is a subject that attracts a lot of speculation and few facts. One of the most obvious reasons why these instruments sound so good is that they are almost always played by brilliant 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 very strong financial incentive to maintain their market price. Consequently, it is usually usually very difficult to get them to agree to double blind tests. Non-owners who have the chance to play one rarely do so blind, and often have expectations that may colour their opinions. So one must be cautious. However, on a FAQ in music acoustics, we cannot get away from this question, so here is my contribution.
Differences among strads. The astonishing thing about Stradivarius is that virtually all of the instruments attributed to him are judged to be excellent, and (at least) comparable in quality with the best modern instruments. To a small extent, 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. 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. Over the intervening years, virtually all have been subjected to the following changes:
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', ie 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.
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.) 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, train him well, give him 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 publishes the solution to the 'secret of Stradivarius', the press 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:
We have begun a long term study on this and related questions.
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.
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 vibration 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.
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
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
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:
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.
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.
This question requires diagrams and a bit of concentration. I've devoted a whole web page to it at Pipes and harmonics.
Let us imagine a pulse of high pressure and therefore high density air travelling down the tube. When it reaches the end of the tube, its momentum carries it out into the open air, where it spreads out in all directions. Now, because it spreads out in all directions its pressure falls very quickly to nearly atmospheric pressure (the air outside is at atmospheric pressure). However, it still has the momentum to travel away from the end of the pipe. Consequently, it creates a little suction: the air following behind it in the tube is sucked out (a little like the air that is sucked behind a speeding truck).
Now a suction at the end of the tube draws air from further up the tube, and that draws air from further up the tube and so on. So the result is that a pulse of high pressure air travelling down the tube becomes a pulse of low pressure air travelling up the tube. We say that the pressure wave has been reflected at the open end, with a change in phase of 180°.
Compare this with what happens when a pulse of high density, high pressure air arrives at a closed end. It collides with the blockage at the end. It could be considered to 'bounce off' it: the high pressure on the blockage pushes air back in the way it came. Here we say that the pressure wave has been reflected at the closed end, with a change in phase of 0°.
Incidentally, a physicist would explain both of these in terms of the acoustic impedance. The acoustic impedance has an infinite value for the closed pipe, a very low value outside the pipe, and an intermediate value inside the pipe. The acoustic impedance is (in a limited sense) analagous to the refractive index for light: going from low to high acoustic impedance, there is reflection of the pressure wave with a phase change of 0°. Going from high to low, there is reflection with a phase change of 180°.
For an expanded explanation, see Open vs Closed Pipes.
First, look at the animation in the section above about reflections at an open end of a pipe. When a pulse of high pressure air gets to the end of the pipe, it spreads out, and that allows the reflection. What happens exactly at the end? inside the tube there is a plane wave, when the wave is radiating externally it is a spherical wave, but between the two there is some complicated geometry. In this phase, the pulse of air is not in the free, unimpeded air away from the pipe, nor in the tightly constrained environment of the pipe. It is somewhere between the two: unconstrained on one side, but constrained by the pipe on the other.
As we explain above, the reflection is caused by suction that results when the momentum of the pulse of air takes it away from the pipe. But this suction doesn't appear immediately when the pulse reaches the end of the pipe, but a little later, as it starts to spread out.
So the reflection appears to occur slightly beyond the open end of the pipe. To a rather good approximation, this effect can be calculated by saying that the effective length of the pipe is a bit longer than its geometrical length. The difference is called the end correction.
For a closed end, there is no such end correction. For a simple cylindrical pipe as shown above, the end effect at the open end is 0.6 times the radius. Note the consequence of this: all else equal, a large diameter organ pipe is a little flatter than a thin one.
If you look closely at the animations above, you'll see that we have included end effects. Although the geometrical lengths of the two pipes are equal, the open-open pipe has two end effects and so its effective length is slightly greater than that of the open-closed pipe. Hence the travelling pulses get successively further out of step with each lap of the pipes.
Metals expand by about .001% to .002% per °C. And in any case, the metal itself doesn't warm up much. So the change in the dimensions of the instrument are negligible.
