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.
Do end corrections, such as those produced by varying the key clearance above a tone hole, have the same effect on different registers?
I'm warning in advance that this question is a bit subtle, and so I'll have to refer you to some technical discussion, but shall try to answer it here.
If the frequency were well below the cutoff frequency, then the end effect would be primarily due to the mass of the air in the open tone hole. (See a brief explanation of cutoff frequency.) So the acoustic impedance would be very small and it would be almost completely an inertance, therefore we could model it as a small extra length, and this would not narrow octaves.
However, on real geometry instruments, the second register is rarely well below the cutoff frequency. That's why cross fingerings produce semitones
on small hole instruments, even in the first register, and why they flatten
notes in the second register of larger hole instruments. (We have a paper on cutoff frequencies and cross fingering on our 'publications' site where you can
get explicit values and examples.)
So, if you are serious about tuning, you can rarely neglect the wave that
is transmitted past the first or second open hole. At low frequencies, this
impedance (in parallel with the open hole) is usually inertive. But as you
get close to the cutoff, it can have either sign. So it is possible for a cross
fingering to sharpen a note, particularly in the second register of a small hole instrument like
recorder or shakuhachi.
To imagine this without doing calculations, it is sometimes possible to
think of the cross fingered note as being in a high register, with the open hole
under discussion functioning as a register hole. However, near cut off, the
situation becomes rather complicated and such qualitative arguments cannot be
trusted.
Of course, musicians naturally prefer such 'arm-waving' explanations, even
if they are not very reliable. So, on our databases for flute, clarinet and saxophone, I tried to give a qualitative explanation, except for the very highest
notes. (Now that will be a challenge when we get to the bassoon!)
How far is 'well below cutoff'? That depends on the precision required. Remember
too that cutoff varies somewhat from note to note.
How important is the air speed in playing musical wind instruments? And how does it relate to pressure? (Teachers do talk about it a lot.)
First let's distinguish two quantities that are often confused. Like other speeds, air speed is measured in metres per second.
Air flow (or air flow rate) is measured in cubic metres per second or, more practically, in litres per second. In playing a musical instrument, the air flow through your mouth is typically in the range 0.1 to 1 litre per second*. The air speed times the cross section through which it flows gives the flow rate: if a wind of 1 m/s flows through a window with area 1 square metre, then one cubic metre of air enters the window per second. Let's do some instrument examples:
Take a the typical wind player's flow range of 0.1 to 1 litre per second, divide by 10 square centimetres (a typical cross section in the mouth) and the air speed in the mouth is 0.1 to 1 metre per second.
Divide the same flow by 1 square centimetre (the cross section of a small instrument bore). The air speed is 1 to 10 metres per second. This is the average+ air speed in the bore of that instrument at that point. (If it were 1 m/s at the entrance to a saxophone, it would decrease steadily down the bore as the cross section increased.)
Divide it by 0.1 square centimetres (a possible average gap between a brass player's or flautist's lips, or a reed-mouthpiece gap) and the speed is 10 to 100 metres per second. A speed of 100 metres per second requires a mouth pressure of about 6 kPa, which is a moderately large but possible blowing pressure. Thus a player could simultaneously have an air speed of 1 m/s in the back of the mouth and 100 m/s between the lips.
The speed of air between the lips can be important functionally. For the flute, the time that the air jet takes to travel from the lips to the sharp edge of the mouthpiece is important to the register (fast jet for high pitch). For brass instruments, fast moving air between the lips exerts a suction that helps close the lips in their vibration.
For reed instruments, the fast moving air between reeds or between mouthpiece and reed can also contribute to closing that aperture (see the pages on flute acoustics, clarinet acoustics, saxophone acoustics and brass acoustics for details).
The speed of the air entering the instrument (or leaving the lips in the case of the flute) is approximately proportional to the square root of the blowing pressure** in the mouth. So blowing pressure determines the air speed into the instrument, and air speed times the cross section(between lips or reeds gives the flow. So, if you keep that aperture constant and the blowing pressure constant, then the air speed in your mouth depends on the cross section of the mouth and vocal tract. So the speed might be fast at a point where the tongue constricts the tract, but slower elsewhere – and usually much slower than the rate at which it enters the instrument. It might also be fast at the larynx, if you play with your vocal folds adducted. Scientists who study wind instruments are much more interested in the blowing pressure and the air speed between the lips or reeds than we are in the varying air speeds in the mouth.
