Double reed acoustics: oboe, bassoon and others

How so double reed instruments work? This introduction to the physics of the double reed family requires no technical knowledge of acoustics. We'll concentrate on oboe and bassoon, but mention in passing the cor anglais, contrabassoon, contraforte, shawm and bagpipe chanter. For background on topics in acoustics (waves, frequencies, resonances, decibels etc) click on "Basics" in the navigation bar at left or go to this website for a physics-based introduction to waves and sound.

  • How do double-reed instruments work? An overview
  • The reed controls the air flow
  • Articulation
  • Playing softly and loudly
  • Control parameters for playing
  • Oboes and bassoons as 'closed' conical pipes
  • Harmonics of a closed conical pipe
  • How the reed and pipe work together
  • Sound spectra
  • Tone holes
  • Register holes
  • Cross fingering
  • Other effects of the reed
  • Cut-off frequencies
  • How the vocal tract is involved
  • More information and links
  • The most common orchestral double reeds are the bassoon (top), cor anglais and oboe. (Figure from The acoustics of woodwind musical instruments. Click on figure to enlarge.)

    photo of bassoon, cor anglais and oboe reeds of oboe, cor anglais and bassoon

    Overview

    The player provides a flow of air at a pressure above that of the atmosphere (several kPa or several percent of an atmosphere: applied to a water manometer, this pressure would support several tens of cm height difference). This is the source of power input to the instrument, but it is a source of steady, continuous rather than vibratory power. In a useful analogy with electricity, it is like DC electrical power. Sound is produced by an oscillating motion or air flow (like AC electricity). The reed acts like an oscillating valve. The reed, in cooperation with the resonances in the air in the instrument, produces an oscillating component of both flow and pressure. Once the air in the instrument is vibrating, some of the energy is radiated as sound out of the bell and any open tone holes. A rather greater amount of energy is lost as a sort of friction (visco-thermal loss) with the wall. In a sustained note, this energy is replaced by energy put in by the player. The column of air in the instrument vibrates much more easily at some frequencies than at others (i.e. it resonates at certain frequencies). These resonances largely determine the playing frequency and thus the pitch, and the player in effect chooses the desired resonances by suitable combinations of keys, with embouchure assistance. Let us now look at these components in turn and in detail.

    The reed controls the air flow

    The photos above show how the reed is made of two symmetric blades, which are springy and can bend. In fact they can oscillate like a spring on its own—for the player this is bad news because it leads to squeaks if the lip doesn't sufficiently damp the high frequency vibration. Normally, the reed's vibration is controlled by resonances of the air in the instrument, as we shall see. But it's also true that the reed vibration controls the air flow into the instrument: the two are interconnected. (Sometimes, resonances of musical instruments are harmonic or nearly harmonic. Musicians often call notes played at the resonances 'harmonics'. To understand the difference between resonances and harmonics, see How harmonic are 'harmonics'?)

    Let's imagine steady flow with no vibration and no resonator, and graph how air flow through the reed depends on the difference in pressure between the player's mouth and the volume inside the reed. If you increase this pressure difference, more air should flow through the narrow aperture between the reed blades. So a graph of flow vs pressure difference rises quickly: it has positive slope, as sketched below on the left of the graph.

    sketch of flow vs pressure

    However, as the mouth pressure gets large, it starts to flatten the reed blades and so tends to close the aperture through which the air is entering. Indeed, if you blow hard enough, it closes completely, and the flow goes to zero. So the flow-pressure diagram looks like that in the (highly simplified) diagram. Different curves are possible, depending on the stiffness and shape of the blades, and on the embouchure forces: vertical lip forces tend to close the aperture and lower the curve overall; 'pushing in' with horizontal forces tends to open it. Adjustment with the reed knife changes the reed mechanics and so also changes this curve in important ways. The curve is complicated by other effects. The air travels rapidly through the reed, which produces a suction that helps to close the blades. Further, as the blades close, their curvature is reduced, which reduces their effective stiffness. So the reed tends to close easily and abruptly at the right and side of the curve, which has important consequences. The solid curve sketched above neglects air resistance in the channel, which can bend the peak of the curve to the right (dashed curve), and reduce the negative resistance region, but allow the reed to oscillate between a closed and open state when matched to a suitable bore. This 'beating reed' (closing completely for a short section of the cycle) is common in normal performance, and contributes high harmonics and thus the bright timbre of double reed instruments.

