A wind instrument, such as a clarinet or trumpet, produces specific pitches, or frequencies, of sound by supporting standing waves of sound inside the body of the instrument. These waves send out vibrational pulses at regular intervals, setting up continuous sound waves, called traveling waves.
To reiterate the distinction: The waves within the tube are standing waves that emit traveling waves moving away from open ends of the tube. Traveling waves reach our ears and are perceived as sound.
Players set up vibrations in wind instruments in various ways, including 1) blowing across an opening (pop bottle or flute); 2) blowing past a sharp-edged side opening (recorder or simple whistle); 3) blowing into a mouth piece past one or more thin-edged pieces of wood called reeds (clarinet, one reed; oboe, two reeds); or 4) blowing into a mouthpiece that allows the player's lips to vibrate (brass instruments).
Watch this video for an explanation of how a tube produces and sustains certain specific pitches (frequencies of vibration), which depend upon its length. In the calculated results at the end, there is an error (correction below), but other than that, this is one of the best explanations I have found, for students at our level, about how tube length determines the pitch of standing waves.
Corrected calculations:
Closed Tube (wavelength = four times the tube length of 10 cm)
frequency, f = speed of sound/wavelength
= (34,300 cm/s)/(40 cm) = 857.5 /s or 857 Hz, one octave lower than the open tube.
Open Tube (wavelength = two times the tube length of 10 cm)
f = speed of sound/wavelength
= (34,300 cm/s)/(20 cm) = 1715 Hz, one octave higher than the closed tube.
Natural fundamentals and overtones
This diagram shows how the fundamental and overtone pressure waves fit into open and closed tubes, including conical closed tubes (middle column), such as the trumpet. Notice that, in every case, the enclosed wave has a node (pressure same as the air outside) at any open end of the tube, and an antinode (peak or trough) at any closed end. Waves that "fit" in a tube, that is, that have a static node at any opening, are supported and produce sustained sound.
This diagram shows how the fundamental and overtone pressure waves fit into open and closed tubes, including conical closed tubes (middle column), such as the trumpet. Notice that, in every case, the enclosed wave has a node (pressure same as the air outside) at any open end of the tube, and an antinode (peak or trough) at any closed end. Waves that "fit" in a tube, that is, that have a static node at any opening, are supported and produce sustained sound.
Read more HERE.
Adding pitches in between the natural tones: Keys (woodwinds) and Valves (brass)
The keys (woodwinds) and valves (brasses) change the effective length of the tubes to introduce notes in between the natural tones (fundamental and overtones).
Which instruments are open, closed, and closed-conical tubes?
Clarinet: closed
Flute, organ flue pipes*: open
Trumpet, trombone, oboe (surprise!): closed-conical
Remember that open means open at both ends; closed means closed at one end, open at one end.
(What if the tube is closed at both ends?) (Trick question.)
Which instruments are open, closed, and closed-conical tubes?
Clarinet: closed
Flute, organ flue pipes*: open
Trumpet, trombone, oboe (surprise!): closed-conical
Remember that open means open at both ends; closed means closed at one end, open at one end.
(What if the tube is closed at both ends?) (Trick question.)
* HERE is a good look at the design of a simple organ flue pipe and an explanation of how passing air into it produces sound vibrations within the pipe. The pipe shown is a flue pipe, and has a sharp knife edge near the the air inlet. Pipe organs also contain reed pipes, which do not produce sound by setting up standing waves in a cylinder. Instead, sound comes by air blown across an enclosed reed, which vibrates to produce the sound. The design is called a capped reed, such as the ancient crumhorn, but is a single reed, whereas the crumhorn has a double reed.
Fundamentals and Overtones
The longest wave, or lowest frequency, that an open tube can sustain is called its fundamental. The higher pitches are called overtones. Their frequencies are simple, whole number ratios of the fundamental. For the first overtone, an octave, the ratio is 2/1; that is the overtone frequency is twice the fundamental. The second overtone has a ratio of 3/2 to the first overtone. (For emphasis: each ratio is related to the previous overtone, not to the fundamental). The amount and relative loudness of overtones of notes played on instruments gives those instruments their characteristic tone color (Copland, Chapter 7).
