suction produced by the moving air. The cycle then repeats. (This explanation is rather simplified because the internal motion of each lip is complicated and because of the behaviour of the air in the bore and in the mouth.)
The more tension you apply to your lips (the harder you pull your lips backwards in a strange smile), the more quickly they spring back into position. If the whole cycle takes a time T (called the period), then there are (one second)/T cycles per second. So the frequency f, in cycles per second, is just f = 1/T. All else equal, high lip tension gives high frequency and so high pitch. In the sound files above (play them again), the lip tension is increased and decreased smoothly.
This frequency of the lip vibration is, of course, the fundamental frequency of the note. For a high trumpet note, the lips may vibrate at more than 1000 vibrations per second. This is fast, but remember that the muscles are not contracting at that rate: the muscles in the brass player's lips exert almost constant tension, and it is the elastic and aerodynamic forces on the lips that produce the vibration.
The film clip below was made using a transparent pipe played as a didjeridu, at a frequency of about 80 Hz. On the left you see a high speed video of the lips, on the right schlieren images of the air jet from the lips. More details on our yidaki (didjeridu) site.
Adding (only) a mouthpiece makes relatively minor changes: it can reduce the amount of the lips that move, and it allows the pressure outside the lips to be a little different from atmospheric. We'll discuss mouthpieces in detail later, but for now here is a sketch of the lips and a horn mouthpiece. Many players position the mouthpiece asymmetrically, so that it covers more of the upper lip than the lower lip, as is shown here.
In this figure, the two upper figures are spectra that were taken over the first and last 0.3 seconds of the sound file. The spectrogram (lower figure) shows time on the x axis, frequency on the vertical axis, and sound level (on a decibel scale) in false colour (blue is weak, red is strong). In the spectra, note the harmonics, which appear as equally spaced components (vertical lines). In the spectrogram, the harmonics appear as horizontal lines. Note that the pitch doesn't change, so the frequencies of the spectral lines are constant. However the power of every harmonic increases with time, so the sound becomes louder. The higher harmonics increase more than do the lower, which makes the timbre 'brassier' or brighter, and also makes it louder. (To re-emphasise our point about loudness: over this crescendo, the fundamental increases by only 8 dB, but the ninth and some higher harmonics--in the sensitive range of your ear--increase by more than 45 dB.)
An interesting point about loudness in brass instruments. In acoustics, we are usually interested in cases in which the amplitude of the sound pressure wave is only a tiny fraction of atmospheric pressure. Consequently, the medium (air) behaves linearly for such waves, which simplifies the mathematics considerably. For example, a sound of 120 dB is painfully loud, but it corresponds to a sound pressure of only 20 Pa, or 0.0002 atmospheres. At the position where the viola player is complaining about the pain in her ears from the trumpets behind her, the sound level is not at 120 dB.
However, inside the narrow bore of the instrument, the sound pressure is much higher, for two reasons. First, it is concentrated in a small cross sectional area, instead of spread out over a much larger area outside the instrument. Second, most of the sound inside the instrument is reflected at the ends to provide the standing waves we have discussed above. So the pressure inside a brass instrument can be a substantial fraction of atmospheric pressure, and so the medium can behave in a non-linear way. This can produce a shock wave in the instrument, which results not only in the conversion of power from low frequency to high, but also to the production of frequencies that are not harmonic. This phenomenon was analysed formally by Mico Hirschberg and colleagues (1996, JASA, 99, 1754-58). It is stronger in instruments with long, narrow sections of bore (trumpets and trombones) than in others (flugelhorn or tuba. Some scientists associate the onset of the shock wave with what some brass players refer to as the sizzle point. (Here is a video showing the emergence of the shock wave.)
While talking about decibels, we should mention that spectra are usually shown on a decibel scale. (See What is a decibel?) This means that one notices easily on the spectrum a harmonic that is say 20 dB weaker than the fundamental, even though it has 10 times less pressure and 100 times less power. What is important is that your ear notices it too, because of the frequency dependence referred to above. However, it is much more difficult to notice the presence of harmonics if you look at the waveform: for this reason, an oscilloscope is only of limited use.
The difference between closed and open pipes is explained in Open vs closed pipes , which has more explanation of the animation below.
The shape of the pipe varies widely among instruments. For the modern instruments, there is a section of tubing with constant diameter (this includes the slide of the trombone and the section containing the valves of other instruments). On the upstream end is a mouthpiece, which has a little cup-shaped section and a constriction. At the other is a long flaring section ending in a bell. In the diagram below, the vertical and horizontal scales are different: the ratio of width to length has been greatly exaggerated to show the variation in diameter more clearly.
