The Saxophone

This essay will look at the design and operation of the saxophone from the point of view of a physicist. The following site has a useful diagram to which I shall make reference (Figure 1): http://www.musicale.co.uk/img/saxophone-diagram.jpg As an introduction, there is a family of saxophones: soprano, alto, tenor and baritone (from highest range to lowest) are the most common and all but the first have a curve in their shape (see fig.1). Others include the sopranino, bass and tubax. All have a similar operation. This essay will follow the process of sound production from initiation (at the player`s mouth) to the emergence of sound from the tone holes, which are mainly under the keys, and at the bell (again in fig.1). Interesting effects will be included in this article too, at the points where they are relevant.

Sound is primarily sourced from the power supply of the instrument which is the air provided by the player. This is approximate to a constant stream (between breaths) at a few thousand Pascals (the standard unit of pressure) above one atmosphere (the normal air pressure at ground level).

If the pressure difference is increased a little from an already small amount (e.g. The player blows slightly harder) more air flows through the small gap between the tip of the reed and mouthpiece. This is initially a linear relationship (see fig. 2). However as a larger difference in pressure causes more air to flow, the reed starts to bend up (see arrow in lower image, fig.2). This prevents airflow and if enough pressure difference is attained, the reed will close entirely and won't reopen until the pressure drops sufficiently (this is easier demonstrated when the mouthpiece is removed from the neck of the instrument).

Figure 2. http://www.phys.unsw.edu.au/jw/graphics/saxreeddiagrams.GIF A graph showing the relationship of airflow to pressure difference in the mouthpiece-reed mechanism and two images of the different stages: above-initial gap size, below-closing of the tip under high pressure difference.

The reed converts the constant stream of air (an analogue of DC flow in electricity) to an oscillatory one (analogous to AC flow). To continue the electricity comparison, the linear element of the graph in figure 2 is similar to the current versus voltage graph of an Ohmic conductor with its constant resistance. This resistance saps power from the instrumental system and causes no acoustic sound only breathy noises from inside the bore. Therefore, the sound of a saxophone playing properly comes from pressures in the second part of the graph in figure 2 and this leads to a minimum and maximum pressure (which varies with the reed).

In addition to the role of the reed as already described, there is also an acoustical effect due to the volume of air between it and the mouthpiece. If the pressure is high in the mouthpiece, the reed moves out and down (fig.2 upper image) leaving a larger volume of air in the mouthpiece than if it were more closed (the situation in a low mouthpiece pressure regime). This extra volume of air adds to that in the bore and lowers its natural frequency (as longer tubes have lower frequencies). This effect modifies higher notes more than lower ones. The choice of reed now has an effect. Reeds (see fig.3) come in degrees of hardness, softer reeds bend more and so the above effect is greater for that type. Therefore, using a softer reed will narrow the intervals between high and low notes more so than a harder reed. This can be applied to solve intonation issues.

Figure 3. http://upload.wikimedia.org/wikipedia/commons/6/65/Saxophone_reeds-alto%2C_tenor.jpeg Two reeds for different saxophones (alto and tenor, left/right respectively). The under and outer side (when in use) is facing, the away turned side is planar.

The reed has a resonant frequency, which is demonstrable by putting the teeth on the reed and blowing as normal. The reed resonates at its high natural frequency producing a high pitched `squeak`. In fact this is the highest pitched sound possible on the instrument. The resonances of the bore are much lower than the reed and can be represented by transverse waves (acoustic waves are actually longitudinal). The saxophone is approximately conical (with an added bend) and the volume of the mouthpiece is almost the same as the smaller cone, which would replace it if the instrument was a perfect cone i.e. without curves. The bend and other variations from the strict cone have minor outcomes in terms of acoustical effect. The cone is effectively closed at the reed end (the gap between reed and mouthpiece is negligible when compared to the open end at the bell). Indeed the mouthpiece end is small enough to reflect internal waves as a closed pipe does. Thus the bore has the resonances of a closed conical bore (as with the oboe and bassoon), which are shown in figure 4.

Figure 4. http://www.phys.unsw.edu.au/jw/graphics/columns.GIF Resonant frequencies in a conical bore. The blue line represents the displacement of air and the red line represents the pressure. There is a full harmonic series. The top diagram is the fundamental (1st harmonic), and then follows the 2nd, 3rd and 4th harmonics going down from the top.

From the vibration of the reed, the air in the bore of the instrument is excited. However, this is just the first stage. A standing wave is set-up in the bore as the pressure variations are reflected at the open end. This standing wave then feeds back into the oscillations of the reed (which is still being forced to vibrate as described above). The true oscillations reconcile (relatively quickly) to form the sound heard.

Mathematically this can be described in the general solution to the wave equation for a forced and damped harmonic oscillation. The two terms of this solution are, firstly, the transient, which is short lived and then the steady state. Initially, it is both terms that cause the transient `attack` at the beginning of any played note. As opposed to the `squeak` described earlier, under normal playing conditions the embouchure (the position of the mouth and lips) is such that the lower lip heavily damps the resonance of the reed. The result is the dominance of the bore`s resonances.

As with most instruments, when a note is played on the saxophone it is made up of a sound spectrum containing various quantities of the harmonic series, the fundamental being the frequency of the apparent sound. When the spectral components are altered by altering the initiation of the vibrations, one fingering can produce different notes to the ear. This means a complete harmonic series is achievable on a single fingering by altering the embouchure and rate of air flow i.e. blowing harder or softer. The series produced this way on the fingering for the lowest note on a tenor saxophone is given in figure 5. Figure 5. http://www.phys.unsw.edu.au/jw/graphics/saxophoneharmonics.GIF The harmonic series playable on the fingering for the longest note on a tenor saxophone {Bflat3}.