Speed means pretty much the same thing to scientists as it does in common use. But it's possible that teachers don't mean the same thing when they say 'maintain air speed'. So what’s it about? They may be using the phrase metaphorically. Or perhaps a teacher's exhortation to maintain air speed in the mouth is an implicit instruction not to vary the blowing pressure too much (useful because the pitch depends on blowing pressure, all else equal). Or it may be an indirect way of promoting a vocal tract shape that has particular acoustic properties. For example, clarinettists and saxophonists tune their vocal tract resonances for altissimo playing and exotic effects. However, when playing the clarino register, clarinettists also adjust the resonant frequency of the tract even when though they are not tuning it to the note played. Or perhaps the discussion of air speed tends to achieve some other effect, in which case I’m interested to hear suggestions about what it might be.
* You can estimate your rate of air flow. Take a big breath of air and exhale all of it and you have probably exhaled about five litres. If you do that in five seconds (a loud note on say trombone or saxophone) then your flow rate is one litre per second. If you exhale it over 50 seconds (a long note on oboe) then your flow rate is one tenth of a litre per second. If you also know the aperture (a flutist can look in the mirror to see the gap between the lips) you can estimate the air speed.
+ This discussion concerns average or DC flow. Inside the instrument, the acoustic or AC flow becomes important: it can have a magnitude larger than the average flow, so that the air at the bell is oscillating in and out of the instrument much faster than it is (on average) flowing through it.
** This result, sometimes known as the Bernoulli effect, comes from Newton's second law via the work-energy theorem: the gain in kinetic energy of the air in this narrow passage equals the work done by the air pressure in the mouth. This kinetic energy is then nearly all lost downstream in turbulence.
Warming up wind instruments. As a saxophone warms up, you need to pull the mouthpiece out to keep it in tune. Thing is, this doesn't seem right to me. As a sax gets warmer, I should think it would expand, making the tube bigger and longer, forcing a player to compensate by pushing the mouthpiece in to shorten it up.
Metals expand by about .001% to .002% per °C. And in any case, the metal itself doesn't warm up much. So the expansion of the metal is not significant when the instrument warms up. Some important effects do occur, however.
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 equals a temperature difference of 1 K. Add 273 K to the temp in °C to convert.)
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.
Countering this, happily, is the effect of CO2.
This is denser than ordinary air and, especially if it's been a while since you last breathed in, the expired air has increasing concentration of CO2, which lowers the pitch. In some circumstances, this effect can be bigger than those of heating and humidity, especially in big brass instruments (e.g. tuba) that require a lot of air and that don't heat up much.
How important are the materials from which instruments are made?
It varies among different instrument families. The type and quality of wood used in string instruments is very important, because the vibrating wood provides the sound of string instruments. In wind instruments, the materials are of much less importance, provided that they are sufficiently rigid. The walls do vibrate – one can sometimes feel the vibration for some notes. In common brass and woodwind instruments, however, the walls radiate at most an extremely small fraction of the total sound, so any wall vibration contributes directly at most a tiny component of the sound. One can imagine that the slight vibration during playing could affect the motion of the air jet, reed or lips driving the pipe, but this would be a very small effect. 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 and horns is measurable and this might have a small effect, probably indirect, on the
sound. Again, a feedback on the lip of the player is a possible mechanism.
The appearance on the market of plastic trombones raises the obvious question: do they sound as brassy as trombones made of brass? The simple answer is that they sound very similar to trombones made of brass. Acoustic differences between two different trombones from two different manufacturers (in whatever materials) can usually be measured but these differences are likely to have more to do with small differences in geometry than with differences in materials.
In wind instruments, especially woodwinds, 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 mandrel or 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 mandrel. 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 comparison 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 the wall material does not appreciably affect 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 theflute" Proc. International Symposium on Musical Acoustics, Perugia. D.Bonsi, D.Gonzalez, D.Stanzial, eds, pp 277-280.
In principle, one could imagine that the acoustic waves in the bore could couple to mechanical modes in the walls of the instrument. However, nearly all wind instruments have circular cross sections (exceptions include square cross-section organ pipes) and this makes such coupling very much weaker, because small changes in the eccentricity of an ellipse have least effect on the area when the ellipse is a circle. Further, instruments are made of materials rigid enough to withstanding handling, they have chimneys and other structures tend to stiffen them and the player's hands tend to damp wall vibrations. Finally, the relevant wall modes occur at frequencies well above those which have substantial acoustic pressure variation. Nevertheless, such coupling can be measured – if you can avoid all the above conditions. Neuf et al.* made an artifical 'instrument' with an elliptical cross section, out of metal much thinner than that used for wind instruments, with low frequency breathing modes, with no chimneys or other stiffening, and with no hand damping. In this very extreme example, they were able to measure coupling between wall vibrations and the acoustic waves at the frequencies of some of the resonant breathing modes of the walls. They conclude, of course, that there would be no such effect on musical instruments, with the possible exception of some organ pipes.