The shapes whose acoustical properties are easiest to understand are the simple conical bore (rather like the cornetto) or the cylinder (rather like the didjeridu). We devote a whole page to the acoustics and harmonics of cylindrical and conical bores, open and closed because the oboe, saxophone and bassoon are closed, approximately conical bores, the clarinet is nearly a closed cylindrical bore and the flute is well approximated as an open cylinder. If you are interested in what finger holes do on the older lip reeds, go to the section on tone holes on saxophones, because a saxophone can be considered approximately as just an ophicleide with a clarinet mouthpiece.
In fact, we shall see that several of the harmonics of the modern brass instruments are not very different from those of a cone. Exceptions are the fundamental or pedal note, and the very highest notes. To understand why, we shall 'build up' a brass instrument, starting with a simple piece of pipe.
In these cases, the lips and the pipe were in a sense cooperating to form the vibration that gives rise to the sound. The lips may have their own natural vibrating frequency, which the player can control with lip tension. The pipe has also its own natural frequencies, which are due to standing waves. There is a whole page devoted to pipes and harmonics, and another on standing waves. We briefly review the results here.
At the far end, the pipe is open to the air, so the pressure there must be close to atmospheric at all times: in other words, the varying part of the pressure (what we call the acoustic pressure) is near zero. We call this a node in acoustic pressure. At the other end, the pipe is sealed from the atmosphere by the player's lips, and the pressure can vary maximally as the lips open and close: indeed, it is the large variation of pressure in the mouthpiece that (usually) forces the lips to vibrate at a resonance of the bore. So at this end we have a pressure antinode. If we look for the simple waves that satisfy these constraints – maximum at the mouthpiece, minimum at the bell – we obtain the set shown below.
These frequencies are in the ratio 1:3:5:7 etc. They constitute the odd members of the harmonic series, and they are what we heard in the sound file above. The even harmonics (dashed in the figure) don't fit the conditions in a closed, cylindrical pipe because they have a node at the mouthpiece. (More detail in Open vs closed pipes .
(Before we go further, we should note that the resonances of a simple pipe are only exactly harmonic if the pipe is very thin. As the pipe diameter increases, the higher resonances are successively flatter. This is quite noticeable in a cylindrical didjeridu.)
The short waves, on the other hand, are better able to travel into the rapidly widening bell. The further they go into the bell, the easier it is for them to escape into the outside air. So the higher frequency waves are more efficiently radiated as sound outside the instrument. This is a characteristic of the bright sound of brass instruments: the bell radiates several of the higher harmonics well, so power of the harmonics does not decrease with frequency the as strongly as it does in the woodwinds. This also makes brass instruments loud, because these strongly radiated high frequencies begin to fall into the range where our ears are most sensitive. To hear the effect of the flare and bell, listen to the sound files below. Notice how the bell raises the pitches of the first three resonances, and also brings them closer together. We'll discuss that in detail below. Notice too that the bell begins to add the 'brassy' sound.
Finally, let's add a mouthpiece. Now we can play the normal harmonic series (see standing waves to revise about harmonics), including the notes used for bugle calls. We'll see why in the next section.
When placed against the player's lips, the enclosed volume is sealed at one end by the lips, and has the constricted part of the pipe at the other end. You might imagine it as a tiny bottle, with your lips at the base of the bottle, and the constriction representing the neck of the bottle. Now, just as the bottle has a resonance that you can excite by blowing over the top, the mouthpiece has a resonance that you can excite by slapping the wide end against the palm of your hand. When shopping for mouthpieces, brass players sometimes do this to compare what they call the 'pop tone'. It is much easier to hear the pitch if you compare two: say a trumpet and a trombone mouthpiece. As you might expect, the larger the volume (all else equal), the lower the pitch of the pop tone. This is an example of a Helmholtz resonator, whose frequency depends on the enclosed volume and the geometry of the constriction.
The mouthpiece does a few things. First, it allows you to connect the pipe to a comfortably large section of lips. Then there are the acoustic effects of the enclosed volume and constriction. One effect is to lower the frequency of the very highest resonances (roughly speaking: those above the pop tone). So in this regard it opposes the effect of the flare and bell, which tend to raise all the resonances. Another is that to strengthen some of the resonances. We'll return to this when we discuss the spectrum of brass instruments. But first, let's look at the combined effect of the mouthpiece, pipe, flare and bell.