    The player provides power via a flow of air at pressure above atmospheric: this is the source of energy, but it is (more or less) steady. What converts steady power (DC) into acoustic power (AC) is the reed. The left side of the graph is something like a resistance: flow increases with increasing pressure difference (DC electrical current increases with increasing DC voltage). Just like an electrical resistance, an acoustic resistor loses power. So in this regime, the instrument will not play, though there is some breathy noise as air flows turbulently through the reed aperture. The operating regime is the downward sloping part of the right hand side of the curve in the sketch above. This is related to why there is both a minimum and maximum pressure (for any given reed) that will play a note. Blow too softly and you get air noise (left side of the graph), blow very hard and it closes up (where the graph meets the axis on the right).

    Consider the red dot as an operating point. Readers with a background in electricity, seeing this region of the curve in which flow decreases with increasing pressure, will recognise this as a negative AC resistance—the inverse slope of the red line. Whereas a positive resistance takes energy out of a circuit, a negative resistance puts energy into the circuit (as happens in e.g. a tunnel diode oscillator). Note that the DC resistance (inverse slope of the dashed straight line) is positive. In the instrument, the positive DC resistance takes energy out of the DC air flow and it is indeed this negative AC resistance that provides the AC energy lost in the rest of the instrument. Most of the energy is lost inside the bore, in viscous and thermal losses to the walls, and a relatively small fraction is emitted as radiated sound.

    Another way of understanding the active role of the reed is to consider a small wave of excess pressure arriving from the bore into the reed. This opens the reed aperture a little, letting in more high-presssure air from the player's mouth. That extra air raises the pressure in the reed still further, so a larger pressure wave is reflected back down the bore. Conversely, a negative pressure wave arriving in the reed closes the aperture a little, reducing the flow and futher reducing the pressure. So, with appropriate blowing pressure, embouchure forces and reed properties, the reed provides positive gain: it tends to amplify sound vibrations in the bore. The stroboscopic video below was made by Leonardo Fuks. (The upper joint of the oboe, finger holes closed, was playing near G4 while air was the joint end was connected to a volume at less than atmospheric pressure.)

    Articulation

    To start a note, the tongue releases the reed, which sends a small pulse of air pressure ino the mouthpiece, where it travels down the bore, reflects, comes back and is amplified—as described in the preceding paragraph. Its amplitude grows until it is comparable with the blowing pressure. (At this stage, the simple linear story described above no longer applies and nonlinear effects dominate. There is much more to this process, so we have studied it in detail and made this page about tonguing and articulation.)

    Playing softly and loudly

    The preceding sketch of flow vs pressure allows us to explain something about the timbre of oboe and bassoon. For a very small variation in pressure and small acoustic flow, the relation between the two is approximately linear: a small section of the red line. A nearly linear relation would give rise to nearly sinusoidal vibration (i.e. one shaped like a sine wave), which means that the fundamental in the sound spectrum is strong, but that the higher harmonics are weak. This is rarely the case with double reeds, because it is very easy to close the reed completely, for reasons described above. This makes the pressure-flow relation strongly non-linear, which in turn means that the high harmonics are strong. Add to this the fact that the higher harmonics are radiated from the bore more effectively than the low, and we understand that the fundamental in the sound of the double reeds is often weaker than some of the higher harmonics, especially in the low range. This gives the double reeds their rich, bright timbre.

    Control parameters for playing

    For any given fingering, reed and mouthpiece set-up, the player can make a range of sounds, varying the pitch, loudness and spectrum. Various control parameters are available to the player: pressure in the mouth can be varied, so can the lip forces in the vertical and horizontal directions, the position at which the lip presses on the reed, and sometimes the configuration of the vocal tract. We have investigated these parameters mainly on the clarinet. One reason is that we understand the (simpler) clarinet reed better than the double reed. Another is that we have a robot clarinet player in which we can vary all these parameters in a controlled way. We report these results in more detail on the clarinet robot site, along with links to scientific reports on that project. A big difference, however, is the application of horizontal lip forces on the double reed.

    Pressure, air flow and speed. It's worth mentioning that, among woodwind instruments, the blowing pressure for oboes is sometimes very high (possibly above 10 kPa for a loud high note). Nevertheless, the rate of flow into the instrument is relatively low, because the gap between the blades of the reed is small. So an oboist can sustain a note for a long time. The low register of the bassoon, in contrast, can accept relatively high rates of flow, especially when playing loudly. We have a separate page on air speed, air flow, pressure and power in woodwind and brass instruments.