* Neuf, G, Gautier, F, Dalmont, J-P and Gilbert, J (2008) "Influence of wall vibrations on the behavior of a simplified wind instrument" J. Acoust. Soc. America, 124, 1320. The choice of elliptical cross-section was made because, for such a tube, wall vibrations have a first order effect on cross section, but no such effect on a circular cross section.
There is a subtle difference for brass instruments, because they have a bell, which is not cylindrical and not so rigid as the rest of the instrument. The vibrations of the bell can change the reflection of the wave in the bore, which would be expected to have some change on the sound and also perhaps on the sensation for the player. One experiment measured the input impedance and transfer function of a trumpet in two such conditions. Would you hear these differences? Some researchers think so.
Kausel, W., Zietlow, D.W. and Moore, T.R. (2010) "Influence of wall vibrations on the sound of brass wind
instruments" J. Acoust. Soc. America, 128, 3161.
This topic will probably be discussed for a long time. Some musicians clearly prefer one instrument (made of material X) to another (made of material Y). The musicians are sensitive judges. The question that scientists will ask is whether the differences are due to some acoustical effect of the materials (which seems unlikely in woodwinds), due to a different shape that is the result of using a different material (which is plausible) or due to some more subtle, indirect effect.
Can you make a flute mute? Is there a way I can practise in the middle of the night without disturbing the whole house and at the same time learning how to play notes and music?
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.
How do clarinet players do that big glissando in Rhapsody in Blue?
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 can also change the position and force of the lower lip on the reed, thereby changing its natural frequency.
Most importantly, however, 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 comparable in strength with 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. This is explained in detail in a recent scientific paper.
Why is a trumpet called a Bb instrument and an alto sax an Eb instrument? I saw your website and I was wondering how you would simply explain to a beginning clarinet or trumpet player why they have transposing instruments.
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.
How can I change a C trumpet to a Bb trumpet and vice versa?
Some trumpets have two tuning slides: the normal U shaped piece with the water key and two straight pieces, spaced with rods, that fit between the U slide and the trumpet. Sometimes, when the two straight pipes are in place, the trumpet plays Bb with no keys depressed. Take it out and it plays C. (In other cases, it converts a high pitch Bb trumpet to a low pitch Bb trumpet, in others an A trumpet to a Bb trumpet. You'll find out easily!)
Caution: The second valve should increase the effective length of the trumpet by 6%, the first by more than 12%, the third by 19%. But 6% of the length is different depending on whether or not the extra slide is in. So, to convert from C to Bb, you not only need to add the extra slide, but also to increase the length of all your valve slides, which may mean new slides.
What are multiphonics? How can you play two
notes at a time on a woodwind instrument?
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?
What happens when you sing or whistle into a musical instrument?
It is usually possible to sing while playing a wind instrument, and this causes one type of multiphonic, where your voice (your vibrating vocal folds) makes one note and the instrument the other. Normally, the pitch of your voice is determined by the pressure in your lungs and the tension and geometry of your vocal folds. (So your voice is not like a musical instrument: the resonances of the vocal tract, which have rather high frequencies, rarely control the vibration of your vocal folds, which have much lower frequency.) This raises the interesting question: why does the instrument control the frequency of vibration of your vocal folds the way it controls the vibration of the reed or your lips, depending on which instrument you are playing? The answer is that, if the frequencies are comparable, it can happen.
The complete answer is a bit subtle and involves impedance matching.
(When you whistle normally, one of the resonances in your vocal tract
controls the frequency of vibration of the air jet and thus determines the pitch. If you seal a musical instrument around the outside of your lips you'll now have two resonators joined to it and rather more complicated conditions. When I tried that, I couldn't whistle.)
What is an undertone in flute playing and what causes it?
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, thi 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 this particular multiphonic discussed in technical detail, download this this paper.
In the sound spectrum for D6, you will see the fundamental of D6 (near 1.2 kHz), which corresponds with its resonance in the impedance graph, but no trace of undertone: the impedance curve shows a resonance near 520 Hz (C5) but there is no signal in the sound spectrum at this frequency. No surprise: the recording is of a professional flutist. However, you can now see how a beginning flutist, with a wide lip opening and not much breath control, might play D6 with an undertone: s/he is playing a multiphonic with a weak lower note.