Resonances and pedal notes.In this diagram we show at left the resonances of a simple cylindrical pipe, like a very narrow didjeridu. It is 130 cm long, and its lowest note is C2. As a closed, cylindrical pipe, its resonances are the odd harmonics of its fundamental frequency F (careful: here F is a symbol for frequency, not the note above E). We now add a mouthpiece at one end, and at the other we replace a long section of cylindrical pipe with a flare and a bell, to obtain a bore much like that of a C trumpet. The resonances all rise in frequency and pitch (flare and bell effect), although the upper resonances rise proportionately less (mouthpiece effect).
The shape of the trumpet is so designed so that the second and all higher resonances have risen so that they have frequencies in the ratios 2:3:4:5 etc. In other words, the resonances are a complete harmonic series, except for the fundamental The lowest resonance of the trumpet is not a member of this series. Further, it is weak and rather difficult to play. However, good players can play the pedal note, whose fundamental frequency does not correspond to a resonance of the instrument! Further, the spectrum of a pedal note has hardly any power at the fundamental frequency. (We show this quantitatively below in Frequency response and acoustic impedance.)
What happens here is that the higher resonances (2f, 3f, 4f etc) combine to help the lips establish a nonlinear vibration at the frequency of the missing fundamental f. (Technically, this is the process that physicis and engineers call mode locking, and is an effect characteristic of nonlinear oscillators. When oscillations at two frequencies f1 and f2 are input to an non-linear system, they produce what we call sum and difference terms: vibration components with a range of frequencies including f1 + f2 and f1 − f2. In the pedal note vibration, there are lots of vibration components whose difference is f: any two adjacent resonances have that difference.)
For instance, with no valves depressed on a trumpet, you can play the lowest note: (written) C4, with a wavelength of the order of twice the length of the instrument. You may also be able to play the pedal note at (written) C3.
|By the way, you can also play this harmonic series on a string, because strings also have the complete harmonic series.|
The natural trumpet and natural horn do just this. The natural or baroque trumpet came in a variety of tunings, but was usually rather longer than the modern trumpet. This transposes the harmonic series down, which brings the high harmonics into (slightly) more playable range. Here is Paul Plunkett, professor of trumpet at the Conservatorium of Winterthur, playing the Prince of Denmark March on a baroque trumpet in D, whose harmonic series we see here, followed by the first two bars of the march.
Many of the intervals between pairs of the higher harmonics are, shall we say, less usual harmonies. In our example, the seventh lies about half way between A and A# (note the half sharp symbol on this and the preceding graphic) in the equal tempered scale. Similarly, the eleventh harmonic is approximately a half sharp fourth on the scale. On baroque trumpets and horns, the player adjusts the intonation to bring these 'out of tune' notes into more familiar scales, as Paul Plunkett does in the example above.
Another way to deal with this is to get used to it. One midsummer's night, in a little village in the French alps, a natural horn ensemble had gathered for the fête de la musique. Their scales and chords had half-sharpened fourths and sixths. Very strange – at first. However, after they had played for an hour or so, the natural harmonies on these natural horns started to sound pretty natural to me. Or perhaps it as the wine. Anyhow, such music is not common, but still alive. Benjamin Britten, in the opening and closing passages of his 'Serenade for Tenor, Horn and Strings', asks the player of the modern horn to forego the use of valves and to play using just the natural harmonics.
However, the resonances of the bore are, for low frequencies at least, narrower in frequency and 'stronger' than those of the lips: this allows the resonances of the bore to 'take control'. To oversimplify somewhat, a lip reed instrument usually plays at a strong bore resonance whose frequency is slightly higher than that of the lips.
The 'taking control' is never complete, however: it is always a compromise. So the player can adjust the pitch of a note by changing lip tension, even if the frequency of the resonance of the instrument does not change. Trombonists may play vibrato by moving the slide slightly, but players of other brass instruments do it by 'lipping'.
Finally, we should mention here that the player's lips actually interact with two resonators: those of the bore of the instrument, and those of the vocal tract. The instrument's resonances are usually much 'stronger' than those of the tract, but the latter are still important. This is an area of great interest to us, and we are studying it first on the didjeridu, where it is most important. We have also studied it on trombone and trumpet, and have published some of the results: see below.
The three individual spectra are for the lowest note played (written C4 = sounding Bb3), and the notes one and two octaves higher. The fourth spectrum is the average over time for the whole scale. Why do the spectra have their characteristic shapes?
In all wind instruments, the higher frequency harmonics are relatively inefficient because they lose energy in viscous losses – a sort of friction with the walls, so this explains (in part) the shape at high frequencies. However, for brass instruments, the radiated power sometimes increases with frequency over the low part of the range, as here. This means that one tends to get maximum radiation at a moderately high frequency (typically around several hundred Hz, although it has a different value in each lip reed instrument), and less power at lower and higher. Further, the enclosed air in the mouthpiece and the constriction beyond it also tend to drive the lips most efficiently near this frequency, further contributing to the peak.