Two things happen when the instrument warms up. First, the air inside becomes warmer. The speed of sound in air is proportional to the square root of the absolute temperature. Normal temperatures in a sax are roughly 300 K, so this effect is worth about 0.17% per °C (K is for Kelvin, the units for absolute temperature. A temp difference of 1°C and 1K are the same, so you convert to K simply by adding 273 K to the temp in °C.)
The other important thing is that the air in the instrument becomes humid: your breath is nearly saturated at 37°C and when it cools in the instrument, water condenses on the metal providing a water reservoir to keep the air nearly saturated in the instrument. The speed of sound is inversely proportional to the square root of the average molecular mass of the air. Water molecules are lighter than nitrogen or oxygen, so humid air is less dense than dry air, all else equal. (Yes, I know that non-scientists talk about humid air being 'heavy', but I think that by this they mean that one sweats less effectively in humid air.) From normal values of humidity to saturated can increase the speed of sound by 0.1 to 0.2%.
So it's easy to have 1% or even 2% or more increase in speed of sound, and therefore in playing frequency. A semitone is only 6%, so this is a lot.
The type and quality of wood used in string instruments is very important. In wind instruments, the materials are of much less importance, provided that they are sufficiently rigid. The walls do however vibrate--one can feel the vibration for many notes. In common brass and woodwind instruments, the walls radiate at most a very small fraction of the total sound, so any wall vibration contributes directly only a very small component of the sound. One can imagine that the slight vibration during playing could affect the motion of the air jet or reed driving the pipe, but this would be a small effect and I don't know of anyone who has measured it. The vibration of the instrument itself is also part of the feedback that the musician receives, and so may be important for psychological reasons. The vibration of the bells of trumpets is measurable and this might have an effect, probably indirect, on the sound. Again, a feedback on the lip of the player is a possible mechanism.
However, what is more important for wind instruments, especially woodwinds, is that the material usually influences the shape, often in subtle ways. For example, different woods give different surface roughness when the bore is made with the same reamer. This is complicated by the effects of humidity and oiling on different textures. Rough surfaces can make a difference not only to the timbre but to the pitch, as well.
Different metals adopt different shapes when they are formed on the same mandrill. Different materials may be easier or harder to shape, and so the sharpness of corners may be different at, for example, the junction of a tone hole with the bore, or the edge of the embouchure hole in a flute. Further, different materials have different prices. If an instrument is to be made from very expensive materials, it is likely that the most talented or experienced maker in the factory will be asked to make the instrument. All of these effects may produce differences in the detailed shape of the instrument, and some of these differences are likely to be at least as important as the effect of any vibrations transmitted to the walls.
For metal flutes, a nice experiment was done by Widholm and colleagues at the Universitat fur Musik in Vienna. For their study, they used seven flutes made by Muramatsu that were solid silver, 9 karat gold, 14 karat gold, 24 karat gold, solid platinum, platinum plated and silver plated. (Although they were the same model, these flutes may not have been identical in shape, for the reasons mentioned above.) Seven flutists (from the Vienna Philharmonic and the Vienna Opera Orchestra) played them, and were among the 15 experienced professional players that formed the listening panel. Two different sets of blind listening tests were conducted. In one, no instrument was correctly identified, in the second, only the solid silver instrument was identified by a significant fraction of the listeners. There was nearly complete confusion* over the quality and identity of the instruments. The authors conclude that there was 'no evidence that the wall material has any appreciable effect on the sound color or dynamic range of the instrument'. (See also How important are the materials from which string instruments are made?)
* Of couse, everyone wants to know:
even if the differences were small, which one did best?
Ranked on a scale of 1 to 5, the solid silver did 'best'
and the 9 carat gold did 'worst' . However, if one rates
the instruments by subtracting the number of 'don't like
it' from the number of 'like it', the 9 carat gold did best
and the solid silver did worst. This apparently paradoxical
result is due to the statistical variations in very similar
rankings.