In the fourth octave, it is much easier to produce an undertone, because the required jet speed is very high, and so it's easy to blow too slowly! For instance, if you look at the sound file for the note F7 (open a new window for F7), there are a few things to observe. First, there is more breath sound because the required jet speed is much higher. Second, the resonances are much weaker (this is because there is more 'friction' loss between the air and the flute tube at high frequency). You will also observe in the sound spectrum that, as well as the strong harmonics at F7 and its harmonic F8 (about 2.8 and 5.6 kHz), there is a component a little above 2 kHz (near C7). Personally, I cannot hear this pitch but, if I could, that would be an undertone. When less accomplished flutists play F7 with this fingering, they are more likely to produce noticeable undertones.
Finally, we should note that Geoff recorded these sound files before we had built the virtual flute (TVF). If we asked him to record F7 today, he would use one of the fingerings that TVF recommends, some of which are substantially easier to play and so yield less breath sound and less chance of an undertone.
What is an end correction for a wind instrument? Can you explain it in layman's terms? And what causes it?
The simplest calculations that we can do to calculate the frequency that a wind instrument plays turn out to be only approximately correct, for reasons we explain below. The 'effective length' is the value of length that, when substituted into these approximate equations, gives the correct (ie measured) frequency. The difference between the 'effective length' and the real length (the one you measure with a ruler) is called the end correction.
For example, suppose that I take a piece of pipe, 170 mm long, sealed at one end, and I excite a resonance. I could do this by playing a pure tone nearby, then varying its frequency until I found the value at which the pipe would resonate -- ie start to produce its own strong vibration. ( I could also excite it by blowing over the end, as one does with a quena or shakuhachi, but then my face would change the end effect. Indeed, this is one of the important performance details of shakuhachi playing: by changing the angle with the face I can change both the end effect and the jet length, and get spectacular changes in pitch. See shakuhachi for details. There is a similar but less variable effect for the flute, discussed below.)
Now to the calculation of frequency. Suppose that the temperature and humidity are such that the speed of sound is 340 ;m/s (to make the numbers convenient). Because the pipe is closed at one end like a clarinet, I look up Clarinet Acoustics and see that the lowest resonance should have a wavelength about 4 times longer than the closed pipe. (If the pipe had been open, I should have looked up Flute Acoustics and seen that the wavelength is twice the length of the open pipe). Now 4 times 170 ;mm is a wavelength of 0.68 ;m , and so the frequency, which is speed of sound divided by wavelength, should be approximately 340/0.68 ;Hz = 500 ;Hz. (Similarly, a pipe open at both ends has a wave of 2 times 170 ;mm is a wavelength of 0.34 ;m and so a frequency of approximately 1000 ;Hz.)
I now take the pipes, excite them acoustically, and the resulting sounds have frequencies of approximately500 ;Hz and 1000 ;Hz for the closed and open pipes respectively. But not exactly. In fact, the measured frequency is slightly lower than what I calculate, and the bigger the diameter of the pipe, the bigger the depression of the pitch. The pipe could be said to behave as though it were a little longer than it really is. (In other words, the 1/4 wavelength (1/2 wavelength for open pipes) of the simplest diagrams is slightly longer than the pipe.) Now the effective length is the length that would give me exactly the measured frequency. Thus, as we said above, the effective length minus the real length is called the end correction. (For the open pipe, there will be two end corrections, one for each open end.)
What causes the end correction was first analysed quanitatively, for the simplest case, by John Strutt (a.k.a. Lord Rayleigh). When the air in a pipe vibrates in a resonance, it does so along the axis, with maximum vibration at an open end. Just outside the open end is some air that must be pushed forward and backwards by the vibration of air inside the pipe. That air has mass and inertia, and its that inertia that lowers the pitch. Outside the pipe, the sound wave radiates in nearly all directions, so the further you go from the open end, the less the effect. So only air very near to the pipe is involved. We can imagine this extra air as making the pipe effectively longer than it really is. So how much longer? For a simple open pipe, the extra length is about 0.6 times its radius.
