Brass instrument (lip reed) acoustics: an introduction

Trumpet, horn, trombone, tuba, serpent, didjeridu... This page explains the physics of brass instruments (technically the lip reed family). It requires no mathematics beyond multiplication and division, nor any technical knowledge of acoustics. For a range of background topics in acoustics (waves, frequencies, resonances, decibels etc) click on "Basics" in the navigation bar at left.

photo of Sam playing trumpet
photo of Claudette playing trombone

    To set the mood, listen to Anthony Heinrichs playing part of the cadenza from the trumpet concerto by Joe Wolfe.

Overview

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  • The player provides air at a pressure above that of the atmosphere (technically, from a few kPa to perhaps ten or so kPa: from a few percent to about a tenth of an atmosphere). This is the source of power input to the instrument, but it is a source of continuous 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).
  • In the lip reed instruments, the lips act as a vibrating valve: technically we say that they form a control oscillator that, in cooperation with the resonances in the air in the instrument, produces an oscillating component of both flow and pressure: it converts some of the DC power of the breath into AC sound power.
  • Once the air in the instrument is vibrating, some of the energy is radiated as sound out of the bell. A much greater amount of energy is lost as a sort of friction (viscous and 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 changes the length of the instrument and thereby the frequencies of the resonances by suitable combinations of inserting extra pieces of pipe via the valves, or by changing the length of the slide in the case of the trombone.
Let us now look at these components in turn and in detail.

    sketch of lip reed bores    

    Above: schematic of the bores of brass instruments (diameter:length ratio exaggerated). At right, a serpent. Photo courtesy of Ra Inta and the Powerhouse Museum. The hands are those of instruments curator Michael Lea. Click on image for a close-up.

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    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.)

effect of bell and mouthpiece on pipe resonances

Effect of bell, flare and mouthpiece.

    harmonics on viola C       
    wav mp3     The resonances of a C trumpet, starting from the second.
By the way, you can also play this harmonic series on a string, because strings also have the complete harmonic series.

You could also play the first several of them on a bass flute (whose lowest note is C3), because a flute (open cylindrical pipe) has a complete harmonic series, or an octave higher on a normal flute.

    harmonic series on D2

    wav (1400k) mp3 (250k)

    The Prince of Denmark March, played by Paul Plunkett on baroque trumpet.

    wav mp3     Playing a phase without and then with a mute.

    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.)

spectra without and with mute

sketch of rotary valve

    In the sketch above, (a) shows the piston and (b) the cylinder into which it slides. (c) shows the valve in the up position, and the fine line below shows the pathway. (d) shows the valve depressed, and the fine line shows how the pathway is elongated. The sketches at right show the valve in the normal position (left) and the effect of rotating the valve 90 (arrow), to include the extra pipe. (Photo courtesy Shaddy Zeineddine.) The sketch at right shows how rotary valves (common on horns) operate.

measured impedance spectrum for a bass trombone on Bb2 measured impedance spectrum for a bass trombone on Bb2

measured impedance spectrum for a bass trombone on Bb2  measured impedance spectrum for a bass trombone on Bb2


 

close up of Yamaha trombone
Our research on wind instruments benefits from instruments lent or given by Yamaha and pBone.
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