Widholm, G., Linortner, R., Kausel, W. and Bertsch,
M. (2001) "Silver, gold, platinum--and the sound of the
flute" Proc. International Symposium on Musical Acoustics,
Perugia. D.Bonsi, D.Gonzalez, D.Stanzial, eds, pp 277-280.
Two ways. Put a small piece of cotton wool in the headjoint. This works well on the low range, but less well at the high pitches where you will wake the neighbours. Alternatively, put a piece of modelling clay on the edge of the embouchure hole, just opposite where you blow.
Several effects are used by the player to achieve this spectacular, smooth ascent in pitch. First, the clarinet has seven holes that are covered by the fingers, rather than keys. By gradually sliding the finger off the hole, one can obtain a smooth transition from one note to the next. This is how the first part of the glissando is achieved. The player also changes the position and force of the lower lip on the reed, thereby changing its natural frequency.
S/he also uses the resonances in the vocal tract. Normally, the resonances of the instrument are so strong (have such high acoustic impedance) compared with those of the vocal tract that the latter make only modest changes to the pitch. However, in the upper range of the instrument the player can produce resonances of the vocal tract that can be stronger than those of the instrument, so the note played tends to follow that of the tract resonances, which the player increases smoothly--with some considerable help from the sliding fingers and the change in the natural frequency of the reed as the player's bite changes simultaneously.
In the time of Bach and Mozart, trumpets and horns had no valves. Instead, the players played notes in the harmonic series, and lipped them into tune. See Harmonics of the natural trumpet and horn on our brass acoustics site. If the piece was in C major, and you had a horn of the right length, you could play C3, G3, C4, E4, G4, A*4, C5, D5, E5, F*5, G5 A5 etc, where "*" means half sharp--the notes you had to lip up or down. You then had a "horn in C". If the piece of music were in Bb instead of C, then you could remove a piece of pipe (called a crook) from the instrument, replace it with a longer one, to give you a "horn in Bb". The series then becomes Bb2, F3, Bb3, D4 etc. So your lowest (normal) note is a Bb, rather than a C, the note above that is F instead of G. And there was a different series to learn for each key. To make it easy for players, the music for the horn was transposed to the key of C and the player was told to insert the appropriate crook. (If the score is in Bb, then the "Bb horn" player's part would be written with no flats, and each note raised a tone.) This tradition continued for horns long after the invention of valves.
Modern brass instruments have valves (or slides) and can play all notes with relative ease. However, the tradition has remained to call C the lowest note played with no valves depressed or the slide completely in. For most trumpets, this is Bb (though trumpets are also made in C, D, A and Eb) and for most horns it is F. An advantage is that, once you have learned the fingering for the trumpet (in Bb), you also know the fingering for the horn in F (at least if you don't use the thumb valve), the euphonium in Eb etc. Or at least this works in military bands. For orchestras, you had better learn to transpose.
A similar situation occurs in woodwinds. The alto flute is mechanically similar to the normal flute, but is 33% longer. So the same fingering plays notes with frequencies a fourth lower in pitch. For that reason, it is called an alto flute in G, and the music is written a fourth higher than it sounds: the flutist can swap without having to learn new fingerings. Similarly, the cor anglais (in F) sounds a fifth lower than the oboe for the same fingering. (In the few nervous moments between the first and second movements of Dvorak's New World Symphony, the player who has put down an oboe and picked up a cor anglais doesn't have to think about new fingerings.)
Clarinets and saxophones come in a large range of sizes, and many of them are in Eb or Bb. You can go from alto to tenor saxophone without learning new fingerings. However, most clarinettists in orchestras will carry a case with two clarinets that share the same mouthpiece: one in Bb and one in A. So why have two instruments only a semitone apart? This is a vestige of the days when clarinets had few keys and it was (more) difficult to play in keys with several sharps or flats. So if the piece had flats, the clarinet part was written for Bb clarinet (two fewer flats), if it had sharps, the clarinet part was written for A clarinet (three fewer sharps).