End corrections are more complicated in real instruments. The end correction at the foot of a flute for the lowest note of the instrument is indeed about 0.6 times its radius. It's different if there is a bell. For example, see this link for the effect of a clarinet bell. It's also complicated at the other end. For a clarinet, the pipe is not completely closed, and the end effect depends on the properties of the reed. For the flute, the pipe is open at both ends, but you can vary how open it is by rolling the embouchure hole towards you (which makes a longer end effect -- the note goes flat) or away (conversely). There is also the small volume between embouchure hole and the cork -- see lipping up and down on our page introduction to flute acoustics for further explanation. For brass instruments, the mouthpiece, the mouthpipe, the flare and the bell complicate things even more: see effects of bell and mouthpiece on our introduction to brass acoustics.
Finally, if we consider a woodwind instrument with several open tone holes, it is not just the air outside the first tone hole that must be vibrated, but air inside the bore, too. So the end correction is longer here. It is also longer to explain, and it becomes longer and more complicated still when there are cross fingerings. If you want to see how this works, download our scientific paper, in which we measure how far waves of different frequencies propagate past the first open hole, both for simple and cross fingerings.
What is the difference between whistle tones and edge tones on a flute?
I was about to try to answer this when I rememberd Benoit Fabre who, as as part of presentation to a scientific conference on music acoustics, gave a brilliant performance on the flute, using whistle tones. Here is his response:
There is considerable confusion between WhistleTone (WT) and Edge Tone (ET), probably because both are played very softly. Two arguments showing that WT are not ET. 1. The frequency of oscillation of a WT is always fixed at a passive resonance of the pipe and jumps from resonance to resonance as the jet speed increases, while the frequency of an ET rises continuously as the speed of the jet increases. 2. The amplitude of oscillation, although very weak to the ear (because one blows very very softly), when one expresses the amplitude in nondimensional form (which approximately means if one expresses it as a fraction of its maximum possible value), is stronger than that of the normal playing regime: in the measures we made, the amplitude is about 100 times stronger than that expected for an ET.
The first is easy for anyone to verify, the second requires some experimental dedication. Now to describe what happens:
In the 'normal' playing regime, the propagation of perturbation on the jet (at a speed about half that of the jet) induces a delay of about a half period (of the fundamental of the note played). When playing WT, the jet speed is much lower, and the delay becomes about 3/2 ; 5/2 ; 7/2 etc periods, so that it arrives with the same phase delay (approximately a whole number of cycles) in all cases. In other words, the jet oscillates through half a hydrodynamic wave in the normal playing regime and through 3/2 ; 5/2 etc waves in WT: all the interest of playing WT is in finding an appropriate combination of bore resonance and hydrodynamic jet mode, by varying the fingering on the flute and the speed of the jet. In normal playing, one always maintains the the first mode of the jet and one uses different modes of the pipe.
Finally, the nondimensional amplitude en a WT is larger than in normal playing: this is an observation. The explanation could be that the amplitude is so weak that mechanisms that saturate the oscillation are different. This is vortex shedding for normal playing, and is not known for the WT.
What is the process of homogenisation? What (if anything) does it do to wind instruments?
It's difficult to be clear about this, because the vendors of homogenisation keep the details secret. However, one practitioner who charged customers money for the privelege of having him hold a vibrator near the instrument is quoted as saying that the molecules at the points of tension in the instrument are equalised or polarised. I am unaware of any double blind comparisons done to test whether the process works.
Is there any scientific background? Well, the electric polarisability of conductors (especially silver and gold) is high and that the value for hydrocarbons (including petroleum) is low. So a mixture of these components could readily be distinguished according to their polarisation in an applied field. Perhaps homogenization was the inspiration for the observation that a fuel and his money are soon parted.
Fat pipes vs thin pipes: why does the pitch of a fat pipe of an organ depend more on the blowing pressure? and
Why does a narrower tube, for the same length, favour the upper register?
It's worth discussing these questions together. The reflection at the open end of a pipe depends on the ratio of the wavelength to the radius of the pipe. This effect is important for the high resonances, and thus for the high harmonics in the sound. Further, the end effects at both ends are functions of frequency.
For the same wavelength, the reflection coefficient is higher for a narrow pipe than for a wide. The greater the reflection, the stronger the resonance (all else equal), and the smaller the band of frequencies for which it will occur. Consequently, the high resonances of a narrow pipe are also narrow: a narrower band of frequencies will cause resonance in a narrow pipe than in a wide pipe of the same length.
The frequency dependence of the end effect is important. This effect is greater for wide pipes, so in wide pipes (such as the flute rank) the higher resonances are more "out of tune" (ie less close to harmonic ratios of the first resonance). Thus the higher harmonics of the jet do not have a resonance to support their vibration, or to provide an impedance matcher to the radiation field. (The jet partials are exactly harmonic -- see How harmonic are harmonics?) So the 'string' stops on a pipe organ just use very narrow flute pipes.