Remember, however, that the spectrum depends strongly on how loudly you play. The larger the vibration of the lips, the more non-linear the vibration, so the more strong high harmonics present in the sound. (See Playing softly and loudly above.) Indeed, if a trombone plays very loudly, the acoustic pressure in that relatively narrow pipe ceases to be tiny in comparison with atmospheric pressure. When this happens, the linear wave equation for air no longer applies, and the spectrum itself begins to contain non-harmonic elements. You'll recognise this harsh, almost unmusical sound if you know a loud trombonist.
One simple way of changing the spectrum of a brass instrument is to modify the performance of the bell with a mute, which is our next topic.
On the right are the time-averaged spectra of the two sections of the sound file. Note that the mute attenuates the lower frequency components, but produces a formant at about 1 kHz. (Because several different notes are played, the harmonic structure of the individual notes is obscured.)
Almost anything that occludes a substantial fraction of the bell will act as a mute. The real difficulty in designing a mute is to arrange the reflections at the bell so that the tuning is little affected. One exception is the hand stopping of the french horn. In this case, not only is the timbre changed, but the pitch is altered by a semitone.
The trombone of course solves this problem by having a continually variable length using the slide. The other brass instruments use rotary or piston valves.
How long should the second valve pipe be? To make the calculation simple, let's imagine that we are designing valves to make a narrow bore, chromatic didjeridu (ie we are going to add valves to a simple cylindrical pipe), whose natural length is 100.0 cm. If the second valve lowers the pitch of the natural pipe by an equal tempered semitone, then it must decrease the frequency by 5.9% and thus increase the length of the instrument by 5.9 cm. So the difference in frequency between − − − and − 2 − is now 5.9%.
However, if we are to use the fingering scheme above, we also want the second valve to lower the pitch from 5 to 6 semitones below the natural frequency. Suppose that the first and third valves do lower it by 5 equal tempered, in other words they decrease the pitch by 33.5% and so must together add a length of 33.5 cm. So fingering 1 − 3 has pipe of length 133.5 cm. If we add our 5.9 cm, we increase this to 139.4 cm. This is an increase of 5.9 cm/133.5 cm, which is only 4.4%, which is only three quarters of an equal tempered semitone. The standard way to fix this problem is to use a slide. The problem is worst when the third valve is used, so many trumpets have a slide on the added pipe for this valve that allows the player to increase the length to solve problems such the one we have just discussed. On the larger instruments, this would require a lot of work for the finger operating that slide, so one or more extra valves is added.
Scaling up. The Italian for trumpet is tromba, and the suffix for 'big' is -one. So a trombone is a etymologically a big trumpet. We have seen that the method of changing the length is different for the two but, apart from that, they are not very different from being scale models, at least in the acoustical sense. The tenor trombone is twice as long as the Bb trumpet, and it has approximately twice the cross sectional area.
This scaling allows us to do an interesting demonstration. In the first pair of sound files, we hear the Prince of Denmark march played again by Paul Plunkett. So that we do not get the acoustical clues from the slide, he is playing it as though it were a very low pitched baroque trumpet: he plays using the high harmonics and does not use the slide. (He also plays slowly, so that we can later increase the playback speed.)
Acousticians characterise many of the properties of musical instruments by their acoustic impedance spectrum. This is a measure of how much sound pressure is generated by a sound wave of a given frequency in the instrument. For a brass instrument, the lips require a strong pressure signal to help the lips vibrate, so brass instruments tend to play notes near the peaks in the graph of acoustic impedance as a function of frequency. The graph below (measured by Jer Ming Chen, a PhD student in our lab) shows such a graph for a bass trombone, with the slide in and the trigger not depressed. The lowest normal note for this position is Bb2 at about 116 Hz – the second peak on the graph.
This graph shows several of the features discussed above. Of the first several resonances, all but the first form a nearly harmonic series – they are nearly equally spaced in frequency. The notes that they play are labelled on the graph. (The first is far too flat, as explained above in Resonances and pedal notes, and is difficult to play. It does not correspond to the pedal note, which is played at about Bb1, half the frequency of the first peak and indicated by a dotted line.)The seventh harmonic corresponds to a note in standard Western scales, hence the half sharp symbol. Note too that, because the bell effectively radiates high frequencies rather than reflecting them, the resonances become successively weaker at high frequency, then disappear altogether as the trombone begins to function as a megaphone, rather than a resonator, above about 700 Hz.