    A warning for oboists: sustained very high pressures are reported to affect circulation in the head and neck, with reports of possible consequences including stroke and eye damage. Be careful, and avoid very hard reeds. Also, the fact that you can play very long soft passages without breathing doesn't mean that you should: if the conductor goes blurry, you are about to faint.

    Oboes and bassoons as 'closed' pipes

    Oboes and bassoons are open at the far end or bell—with the consequence that the pressure there is close to atmospheric and its variations small. But they are (almost) closed at the reed end, where the pressure variations are large. For a sound wave, the tiny aperture in the reed is enough to cause a reflection a bit like that from a completely closed end. The rest of the instrument is approximately conical (semiangle about 0.7° for the oboe and about 0.4° for the bassoon). Of course, bassoons are bent in the boot joint and bocal to fold the bore into a convenient length, but these bends have only a modest acoustic effect. There's also the bell, some of whose effects we shall discuss below.

    For the purposes of this simple introduction to double reed acoustics, we shall now make some strong approximations. First, we shall pretend that it is a simple conical pipe – in other words we shall assume that all holes are closed (down to a certain point, at least), that the bore is conical, and that the mouthpiece end is completely closed. This is a crude approximation, but it preserves much of the essential physics, and it is easier to discuss.

    The natural vibrations of the air in the instrument, the ones that cause it to play notes, are due to standing waves. (If you'd like an introduction to this important concept, see standing waves.) What are the standing waves that are possible in such a tube?

    To answer the question, we must take into account the fact that the instrument is approximately conical. This means that sound waves 'spread out' as they travel down the bell. This means that the amplitude of travelling waves gets smaller as we go from mouthpiece to bell. The fact that the instrument is open to the air at the far end means that the total pressure at that end of the pipe must be approximately atmospheric pressure. In other words, the acoustic pressure (the variation in pressure due to sound waves) is approximately zero (called a pressure node). The mouthpiece end, on the other hand, can have a maximum variation in pressure: it is an antinode in pressure. If we were dealing with a cylindrical pipe (such as a flute or clarinet), where the standing waves are sinusoidal, we would expect the maximum and the zero of a wave to be one quarter wavelength apart. But the variation in the amplitude of the wave due to the variation in cross sectional area complicates the story.

    So we have devoted a whole page to comparing cylindrical and conical pipes and, if you want the details, you should read that page now. However, the result is this: the standing waves in a cone of length L have wavelengths of 2L, L, 2L/3, L/2, 2L/5... in other words 2L/n, where n is a whole number. The wave with wavelength 2L is called the fundamental, that with 2L/2 is called the second harmonic, and that with 2L/n the nth harmonic.

    The frequency equals the wave speed divided by the wavelength, so this longest wave corresponds to the lowest note on the instrument: Bb1 on a bassoon, E3 on a cor anglais and Bb3 on an oboe. (You can check the frequencies of notes on the note table.) You might want to measure the length of your instrument, take v = 345 m/s for sound in warm, moist air, and calculate the expected frequency. The calculation setting L = v/2f doesn't give you the real length of the instrument. Rather it gives the length of the cone made by removing the reed and extrapolating the taper of the bore back to a point, which is considerably longer than the geometrical length.

    An important point about the bassoon is that its bore can be modified by the user to a substantial extent by replacing the crook (bocal), which is a non-negligible fraction of the effective length, particularly for high notes, and whose diameter significantly affects performance. Professionals often have several crooks and change among them, sometimes in the same piece: one may play more safely in the high passages while another sounds better or is easier to play in tune.

    Harmonics

    So, with all of the tone holes closed, an oboe or bassoon can play the lowest note (Bb3 or Bb1) with a wavelength somewhat more than twice the length of the instrument. With this fingering, however, it can also play other notes by overblowing – by changing your embouchure and blowing pressure and selecting a different resonance. The harmonic series on Bb3 is shown below. For the bassoon, transpose this down two octaves. (Note that the 7f doesn't fit into the usual scale; G half-sharp is a reasonable approximation.)

    harmonics on Bb3

    How the reed and bore work together

    To sum up the preceding sections and the page on cylindrical and conical pipes: the bore of the instrument has several resonances. For the fingerings of the low notes, the frequencies of these resonances are approximately in the ratios of the simple harmonics, 1:2:3, but successively more approximate with increasing frequency–we'll see why below under frequency response. The reed has its own resonance—which is usually higher in frequency than the notes you play (though this is not true in the highest ranges, especially in double reeds). In normal playing, with your lips touching the reed, you damp (ie reduce the strength of) the reed's resonance(s) considerably. This allows the resonances of the bore to 'take control'. To oversimplify somewhat, the instrument usually plays at a strong bore resonance whose frequency is lower than that of the reed. (We shall see below how register holes are used to weaken and to detune the lower resonance or resonances and thus make one of the higher resonances the strongest.)