As another consequence, one can 'bend' the note more on a wide pipe: one can drive it at a frequency further from its resonance.
A cone overblows at the octave, a cylinder at the 12th. What happens as we make our cone closer and closer to being a cylinder? Is there a point where it will suddenly change (if so, where), or is there an 'almost cylinder' shape which is unstable as between the two options?
The change in the frequency of the resonances is a continuous function of the cone angle. The didjeridu provides an example of a range of dofferent instruments covering quite a large range of cone angles. Some are nearly cylindrical and overblow close to a twelfth. None are ever conical, but some approximate truncated cones with varying angles. The higher the angle (for a given length and mouthpiece diameter) the lower the note of the next register. A typical didjeridu overblows about a tenth.
An interesting point: most wind instruments have harmonically related resonances, at least for notes in the lowest range. These support higher harmonics and add to the timbral brightness, loudness and pitch stability. For the didjeridu, harmonically tuned resonances are not advantageous and might in some cases be disadvantageous. The reasons are subtle, and are discussed here.
Changing the cone angle of a bore changes the mode relationships continuously. So Why doesn't the cone angle affect the overblowing interval on woodwinds?
Let's begin with the didjeridu, which is often approximately a truncated cone. The angle varies considerably among instruments. If you use a plastic pipe as a didjeridu, the hoot note (the first overblown note) of this cylindrical instrument is nearly a twelfth above the drone. Most real didjeridus are somewhat flared, however, as mentioned above, and they overblow typically a tenth.
So the question is a good one: how is it that the narrow angle of a bassoon (0.8°) and the wide angle of a soprano sax (3.5°) overblow octaves? There's a related puzzle: the wavelength of the note that these instruments play is not twice the length of the instrument, but rather about twice the length of the complete cone, i.e. (about twice) the length obtained by extrapolating the cone to a point, well beyond the mouthpiece. (See Pipes and harmonics for the sounding frequencies of conical and cylindrical pipes.)
The explanation is that woodwind instruments are not simply truncated cones. The inside the saxophone mouthpiece is not a continuation of the conical bore: there is an extra volume in the mouthpiece. Informally, we could can describe it this way: imagine a pulse of air flowing up the bore towards the reed, in both a complete cone and a real saxophone. In the saxophone, the air arrives at the mouthpiece and starts pouring into it, gradually raising the pressure. When the pressure is high enough, it forces the air back, and a reflection has occurred. Of course, the bigger the volume of the mouthpiece, the longer it takes before the pressure builds up and the reflection occurs. Meanwhile, the pulse in the complete cone is completing the longer path to the end of the bore, and then its reflection occurs. (Technically, we would say that this volume is an acoustic compliance and, using a perturbation method due to Helmholtz, this compliance flattens all of the resonances that have a pressure antinode there.) It turns out that, if you make the volume of the mouthpiece equal to the missing volume of the cone, the frequency of the first resonance is about the same, although this approximation only works for low frequencies and if the truncation is a reasonably small fraction of the total length.
A complication arises because the reed is also compliant: it takes a bit of air flow to deform it and thus to increase the pressure. So the reed's passive operation behaves somewhat like an extra air volume, in parallel with the air in the mouthpiece. (About 1 ml for a clarinet, about 3 ml for a tenor sax. Soft reeds and a more relaxed bit both give greater compliance, so you then have to push the mouthpiece in and reduce the air volume to get the instrument and its octaves in tune.)
For the double reeds, the extra volume inside the reed is not very large, but neither is the truncation. Further, the compliance of the reed itself is more significant, and the viscothermal losses near the reed complicate the tuning.
Why does the clarinet produce even harmonics?
The reeds of clarinet and saxophone are fairly similar. In general, the vibration of the reed produces both odd and even harmonics for both instruments. For notes in the lowest range of the clarinet, however, the resonances of the bore fall near 1, 3, 5, etc times the fundamental frequency, and so are more effective at radiating odd harmonics than even harmonics.
So the even harmonics are weaker for these notes. For notes in the lowest range of the saxophone (or flute, oboe etc), the resonances of the bore fall near 1, 2, 3, 4, etc times the fundamental frequency, and so are effective at radiating both odd and even harmonics.
For notes in the middle and high range of both instruments, there is a resonance near the fundamental, but not consistently near any of the harmonics, so there is no pattern of even or odd harmonics. See Open and closed pipes (flutes vs clarinets) for more discussion.
