Introduction to Acoustics

section. The player selects the note to be sounded by
buzzing the lips, usually at a frequency corresponding
to one of the natural resonances of the coupled air column.
The essential nonlinearity of this excitation process
also excites multiples in frequency of the playing pitch.
Ideally, for ease of playing, these harmonics should coincide
with the higher modes of the excited air column.
As already described, brass instruments are therefore designed
to have a full harmonic set of modes. However,
because of their shape and outward flare, it is impossible
to achieve this for the fundamental mode (Fig. 15.77).
By adjusting the pressure and the tightness of the
lips in the mouthpiece, the player can pitch notes based
on the n =2,3,4... modes – the n = 2 mode, a fifth
above, an octave above, an octave and a fourth above, etc.
Trumpet players typically sound notes up to the 8–10 th
mode, while skilled horn players can pitch notes up to
and sometimes above the 15th. In the higher registers,
the instruments can therefore play nearly all the notes of
a major diatonic scale. A few of the notes can be rather
badly out of tune, but a skilled player can usually correct
for this by adjusting the lip pressure and flow of air
through the mouthpiece. The low notes are based on simple
intervals: the perfect fifth, octave, perfect fourth, etc.
Trumpets and horns were therefore often used in early
classical music to add a sense of military excitement and
to emphasise the harmony of the key in which the music
is written. However, in later classical music and music
of the romantic period, all the notes of the chromatic
scale were required. To achieve this, brass instruments
such as the trumpet and horn were developed with a set
of air valves, which enabled the player to switch in and
out various combinations of different lengths of tube, to
change the effective resonating length of the vibrating
air column and hence playing pitch. Uniquely, the pitch
of the trombone is changed by the use of interpenetrating
cylindrical sliding tubes, which change the effective
length. Modern instruments generally use a folded or
coiled tube structure to keep the size of the instrument
to manageable proportions.
The trombone can sound all the semitones of the
chromatic scale, by the player sliding lengths of closely
fitting cylindrical tubing inside each other. In the first
position, with the shortest length tube (Fig. 15.112), the
B-flat tenor trombone sounds the note B-flat at ≈115 Hz,
corresponding to the n = 2 mode, an octave below the
lowest note on the B-flat trumpet. To play notes at
successive semitones lower, the total length has to be
extended sequentially by fractional increases of 1.059.
From the shortest to longest lengths there are seven such
increasingly spaced positions. When fully extended, the
Musical Acoustics 15.3 Wind Instruments 639
trombone then plays a note six semitones lower (E) than
the initial note sounded. One can then switch to the n = 3
mode to increase the pitch by a perfect fifth, to the note
B a semitone higher than the initial note sounded. Using
the closer positions enables the next six higher semitones
to be played. Higher notes can be excited by suitable
combinations of both position and mode excited. The
trombone is one of the few musical instruments that can
slide continuously over a large range of frequencies, simply
by smoothly changing its length. This is widely used
in jazz, where it also enables the player to use a very
wide, frequency-modulated, vibrato effect and bending
of the pitch of a note for expressive effect.
The fully extended length of the vibrating air column
in the first position is ≈ 2.5 m. Two-thirds of the length
is made up of 1.3 cm-diameter cylindrical tubing with
the remaining gently flared end-section opening out to
a bell diameter of 16–20 cm.
The trumpet achieves the full chromatic range by
the use of three piston valves, which enable additional
lengths of tubing to be switched in and out of the resonating
air column. In the inactive up position, the sound
travels directly through a hole passing directly through
the valve. When the piston is depressed, the valve enables
the tube on either side of the valve to be connected
to an additional length of tubing, which includes a small,
preset, sliding section for fine tuning. The pitch is decreased
by a tone on depressing the first piston and
a semitone by the second. Pressing them down together
therefore lowers the pitch by three semitones (a minor
third). Depressing the third valve also lowers the pitch
by three semitones, so that when all three valves are
depressed the pitch is lowered by six semitones. However,
the tuning is not exact, because whenever any single
valve is depressed the effective tube length is lengthened.
Therefore, when a second (or third) valve is depressed,
the fractional increase in effective length is less when
the second valve alone is used. This is related to the need
to increase the spacing of the semitone positions on the
trombone as it is extended. Similar mistuning problems
arise for all combinations of valves used.
To circumvent these difficulties, compromises have
to be made, if the instrument is to play in tune
(Backus [15.132], pp. 270–271). The added length of
tubing to produce the semitone and tone intervals are
therefore purposely made slightly too long, giving semitone
and tone intervals that are slightly flat, but which in
combination produce a three-semitone interval which is
slightly sharp. Similarly, the third valve is tuned to give
a pitch change of slightly more than three semitones.
This allows the full of range of semitones to be played
Part E 15.3