More on overtones
If the fundamental is C, then the overtones are
#1, C (interval of an octave)
#2, G (interval of a major fifth above #1, or 7 semitones)
#3, C (interval of a major fourth above #2, or 5 semitones)
#4, E (up a major third, 4 semitones)
#5, G (up a minor third, or 3 semitones)
#6, Bb (up another minor third), the first overtone that is not in the diatonic major scale of C (first black key).
In a nutshell:
• Listen now to Bach's Brandenburg Concerto #2, paying particular attention to the trumpet part (not hard to do). Why did Bach write this part so high in the trumpet range? For a hint, read about the natural trumpet at Wikipedia.
Questions: Do traditional harmonies sound “good” to us because they mimic natural sounds, such as fundamentals of natural vibrations and their overtones? Are they somehow “good” for us in some adaptive way? Or does this "natural" preference simply come from our cultural history?
From https://graham.main.nc.us/~bhammel/MUSIC/ovrtns.html:
If we tune the C major scale according to the overtones of [middle] C, using also the overtones of the F (a perfect fifth below) and G (a perfect fifth above)[, the ratios are as follows]:
C = (1) x C [where C equals the frequency of middle C, which is 261.626 Hz]
D = (9/8) x C
E = (5/4) x C
F = (4/3) x C
G = (3/2) x C
A = (5/3) x C
B = (15/8) x C
C' = (2) x C
This defines the relations between the frequencies of a justly tempered major scale: C major when the frequency of C is inserted into the formulas. This pattern of ratios is extensible in both directions to tune all the white keys of the piano.
Comments and Questions from Students
Fundamentals and Overtones
The longest wave, or lowest frequency, that an open tube can sustain is called its fundamental. The higher pitches are called overtones. Their frequencies are simple, whole number ratios of the fundamental. For the first overtone, an octave, the ratio is 2/1; that is the overtone frequency is twice the fundamental. The second overtone has a ratio of 3/2 to the first overtone. (For emphasis: each ratio is related to the previous overtone, not to the fundamental). The amount and relative loudness of overtones of notes played on instruments gives those instruments their characteristic tone color (Copland, Chapter 7).
More on overtones
If the fundamental is C, then the overtones are
#1, C (interval of an octave)
#2, G (interval of a major fifth above #1, or 7 semitones)
#3, C (interval of a major fourth above #2, or 5 semitones)
#4, E (up a major third, 4 semitones)
#5, G (up a minor third, or 3 semitones)
#6, Bb (up another minor third), the first overtone that is not in the diatonic major scale of C (first black key).
In a nutshell:
• Listen now to Bach's Brandenburg Concerto #2, paying particular attention to the trumpet part (not hard to do). Why did Bach write this part so high in the trumpet range? For a hint, read about the natural trumpet at Wikipedia.
Questions: Do traditional harmonies sound “good” to us because they mimic natural sounds, such as fundamentals of natural vibrations and their overtones? Are they somehow “good” for us in some adaptive way? Or does this "natural" preference simply come from our cultural history?
From https://graham.main.nc.us/~bhammel/MUSIC/ovrtns.html:
If we tune the C major scale according to the overtones of [middle] C, using also the overtones of the F (a perfect fifth below) and G (a perfect fifth above)[, the ratios are as follows]:
C = (1) x C [where C equals the frequency of middle C, which is 261.626 Hz]
D = (9/8) x C
E = (5/4) x C
F = (4/3) x C
G = (3/2) x C
A = (5/3) x C
B = (15/8) x C
C' = (2) x C
This defines the relations between the frequencies of a justly tempered major scale: C major when the frequency of C is inserted into the formulas. This pattern of ratios is extensible in both directions to tune all the white keys of the piano.
Comments and Questions from Students
Hi Gale:
While I was watching the organ flue pipe video, I thought of the didgeridoo and found these expositions of its acoustics from the University of New South Wales and the University of Victoria. Thought you might be interested. I added a couple of Youtube videos as well.
http://newt.phys.unsw.edu.au/jw/didjeridu.html Didgeridoo acoustics
http://web.uvic.ca/~stucraw/didgephysics.html Physics of the Didgeridoo
https://www.youtube.com/watch?v=9g592I-p-dc short video of traditional playing
https://www.youtube.com/watch?v=cLu9GmV2vF0 Didgeridoo Meets Orchestra
Tom Werley
Thanks, Tom!
I'll have a look.