Of course, one could manage without transposing instruments. Recorder parts are not transposed, and the player learns that all holes closed is F on an alto recorder and C on a tenor. Further, while an orchestral tenor trombonist calls first position Bb and reads bass clef, a brass band trombonist may learn that first position is C and reads treble clef, displaced by an octave.
In hindsight, it is a messy result. However, the complications of overcoming it are considerable, so we are likely to retain it for a long time. And yes, you'd better learn how to transpose at some stage.
Let's use the flute as an example, but the principle is similar for all woodwinds. In a 'normal' note played on a woodwind instrument, especially in the low range, all (or nearly all) of the tone holes downstream of a certain point are open, and all those upstream from that point are closed. The pipe behaves approximately like a pipe that stops near the first open hole. (For more detail, see tone holes on our introduction to flute acoustics.) Multiphonics are usually produced by opening a single tonehole, usually a small one, somewhere in the line of closed tone holes, but not at 1/2 or 1/3 of the way along the pipe. (At these positions, the hole would function as a register hole.) A wave travelling down the closed-hole part of the pipe can be partially reflected at the first open hole. And part of the wave can travel down to the first open hole in the series of open holes, where it is reflected. This gives rise to two standing waves, with different wavelength, and therefore different pitch. Figure 1 in our downloadable paper The virtual Boehm flute--a web service that predicts multiphonics, microtones and alternative fingerings" shows this diagrammatically and gives further explanation. Technically, we could also refer to the acoustic impedance spectrum or "frequency response" of the instrument for a given fingering. Two non-harmonic minima in this function can give rise to multiphonics. You can inspect many such spectra for multiphonics with the virtual flute. There is a further discussion of multiphonics on the topic what is an undertone?
Sometimes when a flutist plays a high note, one hears a faint note at a lower pitch. Normally there is no exact harmonic relation between the two. This is what I call an undertone. (This is not to be confused with Tartini tones or other combination tones, in which two notes, often from different instruments, interact.)
The undertone is a special case of a multiphonic, usually produced accidentally. Let's take a simple example. Play the note D6, but play it with a relatively large opening between the lips. Then, still with large lip opening, reduce the air speed gradually. When the jet is slow enough, the note will drop down to C5 with a rather breathy tone. However, on the way you will pass through the multiphonic C5&D6. By adjusting the jet speed, you can vary the proportions of the two notes: when they are about equally loud, you have a standard multiphonic. (If you have never played multiphonics before, this is an easy one with which to start, although it's not very interesting. See multiphonic fingerings for more.)
Let's now see why this works by consulting the database on flute acoustics. The fingering for D6 is a bit like that for G4, except that you raise your left index finger and open one of the small holes on the flute. The harmonics of G4 are G4, G5, D6 etc, and all can be played on the G4 fingering. When you open the index finger key, you create a register hole that both weakens substantially and mistunes the second resonance, the one that supports G5, so that G5 becomes unplayable. This is the object of a register hole: you don't want your D6 dropping down to G5 when you decrescendo. The lowest resonance is also very much changed: it is weakened a little, and its frequency is raised. You can think of this lowest resonance as a cross fingering for C5.
To understand this, open a new window for G4 and another for D6. The impedance curves tell us (approximately) which notes the flute can play for any fingering. The flute will usually play a note where it has a (sufficiently strong) resonance, and the resonances are the deep minima in the curves of impedance. The first three of these for G4 are at about 400, 800 and 1,200 Hz, and they play G4, G5 and G6. They also support the first three harmonics of G4, so these frequency components are consequently strong in the sound spectrum of the note played. The first three minima for D6 show a reasonably strong minimum for C5 (near 520 Hz), a shallow (unplayable) minima above that (near 900 Hz), then the strong minimum for D6 at 1.2 kHz. There are no minima to support harmonics of C5, which is why this (and other crossfingered notes) sounds darker and weaker than a normal fingering. (In fact, this is the normal fingering for C5 on a baroque flute and explains why that note is darker than its neighbours.) If you found this paragraph heavy going but are still keen to understand it all, you might want to read our Introduction to flute acoustics. If you would like to see