Below, we compare the impedance curves for a Bb bass trombone (first position, valve not depressed), a Bb trumpet (no valves depressed) and a horn in the open Bb and F states (no finger valves depressed). The trumpet is shown on a frequency scale that is twice that of all the others. This facilitates comparison with the trombone, and shows that the trumpet is rather like a one-half scale model of the trombone. (Etymologically, the reverse is true: trombone means big trumpet.) The narrower bore and especially the narrower mouthpiece constriction in the trumpet and horn give them, overall, higher values of impedance than those of the trombone. (These impedance spectra were made by Jer Ming Chen, a student in our lab, using a technique described in this scientific paper. If you want to cite them, the reference is: Chen, J.M. (2009) Vocal Tract Interactions in Woodwind Performance, PhD thesis, UNSW, Sydney.)
The third graph shows Z(f) for a horn playing open Bb. As with the trombone and trumpet, the bell radiates high frequencies well, rather than reflecting them, so the resonances become weak at high frequencies. However, the horn is played with the hand positioned in the bell. This increases reflection at high frequency and the stronger peaks in Z make high notes easier to play. The hand also lowers the frequencies of each resonance. (Before the introduction of valves, and occasionally still, different extents of hand stopping were used to vary the pitch).
Most horns are double horns: the player has a valve for the thumb that introduces extra tubing to convert between a Bb horn (lowest playable resonance Bb2 and pedal Bb1, like the trombone) and an F horn (lowest playable resonance F2 and pedal F1, like the bass trombone with the valve depressed.) The horn also has three keys, each of which is connected to two valves, one in the F and one in the Bb horn. The keys add appropriate lengths of tubing to lower the pitch by about 2, 1 and 3 semitones respectively.
Horns are occasionally required to play pitches above 500 Hz. The increase sharpness of the impedance peaks provided by the hand make this easier to do. Comparing the last two curves, we can see why horn players often prefer to play the Bb horn for the high range. The player must select which impedance peak will determine the note by adjusting his/her embouchure. On the Bb horn, the peaks are further apart and one is less likely either to select the wrong one, or to crack a note by involving two neighbouring peaks.
Note that the seventh and eleventh peaks are not very close to a note in the equal tempered scale, but are quite close to the natural harmonics of the pedal. We discussed this above in Intonation in the natural harmonic series.
As mentioned above, the player's lips are loaded by two resonators: those of the bore of the instrument, and those of the vocal tract. Acoustically, the vocal tract is in series with the bore: their acoustic impedances just add up (remember, however, that the sum of complex numbers must include the phase). Further, in a simple model, the series combination is approximately in parallel with the passive impedance of the lips themselves. From these simple models, one might expect that different vocal tract configurations might have relatively small effects on a given playing regime. The graph below (from this paper) shows the frequency of notes 'played' by a simple mass-on-spring cantilever valve. Downstream from the valve, there was a trombone. Upstream, there were two different geometries acting as artificial vocal tracts. The 'high tongue' configuration (with a constriction upstream from the valve) played a somewhat higher pitch for most slide positions. However, it also caused the register transition to occur at a higher pitch.
Now, as we've seen above, the resonances peaks for brass instruments are very strong at low frequencies, but less strong at high frequencies. Is it possible, we wondered, that trumpet players 'tune' the vocal tract to the desired note when they played in the high range? (We had previously demonstrated such an effect in the saxophone. Like the trumpet, the high register – called altissimo – has rather weak impedance peaks and these notes are therefore hard to play. On this page, we show how experienced saxophonists tune the resonances of the vocal tract to select these high notes, and in other techniques as well.)
In measurements of the impedance spectra in the mouths of trumpet players, we find that the peaks in impedance have magnitudes that are sometimes comparable with those of the higher resonances of the instruments. However, we saw no consistent tuning. The plots below (from this paper) show the acoustic impedance measured just inside the mouth of a trumpet player playing a medium and a reasonably high note. They show strong impedance peaks, but their frequencies are not related to those of the note being played.
Chen, J.M., Smith J. and Wolfe, J. (2012)"Do trumpet players tune resonances of the vocal tract?" J. Acoust. Soc. America. 131, 722-727.
If you have questions, send them to us: we often post answers on our FAQ in Music Acoustics.
We shall continue to add to this site as we find time and as our research results are published. We are have published papers on the effects of the player's vocal tract on the sound of lip reed instruments, especially the didjeridu and also on reed instruments. We have published some other papers investigating similar effects on orchestral brass: