The following cartoons show a very short
pulse of high pressure, high density air travelling in a pipe. They are not to scale: sound waves involving an r.m.s. density variation of even one percent would give a sound level of 157 dB. So the variation shown here
is vastly exaggerated. To make it visible, we also exaggerate the amplitude
of the wave that leaves the tube: in practice, only a small
fraction of the energy is lost this way.
We do include end effects: the open end has a non-zero radiation impedance so the reflection effectively occurs slightly
beyond the end of the pipe. This is why the next two animations gradually become out of phase. End effects and reflections at an open end are discussed below.
The first animation is for a cylindrical pipe open at both ends (an approximation of a flute).
Next, let's look at reflections in a cylindrical pipe closed
at one end (an approximation of a clarinet or a cylindrical pipe used as a didjeridu).
The reflection at an ideally closed end can be described qualitatively: the high pressure (and high density)
pulse pushes against the closed end, which pushes back (Newton's third law) and the pulse 'bounces' off the closed wall. The flow and momentum have a change change of sign, but the high pressure pulse is reflected as a high pressure pulse,
with no change in sign.
Now compare the periods of the oscillation in these two
examples and note something important: one complete cycle
in the closed-open pipe (below) requires the pulse to traverse four lengths of the tube,
which takes almost twice as long as that in an open-open pipe (above),
in which the pulse covers only two lengths.
A cylindrical pipe closed at both ends has a period that
is slightly shorter than does an open-open pipe. I say 'slightly'
because it has no end corrections.
Such a pipe is rarely
used deliberately in music, but I give a musical example here.
These animations were made by George Hatsidimitris.
Frequency, pitch and the cylindrical pipe resonances
Now, back to comparing cylinders that are open at one end, and
either open or closed at the other, like the flute and clarinet
in our example.
In the Acoustics Today article, the impedance spectra are for instruments with most of the tone holes open (a more typical state). Here, for simplicity, let's close all the tone holes, as though to play the lowest note.
The graph at right is the measured acoustic impedance of a simple cylindrical tube of length 597 mm – between the length of a flute and a clarinet – and an internal diameter of 15 mm, which is similar to that of a clarinet, and a little smaller than that of a flute (19 mm). For both, we measure at the mouthpiece end, and the far end is open.
At low frequencies, this curve looks somewhat similar to the measured impedance of a flute with all the holes closed (Open a new window for flute lowest note impedance spectrum). This curve also looks somewhat similar (again, at low frequencies) to the measured impedance of a clarinet with all the holes closed (Open a new window for clarinet lowest note impedance spectrum). This is not surprising because both flute and clarinet are approximately cylinders and both are open at the far end. The difference is not so much in the graph as in the application of it: the flute is open to the air at the embouchure and therefore operates at or near the minima of Z (the acoustic impedance), while the clarinet is closed, so it operates at or near the maxima of Z.
This figure is taken from a paper that explicitly compares the measured properties of clarinets and flutes with those of simple cylindrical pipes, and explains many of the complications that are discussed only briefly here.
First, let's think of this curve as the impedance spectrum for a flute. (See introduction to flute acoustics.) Looking at the minima of the spectrum, we expect that the (C foot) flute with all keys closed will play at the frequencies of about 260 Hz, 520 Hz, 780 Hz, 1040 Hz, 1300 Hz etc. You can try this and find that it does: these are approximately the frequencies of the notes C4, C5, G5, C6, E6 etc - we say that it plays the harmonic series f, 2f, 3f, 4f etc where f is the fundamental frequency (here about 260 Hz) and the others are higher notes, almost exactly at harmonic frequencies. Note that the seventh harmonic and the seventh resonance lie between A and A#, as indicated approximately by the half-sharp. If you don't have a flute or clarinet handy, you can listen to the sound files below.
Let's now think of the graph at right as the impedance spectrum of a clarinet with all tone holes closed. (See introduction to clarinet acoustics.) The pipe used is slightly longer than the cylindrical part of the bore of the clarinet, so this calculation will be approximate. Looking at the maxima of the spectrum, we expect that it will play at frequencies of about 130 Hz, 390 Hz, 650 Hz. So this hypothetical (cylindrical) clarinet with all its holes closed plays a series of notes C3, G4, E5 and some squeaks above that. In other words the series F, 3F, 5F etc where F is the fundamental frequency, which is about 130 Hz for the hypothetical clarinet with all holes closed. (In practice, the A clarinet plays a series starting on C#3 and the Bb clarinet D3 and higher notes, because their cylindrical sections are shorter than the flute. This issue is further complicated because A and Bb clarinets are transposing instruments, and so clarinettists give these notes names a minor third or a major second higher, respectively.) There are complications due to end corrections, and also to the fact that the instruments are not completely cylindrical: the clarinet has a flare and a bell at one end, and a strong taper in the mouthpiece at the other. (For a discussion of these differences, see the scientific paper.)
The harmonic overtone series on the flute and clarinet
Both the following graphs show the harmonic series.
Compare:
Overblowing the lowest note on a flute.
For the flute, the notes played by overblowing fall close to the harmonics of the lowest note. This is not the case for the clarinet.
Contrast:
Overblowing the lowest note on a clarinet.
The written notes are the harmonics, with, for the clarinet, missing even harmonics in parentheses. (The Bb clarinet is a transposing instrument: written E3 (the first note) sounds D3.) The notes in the sound file are the notes played by overblowing, without using register keys. With the exception of f0 and 3f0, the played notes are considerably displaced from the harmonic frequencies. There are a few reasons: the instrument is not a simple cylinder, the reed is not a simple termination and I had to change my mouth shape and lip forces to play the high notes. There's more discussion in Introduction to clarinet acoustics and in the paper whence these figures come.
End effects and the 180° reflection
In the animation in the section above, end effects in reflections at an open end are approximately included. The acoustic impedance at the open end is not zero: the air just outside has non-zero inertia.
The inertance at the open end is equivalent to that of a cylinder of air with the same diameter as the pipe and length about 0.6 times the radius. This is the end correction: the pipe behaves like an ideally open pipe 0.6r longer.
Note the consequence of this: all else equal, a large diameter pipe plays a little flatter than a thin one.
For a rigidly closed end, there is no such end correction.
I'm often asked why there is a reflection at an open end. Why does an inverted pulse return? In the language of introductory physics, it's easy to explain, which I do here. For musicians without physics, here is a qualitative explanation. When the pulse of high pressure air gets to the end of the pipe, it spreads out in all directions, and its pressure falls very quickly to nearly atmospheric pressure. However, it still has the momentum to move away from the end of the pipe. Consequently, it creates a little suction: the air following behind it in the tube is sucked out, a little like the air that is sucked behind a speeding truck.
A suction at the end of the tube draws air from further up the tube, and that in turn draws air from further up the tube and so on. So the result is that a pulse of high pressure air travelling down the tube is reflected as a pulse of low pressure air travelling up the tube. We say that the pressure wave has been reflected at the open end, with a change in sign (or change in phase of 180° for a sine wave). In the open-open pipe, there is such a reflection at both ends. (This is what physicists call an 'arm-waving argument': it's neither rigorous nor quantitative. If you'd like a more serious explanation, see Standing waves in pipes and Reflection at an open pipe.)
The inertia of the air near open tone holes can produce considerable end effects. Especially at high frequencies, this inertia partly seals the bore and allows a fraction of an incident wave in the upper bore to be transmitted past the open tone hole. As mentioned in the Acoustics Today article, the tone holes on folk and baroque instruments are often small enough to allow players to use cross fingerings to play chromatic notes. A fingering chart for an alto recorder illustrates this.
This scientific paper on cross fingering and cutoff frequencies explains further.
Real instruments: further complications
The bodies of real flutes and clarinets are of course more complicated than simple cylinders: they have tone holes and keys. To a crude approximation, opening all or most of the holes below a certain point on the instrument shortens the pipe available for standing waves. So, at least at low frequencies, the instrument behaves somewhat like a cylindrical pipe with an 'effective length' not very far from the position of the first open key.
That real life is more complicated is demonstrated by the two sound spectra at right. One is a flute playing the note D5, the other is a clarinet playing the note D5 (called E5 on a Bb clarinet). In each case, the microphone was at 3 cm from the first open tone hole. Can you tell which is flute and which is clarinet? This figure is taken from a paper, which deals with some of these complications, and explains some of the differences and similarities between flutes and clarinets, and compares them in detail with simple pipes – and answers this riddle.
Higher resonances in the time domain
Taking the (corrected) pipe length as L and the speed of sound as c, the open-open pipe has its first resonance at c/2L and its second resonance at c/L. The closed-open pipe has its first two
resonances at c/4L and 3c/4L. So it's nice to see animations of these in the time domain, to compare with those above. Here they are for the second resonances of open and closed pipes.
Links with more information on other topics in the article
A classical example contrasting orchestral wind timbres: Maurice Ravel’s Bolero. But before you watch and listen to this performance by the Vienna Philharmonic Orchestra with Gustavo Dudamel, here is a puzzle. What instrument or instrument has the solo after 7 minutes and 47 seconds? Set the scrub bar at this point, press play and close your eyes.
Bolero consists of two similar tunes (the “A” and “B” tune) played 9 times each over a simple repeated bar of C major in 3/4 time. Beginning with the flute, a variety of instruments, mainly woodwinds solo and in combinations, are introduced over an increasingly loud accompaniment. See the table below.
Time
Tune
Instrument or combination
Comments
0:32
A
flute
the flute in its lowest range: soft melody over a very soft accompaniment.
1:26
A
clarinet
a comfortable range for the clarinet, crossing the 'break' between the first and second registers (first and second modes).
2:20
B
bassoon
in the high range: only a semitone lower than the famous solos in Rite of Spring and Shostakovich IX.
3:16
B
sopranino clarinet (Eb clarinet)
it plays high on this little instrument
4:10
A
oboe d’amore
this relatively rare instrument is midway between an oboe and a cor anglais
5:04
A
muted trumpet and flute in octaves
it sounds more like a trumpet, with the flute reinforcing the trumpet's second harmonic (a flute is fairly often used like this, an octave above what one might hear as the tune)
6:00
B
tenor sax
what should a saxophonist do? Play 'classically' because it's in an orchestra? But it is the raunchier version of the tune, and it is saxophone...
6:55
B
soprano sax
the score calls for sopranino sax, then soprano, but it's usually (as here) played on soprano.
clarinet 2 and cor anglais play at the tonic, oboe d’amore at the fifth, clarinet 1 and oboe at the octave
9:36
B
trombone
this is pretty high on the trombone: a challenge for most players
10:30
B
woodwinds (except bassoons) but including tenor sax
parallel chords
11:23
A
1st violins and upper woodwinds in octaves
finally some strings join the tune
12:16
A
1st and 2nd violins, divisi, and upper woodwind
this time in parallel chords
13:07
B
1st trumpet, uppper winds and violins
in octaves
14:00
B
trombones, upper woodwinds, violins violas and cellos
in parallel chords. Finally the celli and viole can stop that repeated, finger-blistering pizzicato bar!
14:50
A
trumpets, flutes and pic, saxes, 1st violins
in parallel chords
15:41
B
trumpets, trombone, flutes and piccolo, saxophones, 1st violins
in parallel chords
16:27
A'
trumpets, trombone, flutes and piccolo, saxophones, 1st violins
wow, a key change to E major never sounded so novel!
16:49
the finale
back to C major, and congratulations to the snare drummer, who has played two alternating bars in a steady crescendo throughout.
After 7:49, the first horn and two piccolos are in three different keys, for a good reason. The horn is playing the tune at the fundamental pitch. The tune is in C major (no sharps and flats, though it looks like G major on the score because the horn in F is a transposing instrument). The second piccolo is playing the same tune, exactly one chromatic twelfth above, i.e. at three times the f0, so he is in G major (one sharp). The first piccolo is playing the same tune, exactly one chromatic seventeenth above, i.e. at five times the f0, so he is in E major (four sharps).
Thus Ravel has made the two piccolos exaggerate the third and fifth harmonics of the horn, making it sound like a new instrument, with the strong odd harmonics perhaps giving it some low clarinet flavour. (Organists may hear it as an organ registration, with the horn being the 8 foot stop and the two piccolos being 8/3 = 22/3 foot and 8/5 = 13/5 foot stops.) For this to work, the piccolos must play quietly, and blend with the horn. Playing quietly is difficult to do up high on the piccolo, especially for the high F# in the 9th bar of the tune – a difficult note on the instrument. And by the way, they are also doubled by the celeste, adding a little extra attack to each note.
Clever writing from a masterful orchestrator. (In the score, the instruments are shown at written pitch. The sounding pitch is one octave higher for the piccolos and a fifth lower for the horn.)
Compare:
Overblowing the lowest note on a flute.
Contrast:
Overblowing the lowest note on a clarinet.
The bodies of real flutes and clarinets are of course more complicated than simple cylinders: they have tone holes and keys. To a crude approximation, opening all or most of the holes below a certain point on the instrument shortens the pipe available for standing waves. So, at least at low frequencies, the instrument behaves somewhat like a cylindrical pipe with an 'effective length' not very far from the position of the first open key.
After 7:49, the first horn and two piccolos are in three different keys, for a good reason. The horn is playing the tune at the fundamental pitch. The tune is in C major (no sharps and flats, though it looks like G major on the score because the horn in F is a