    When the instrument is playing, the reed is vibrating at one particular frequency. But, especially if the vibration is large, as it is when playing even a little loudly, it generates harmonics (see What is a sound spectrum?). These set up, and are in turn reinforced by, standing waves. Consequently, the sound spectrum has strong components at harmonics of the fundamental being played. For notes in the low range of the instrument, there are usually resonances at the frequencies of some of the harmonics. These result in greater radiation of the sound at those frequencies. In the high range of the instrument, the higher resonances do not systematically fall near harmonics of the note being played.

    Sound spectra

    In the section on standing waves, each wave is considered as a pure sine wave. The sound of a wind instrument can bea little like a sine wave when played very softly, but successively less like it as it is played louder. To make a repeated or periodic wave that is not a simple sine wave, one can add sine waves from the harmonic series. So the played note C4 contains some vibration at C4 (fo), some at C5 (2fo), some at G5 (3fo) etc. The 'recipe' of the sound in terms of its component frequencies is called its sound spectrum.

    For double reeds, the low harmonics are all strong, and some may be stronger than the fundamental. Above the cutoff frequency, the amplitude of the harmonics decreases with increasing frequency. (The researcher Lehman has identified two formants in the bassoon spectrum, at frequencies of 440-500 Hz and 1220-1280 Hz, which is a point near the middle of the (human) vowel plane.)

    Opening tone holes

    If you open the tone holes, starting from the far end, you make the pressure node move up the pipe, closer to the mouthpiece—you are effectively making the pipe shorter. Starting near the bell, each opened tone hole raises the pitch by a semitone, which requires a pipe that is about 6% shorter.

    For the moment, we can say the an open tone hole is almost like a 'short circuit' to the outside air, so the first open tone hole acts approximately as though the instrument were 'sawn off' near the location of the tone hole. This approximation is crude, and in practice the wave extends somewhat beyond the first open tone hole: an end effect. (More about this below.)

    (For the technically minded, we could continue the electrical analogy by saying that the air in the open tone hole has inertia and is therefore actually more like a low value inductance. The impedance of an inductor in electricity, or an inertance in acoustics, is proportional to frequency. So the tone hole behaves more like a short circuit at low frequencies than at high. This leads to the possibility of cross fingering, which we have studied in more detail by comparing modern, classical and baroque flutes.)

    The frequency dependence of this end effect means that the higher note played with a particular fingering has a larger end effect than does the corresponding note in the lower register. If the instrument really were a perfect cone with tone holes, then the registers would be out of tune: the intervals would be too narrow. This effect is removed primarily by subtle variations in the shape of the bore, the volume of air in the reed and the flexibility of the reed blades.

    The bassoon's tone holes are unusual in that, over most of the range, they are long. This photo of the tenor joint shows the 'wing': a thickening in the tube wall. I have inserted straws in the three left hand fingerholes to show the angle at which they are drilled (dashed lines superposed). The length and these angles allow holes that are widely separated in the bore to be brought within reach of the fingers. Their extreme length lowers the cut-off frequency and allows cross fingering in the first register (e.g. LH XOX for Eb3) —which is not possible on other modern woodwinds but normal in baroque instruments. This photo also shows that the cone angle is small and that an extrapolation of the cone to a point would go well past the reed.

    wing joint on bassoon

    Register holes

    A small hole can serve as a register hole. For instance, if you play E4 on an oboe (call this frequency fo) and then push the first octave key, you are opening a hole part way down the (closed part of the) instrument. This hole disturbs the resonance that supports the fundamental of E4, but has little effect on the higher harmonics, so the instrument 'jumps up' to E5 (2fo). The oboe has a further one or two register keys, and the option of part opening the LH index finger as a further register hole. The bassoon also has three register keys for the left thumb, and also uses part uncovering ('half-holing') of a tone hole by the LH index finger and the whisper key.

    The first mode of vibration is used for the range Bb3 to C5 on oboe and Bb1 to F3 on bassoon. The second mode is used for C#5 to C6 and F#3 to D4 on those instruments. As register keys, the first few notes use the LH index (which is conveniently near the effective half-way position and so works well, especially if the finger can vary the opening size) then the octave key(s) (oboe) and whisper key (bassoon) (which work as compromises over a larger range of pitch). For the third and higher modes, it becomes complicated: tone holes contribute as register holes (e.g. LH1 for C#6 on oboe and D#4 for bassoon), along with dedicated register keys.

    Where to put a register key? The acoustically obvious place to put a register hole is at a pressure node of the upper note which is also a region of large pressure variation for the lower note. Opening the bore to atmospheric pressure at a pressure node makes no difference to that note. One problem is that each note in the upper register has its pressure node at a different position. One can imagine an instrument that had a separate register hole for each note, but that would be a lot of keys. Counting the half-holing of the LH index finger, the oboe has three or four register holes, and the bassoon five, while tone holes also serve in this role. (Arlen Fast is working with the Fox company to add more to the contrabassoon, and spent time in our lab on this project.) Using one register hole for several notes is a compromise: it is not quite at the pressure node of the standing wave of the required upper note. This is not a big problem in practice. The register hole is small, so it is not really a 'short circuit', except at low frequencies. So it does not too much affect the higher frequency standing waves. (We explain how the mass of the air seals the hole at high frequencies below.) It does however disrupt the fundamental (or other low mode), and that is its purpose: to stop the instrument dropping down to its bottom register.

    Ideally, there is one obvious place to put a register hole for the second mode: halfway along the effective acoustic length. For the third mode, one third and also two thirds of the way are possible. The latter solution is less attractive, however, in that it will work for a smaller range of notes than will a hole at one third of the way along.

    The bassoon regularly uses a technique called flicking, which is rarely used on other winds. For notes in the third octave and higher, a register key that would perturb the intonation or timbre of a sustained note is flicked open briefly as the note is started to avoid having the note 'drop' to a lower resonance.

    (The operation of two of the octave octave keys is partially automated on some oboes (and fully automated on the saxophone and the Fast system contrabassoon): one key opens one or other of two register holes, according to whether or not the third finger of the left hand is depressed. This is an example of a mechanical logic gate. Register holes are discussed in more detail on the page about clarinet acoustics. See more about register holes on the clarinet page.)

    Cross fingering

    On the oboe, successive semitones are usually played by opening a tone hole dedicated to that purpose. Being a closed, conical pipe, the oboe and bassoon overblow an octave, and so one would need twelve tone holes to cover the keys in one octave before repeating fingerings using the octave key for the register holes. Because players don't have this many fingers, the keys and clutch mechanisms are supplied, so that one finger can close or open two or more holes. The oboe fingering system is somewhat similar to that developed by Theobald Boehm for the flute (and largely followed on the clarinet and saxophone). Where cross fingering is used on the oboe, it is usually used to control other keys. For instance, one of the fingerings for the note F4 or F5 is the same as that for E4 or E5 plus one other opened tone hole. Another fingering 'feels like' a cross fingering (XXX|XOX). However, in that fingering the RH ring finger key has a link that closes another key above the open middle-finger hole. (The Viennese oboe uses cross fingering a little more than the standard oboe, beginning at Bb5 rather than C#6. Bassoon cross fingerings also commence near the top of the second register, at Eb4.)

    The bassoon is more complicated and doesn't follow the Boehm system. First, the tone holes are often small in diameter and long. Further, in the boot joint, some of the key pads close two or three separate holes that lead to different points on the bore. (This odd complication allows the maker to improve tuning of higher resonances for some notes.) Further, because of the relatively small tone holes, bassoonists can and do use cross fingerings as alternatives rather more than do other woodwind players, to vary loudness and brightness and to adjust intonation.

    An open tone hole connects the bore to the air outside, whose acoustic pressure is approximately zero. But the connection is not a 'short circuit': the air in and near the tone hole has mass and requires a force to be moved. So the pressure inside the bore under a tone hole is not at zero acoustic pressure, and so the standing wave in the instrument extends a little way past the first open tone hole, particularly for long, narrow tone holes such as those on the bassoon. (There's more about this effect under Cut-off frequencies.) Closing a downstream hole extends the standing wave even further and so increases the effective length of the instrument for that fingering, which makes the resonant frequencies lower and the pitch flatter.

    The effect of cross fingerings is frequency dependent. The extent of the standing wave beyond an open hole increases with the frequency, especially for small holes, because it takes more force to oscillate the air in the tone hole at high frequencies. This has the effect of making the effective length of the bore increase with increasing frequency. As a result, the resonances at higher frequencies tend to become flatter than strict harmonic ratios. Because the oboe's tone holes are not very much smaller than the bore, cross fingerings have relatively little effect on the pitch in the first two registers. They have much greater effect on the bassoon.

    A further effect of the disturbed harmonic ratios of the resonances for cross fingerings is that the harmonics that sound when a note is played will not 'receive much help' from resonances in the instrument. (Technically, the bore does not provide feedback for the reed at that frequency, and nor does it provide impedance matching, so less of the high harmonics are present in the reed motion and they are also less efficiently radiated as sound. See Frequency response and acoustic impedance. To be technical, there is also less of the mode locking that occurs due to the non-linear vibration of the reed.) As a result, cross fingerings in general are less loud and have darker or more mellow timbre than do the notes on either side.

    We have studied cross fingerings more extensively on flutes than on other instruments, by comparing the acoustical response of baroque, classical and modern instruments. See cross fingering on flutes or download a scientific paper about cross fingering.

    Other effects of the reed

    An important effect of the reed concerns the fact that the bore is not a simple cone: it is a cone truncated at a comfortable diameter to take the reed, whose effective cone angle is much greater than that of the bore. This truncation affects the tuning: informally, we can think of it as making the pipe slightly like a cylinder, which stretches the frequency gap between resonances. This means that, unless compensated, it would stretch the interval between registers to over an octave. Now the geometry of the reed is a little complicated, but one contribution to the acoustic response is that it compensates for the 'missing volume' of the cone. Indeed, its volume (including an effective volume due to the flexibility of the reed blades) is comparable with that of the missing cone.

    So, as well as controlling the flow of air, the reed has a passive role. When the pressure inside the reed, the blades are pushed outwards. Conversely, suction draws the reed in towards the bore. Thus deformation of the reed blades increases and decreases the reed's air volume with high or low pressure. (Techncially, we say it is a compliance in parallel with the bore.) Indeed, it behaves a bit like an extra volume of air, which could also be compressed and expanded by changing pressure in the mouthpiece. It has the effect of lowering the frequency of each resonance a little. However, soft reeds move more than hard reeds, so soft reeds lower the frequency (and thus the pitch of the note) more than do hard reeds. Further, this effect is greater on high notes than on low, so soft reeds make intervals slightly narrower. This is useful to know if you have intonation problems. However, compared with single reeds, double reeds have the complication that flattening the blades not only decreases the reed air volume (which reduces compliance) but also decreases the blade stiffness (which increases compliance). Squeezing the blades flatter and closer together reduces the volume, which on its own would raise the pitch. Which of these effects wins can depend on the reed scrape. (This is discussed in more detail under the "effect of hardness" section on the clarinet page.) The effect of reed shape is important and complicated. However, the flat blade is stiffer (less easy to bend), which tends to raise the pitch. Reed mechanics is a very important topic and needs more explanation — another time.

    Cut-off frequencies

    When we first discussed tone holes, we said that, because a tone hole opens the bore up to the outside air, it shortens the effective length of the tube. (Though not to the tone hole: for simple fingerings, the effective length goes beyond the first open tone hole.) For low frequencies, this is true: the wave is effectively reflected at a point near the hole because the hole provides a low impedance 'short circuit' to the outside air. For high frequencies, however, it is more complicated. The air in and near the tone hole has mass. For a sound wave to pass through the tone hole it has to accelerate this mass, and the required acceleration (all else equal) increases as the square of the frequency: for a high frequency wave there is little time in half a cycle to get it moving.

    sketch: inertance of air in open tone holes

    So high frequency waves are impeded by the air in the tone hole: it doesn't 'look so open' to them as it does to the waves of low frequency. Low frequency waves are reflected at the first open tone hole, higher frequency waves travel further (which can allow cross fingering) and sufficiently high frequency waves travel down the tube past the open holes. Thus an array of open tone holes acts as a high pass filter: something that lets high frequencies pass but rejects low frequencies. (See filter examples.) The cutoff frequency is roughly 1500 Hz* for the oboe and 600 Hz* for the bassoon. In the sound spectra of most low notes, the strength of harmonics falls off above this frequency. These cutoff frequencies correspond roughly to the highest pitches in the standard repertoire (though modern pieces go higher). So for the very high notes, the tone holes farthest from the reed can have a large effect on the pitch because the standing wave penetrates a long way down the bore. (As noted above, some of the bassoon's tone holes are rather long, particularly in the wing of the tenor joint and at the bottom of the boot.)

    * These are very approximate values and vary considerably over the range of the instrument. Also the cutoff frequency has risen over the centuries, especially on instruments influenced by Boehm. This is a subtle topic area: see cutoff frequencies. In general, however, smaller-diameter and longer tone holes are less good at radiating high frequencies and so make for a darker timbre. This is one reason why modern instruments sound brighter than baroque. Here is a scientific paper where we compare this effect in baroque, classical and modern flutes.

    Effect of the bell. A flaring bell like that of the oboe 'helps' the sound waves in the bore to radiate out into the air. (Incidentally, the presence of a large, effective bell is what makes brass instruments loud: try playing a trombone with the tuning slide taken off.) More sound radiated means less sound reflected, so the standing waves are weaker. However, this effect is only strong for high frequency: as the frequency increases over this range, the resonances are more weakened by the bell at high frequency than at low. This is because the bell is much smaller than the wavelengths of the low frequency waves, and so is less effective at radiating these waves. So we expect the oboe bell to make the timbre brighter. Doing the experiment will tell you that the effect is not huge. The cor anglais bell has only a modest effect on the timbre and the bassoon 'bell' very little.

    The effect of an oboe bell is that of a high pass filter. We could say that this is rather like the cut-off frequency effect of a series of open tone holes. In fact, one purpose of the instrument's bell chiefly is to provide a high pass filter for the lowest few notes, so that they have a cut-off frequency and so behave more similarly to the notes produced with several tone holes open. As far as fundamental frequencies go, the bell is really only important to the lowest notes. However, it is important to the high harmonics of most notes.

    The bell of the bassoon is the name of the last joint, which doesn't have a widening flare like oboe, clarinet or saxophone.

    Notes at the very bottom of the range are difficult to play softly. We saw in How the reed and pipe work together that the higher harmonics of the reed motion could be 'helped' by resonances of the bore. When you play loudly, you generate proportionally more power in the higher harmonics (this is due to increasingly non-linear motion of the reed). So the lowest notes are not so difficult to play loudly, where the strong, high harmonics of the reed are supported by resonances of the bore, compared with soft notes, where you are relying on just the fundamental resonance. (Note, however, that the lowest notes are also most affected by leaks, for two reasons: first, they involve more keys closing and so more opportunity for leaks. Second, low frequencies have more time to accelerate air through leaks. So if the low notes are very hard to play, check for leaks.)

    Vocal tract effects

    In the simplest model, the mouth is considered as a source of high pressure air, and the reed is loaded acoustically only by the acoustic impedance of the bore of the instrument, which is downstream. The acoustical effects of the vocal tract (upstream) are often small. However, they can be important for the clarinet and saxophone, and so might be important for some double reeds—we haven't looked yet. In clarinet and saxophone, they can be important and even dominant if the player produces a resonance in the vocal tract whose acoustic impedance is comparable with that of the bore. This is easier to do in the higher registers, where the instrument's resonances are weaker. Vocal tract resonances are important in various effects including pitch bending and the famous glissando from Rhapsody in Blue (see this link). We have also published a scientific paper on vocal tract effects in the saxophone. A popular version is given in this link.

    More to come

    This is a draft document and I'll extend and improve it as time permits. Our Introduction to saxophone acoustics has rather more detail about a conical bore interacting with a reed, because we have studied that instrument in detail.

    For further reading, we recommend

    Also in this series:

    Acknowledgment

    Our research is supported by the Australian Research Council, by Yamaha Music Australia and by Legere reeds.

    Yamaha logo

    This photo shows the common orchestral woodwinds in ascending order of pitch. The double reeds are the bassoon (top), cor anglais (third) and oboe (fourth). (d) and (e) show their double reeds (oboe at the top). (Figure from The acoustics of woodwind musical instruments. Click to enlarge.)

    photo of the main orchestral winds

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