On the neck of the saxophone lies one of the register holes and its covering pad. There is another pad and register hole close to the top of the main body. Both pads are lifted and hence the hole opened by pressing the same `register` key but the automated mechanism ensures only one is open at a time. For the lower notes in the upper register, the hole on the neck is opened, then at G#5 there is a turn-over and notes above this use the lower register hole. The register holes are positioned so that when open they create a low pressure point without being so low as to stop the air flow traveling further down the tube. The low pressure point disrupts the fundamental frequency (as it should have high pressure there-see fig.4). The other harmonics aren`t affected as much (the low pressure point is near the second harmonics pressure node- again in fig.4). Thus the sound heard is now dominated by the second harmonic resonance of the bore and as such is an octave up from the note played without the register key. Note that the low pressure point is only near to the pressure node of the second harmonic as the same register hole serves several different notes. If it were to be precisely at the pressure node, a different register hole would be needed for each note, which would cause practical problems in the design of the instrument.

The rest of the holes are primarily tone holes, however some are occasionally used as register holes. An example is the alternate F6 hole, which is slightly closer to the mouthpiece end of the instrument than the lower of the two main register holes. It is designed as a tone hole, which are larger than register holes, so it is not as effective as a proper register hole. Some players choose to close the pad mechanism slightly to reduce the gap created when the button is depressed making it more like a register hole. This works for the register purpose but reduces the standard of the F6 produced when used as a tone hole.

When tone holes are used for their normal purpose, the keys are employed so that all pads (and hence holes) are closed from the mouthpiece down to a certain point. At this point and from it down (and round the bend) to the bell, all the holes will be open. This creates low pressure at the end of the closed section of the bore, close to atmospheric pressure. The end is now effectively open and the bore resonates as if it was shortened to the open/closed boundary point. A shorter air column produces a higher pitched note, so the tone holes are placed so to produce all the notes between the fundamental and the second harmonic an octave up. This is the reason the saxophone has less keys than the clarinet which (as a closed cylindrical pipe) over blows (produces a difference in pitch between first and second harmonics) at an octave plus a perfect fifth. The extra keys on the clarinet are to fill in the notes making up that additional fifth.

The wave does not however end at the open tone hole, it goes slightly further. This is called the end correction and is due to the air at the open end being pushed forward and back again by the wave. The effect decreases further away from the end of the closed pipe as the wave radiates in all directions and so only the nearby air is affected. The extra air due to this correction causes the pitch to be lower than it would have been without it.

The extra end correction down the bore beyond the first open hole allows what is called cross-fingering. This is where after the first open hole there is another set of closed holes (see figure 7). This is used as standard for several notes. The closed hole after the open one causes the wave in the bore to go slightly farther down than it would have. This has the effect of lowering the pitch but not as much as if the open hole in between the closed pads was shut.

Figure 7. http://www.phys.unsw.edu.au/jw/graphics/saxFsharp4cross.GIF An example of cross fingering on a soprano saxophone. The black keys and pads and closed and the white pads and shaded keys are open. The upper image shows the fingering for a simple F# whereas the lower is a cross fingering for the same note.

Cross fingering only works well for a range of notes as the end correction is frequency dependent. The sound escapes the bore by accelerating the air around the tone hole as previously described. The acceleration (and therefore force) required to match the air outside increases as the square of the frequency. There is only a short amount of time for a high frequency wave to get the air moving as it passes. So for high frequencies, the wave travels further down the bore i.e. the end correction is longer (a feature increased for smaller holes). Therefore the higher frequency wave in an upper register is longer and the pitch flattened more than the octave below. This can spoil the harmonic ratio and prevents the use of some cross fingerings in upper registers. However, the cross fingering given in Figure 7 is used for both the lower and upper registers.

The same frequency dependence for end corrections gives the saxophone another characteristic. Under normal (not cross) fingering, lower frequencies are reflected at the of the closed hole regime as stated above and find it easier to leave the bore via the tone holes. However, higher frequencies have this longer end correction. Thus the open holes act as a high pass filter- they allow higher frequencies through (see fig. 8). This inhibits the highest pitch playable along with the reed (its resonant frequency is still the highest note possible). This also means that when using microphone equipment, the microphone should be placed far enough back so that the correct sound balance (bass and treble) can be amplified. The bell helps with this as detailed further below.

Figure 8. http://www.phys.unsw.edu.au/jw/graphics/cutoff.gif A diagram showing the high pass filter property of a series of open holes.

Whilst in the bore, much of the energy supplied is lost to a frictional-like viscous loss. This is caused by the air flow not being constant along the tube walls. Towards the middle, the air is slowed down by the rubbing. The drag is proportional to the viscous thickness on the walls. Similarly, heat loss from the air to the body of the instrument is another tap on the energy provided. Again it is dependent on the thickness of the boundary layer.

Any sound reaching the end of the bore is radiated from the bell, which is the matching transformer of the instrument. The main purpose however, is to act as a high pass filter in order for the notes radiated al the bell (mainly the lower ones) to have a consistent sound with the higher notes radiated at the tone holes. The filter effect is due to the diameter of the bell, which is closer to the wavelength of higher frequencies and so radiates them better.

Source Material

N. H, Fletcher and T. D. Rossing (1991) Springer The physics of musical instruments Johan Sundberg (1992) Academic Press The science of musical sound University of New South Wales acoustics pages: http://www.phys.unsw.edu.au/~jw/acoustics.html and links.