Why does a flute not need a bell?
One obvious answer is that quite a lot of the radiation from a flute comes from the embouchure hole. Other woodwinds and brass don't have open embouchure holes.
Next, pull the bell off an oboe or clarinet and you'll hear that, except for the tuning of the lowest few notes of the bottom two registers, it doesn't make a big difference to those instruments.
The main purpose of the bell on a woodwind is to provide for those notes a cut-off frequency (see the pages on individual woodwind instruments for a discussion) like that produced for the other notes by the array of open tone holes. The flute's cut-off frequency is already rather high, because of the relatively large holes. So it is less necessary. (Incidentally, I have seen a strange bass flute with a bell. A bass flute needs as much radiation as it can get!)
In woodwinds, how do you get the higher resonances in tune? Playing the whole-tube note of let's say an oboe, presumably the partials are not perfectly harmonic because the flattening effect of the reed has greater effect on the higher frequencies. And the end correction will be different for the higher frequencies. And the chimneys of closed holes also affect lower and higher frequencies differently?
That's a tricky question, involving lots of compromises, and makers have answered it by changing the shape slightly in an evolutionary way. Note that the instruments are never exactly cylinders or simple cones. The taper in the head of the flute and the volume between cork and embouchure hole are there to improve tuning between registers. The extra volume in the mouthpieces of saxophones compensates for the fact that they are truncated cones, rather than complete cones.
What harmonics are present when you jump a register? Playing the high register note, will you produce exactly the same set of partials, only minus the fundamental? Or are you producing a new set of partials whose lowest element is not necessarily exactly the same as the second partial of the low register note?
In the higher register, the reed or lips vibrate at the fundamental frequency of the note played. This vibration is not purely sinusoidal, so they produce higher harmonics of that new fundamental frequency. This means that only some of the resonances of the instrument may be excited. eg, for the flute in the second register, all of the resonances of the lower register harmonics are available, but only the even ones are used. For the clarinet, only every third one is used. Don't take my word for it, see the impedance curves and sound spectra for individual notes on the flute and clarinet. See also How harmonic are harmonics?
Why does an oboe jump registers more easily than say a bagpipe chanter?
The oboe reed is directly influenced by your lips and also by your vocal tract.
On a woodwind, why do the 'short tube' notes bend more easily than the 'long tube' notes?
If you look at the impedance curves and sound spectra for individual notes on the flute and clarinet, you'll see that long tube notes have several sharply-tuned resonances that fall at harmonics of the note played. Short tube notes have fewer resonances that fall in tune with the harmonics, and the resonances are in general broader. So in bending a long tube note, you are working against several sharply-tuned resonances.
When you jump a register, are the reeds (or lips) moving in the same mode, or do they get modes with a node, too? Are the reed blade(s) moving roughly the same way as for the lower register, only at twice the frequency, or are the reed blade(s) also moving in mode 2 fashion - ie like a plate with a displacement node roughly in the middle?
Your first answer is correct. When you go across the break between registers, there is only a small modification in the lip or reed motion (apart from the frequency). The small difference is due to the presence of more resonances for the long tube note, as discussed in the preceding question. See also How harmonic are harmonics?
On saxophone and clarinet, is there a correct note that the reed and mouthpiece (alone) should sound?
One might answer: If you can play in tune, with a good sound and you are comfortable, who cares what note comes out when you blow the mouthpiece? Of course, not all of us do satisfy those conditions, and some teachers use the mouthpiece pitch exercise as a diagnostic. So, why might this exercise have a diagnostic value? There is an approximate answer to this question.
In a simple but very useful model probably originating with Benade, the reed of the sax or clarinet is loaded by the acoustic impedance of the bore Zbore in series with that of the player's vocal tract Ztract. To a rough approximation, the sax or clarinet plays at a peak in Zbore + Ztract. You may have learned that, in electricity, we often describe a real generator or battery as an ideal source together with a finite internal resistance or internal impedance. In much the same way, we can conceptually divide the reed into the reed generator (the idealised component that converts breath into sound) and the reed's own passive impedance Zreed. With a little algebra, you can see that this reed generator is loaded by the parallel combination of Zreed and Zbore + Ztract, which we'll call Z||. To a rather better explanation, the instrument plays at one of the strongest peaks in Z||. (I give more explanation in this paper.)
Now the reed is flexible, so its passive response at low frequencies is to blow out or into the mouthpiece with positive or negative pressure in the mouthpiece. The volume of air inside the mouthpiece is compressed or expanded by positive or negative pressure in the mouthpiece. (Technically, this means that both are largely compliant at low frequencies.) So the effective volume of the mouthpiece is its real volume, plus an extra bit due to the presence of flexible reed. (The effective volume of the reed is about 1 ml for a clarinet and several ml for larger saxophones, and it is smaller for harder reeds and tighter embouchures but larger for soft reeds and relaxed embouchures.)
(Also to a reasonable approximation, the tuning of the resonances requires that the effective volume of mouthpiece (including the contribution from the reed) should replace the volume of the truncation of the conical bore. That's another (but related) story.)
The instrument maker determines Zbore. Your bite and the reed hardness largely determine Zreed. Your mouth configuration determines
Ztract (which is usually small, except when bugling, pitch bending or playing altissimo).
Zreed and Ztract also largely determine the frequency at which the mouthpiece plus reed will play, though the pitch may be high enough that we need to worry about the (inertive) contribution to Zreed from the mass of the reed and that of the air in the mouthpiece (the inertive and compliant terms are approximately equal at the frequency of a reed squeak).
So, putting all of that together, if you have adjusted Zreed and Ztract so that Z|| has peaks at a particular frequency for a particular fingering, then the
mouthpiece will play a particular frequency. Or, to put it the other way round: if you have adjusted Zreed and Ztract so that the
mouthpiece plays a particular frequency, Z|| will have peaks at a particular frequency for a particular fingering.
(This frequency could be different for different mouthpieces.)
One more step. I'm told that a player called Santy Runyon told a story about driving a saxophone with an electric gadget (whose exact details I don't know) that acted as a regenerative source, somewhat like a reed, except that one could readily vary its natural frequency, whereas changing the mass and stiffness of the reed is more difficult. If we knew the details of that generator, we could probably draw analogies. So, adjusting one of its component values would be like, say, changing bite or reed hardness. Even though I don't know the details of the electronic gizmo, I think that the analogy is far from perfect: for instance, there would probably be no analogy for Ztract. It is quite plausible, however, that some particular values of its parameters covered different parts of the instrument's range better or more completely than others, for fairly similar reasons.
See our sites on clarinet acoustics and
sax acoustics for details, or this paper on the acoustics of wind instruments and of the mouth of the player.
Inside the mouth of saxophonist or clarinettist, is there a pressure node or antinode near the reed?
Nodes and antinodes apply to standing waves, not to travelling waves. When the vocal tract resonance is tuned to the note being played (as in bugling, pitch bending and altissimo playing), the wave in the mouth is a strong standing wave and the reed is very near a pressure antinode. In ordinary circumstances, however, that wave is not usually a standing wave, so it doesn't have clear nodes or antinodes.
Why are clarinet tone holes different sizes?
As background, the flute has tone holes that are mainly the same size: the three register/trill holes are small, but all the tone holes are the same size, with the exception of the three or four on the foot joint, which are slightly larger.
There are several reasons. The clarinet is not cylindrical. Most of the lower joint is flaring and so, with the larger bore diameter, the holes have bigger diameter. But there are still variations in the cylindrical part. I think that the reason is this: if one can change both the size and the position, one has twice as many parameters to vary in order to get the instrument in tune.
How does an ocarina work? How can such a small instrument produce such a low pitch?
An ocarina is an example of a Helmholtz resonator. Think of blowing over the neck of a bottle: the air in the neck of the bottle is a mass (strictly an inertance) that oscillates on the 'spring' (strictly a compliance) of the air inside the bottle. The ratio of compliance to inertance determines the frequency, but there is no simple relation between the instrument size and the wavelength it produces.
On a ocarina or similar instruments, tone holes can be added to change the frequency of the resonance. The tone holes in the ocarina are also inertances that can also oscillate on the compliance of the enclosed air. Opening successive tone holes puts these other inertances in parallel with that of the blow hole. The total inertance of two or more inertances in parallel is less than that of any one of them, so the compliance/inertance ratio is larger and the pitch is higher.
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.
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 
However, there is more to the story than this. First, mass producing some components makes a big difference to the price of cheap instruments, a useful difference to mid-range instruments, and very little to high priced instruments. If you are paying a maker's wages for a month or two to make you an instrument, you might well think about paying for an extra day to carve some components that s/he could buy ready made. Second, there is the tradition: it is still possible for one person to make (nearly all of) a violin in a simple workshop, and keeping this tradition alive is appealing in an age in which, for example, cars and computers are no longer thus made. (The photo shows the celebrated maker of more than 500 instruments 
