Introduction to Acoustics

transient, a period with a quasi-constant amplitude for
a continuously bowed or blown instrument, and a period
of free decay, when the instrument is no longer being
excited. The sound produced by musical instruments is
also significantly affected by the acoustic environment in
which the instrument is played, but this will be ignored
for the moment. Typical initial transients and overall envelopes
of single notes played on a violin, clarinet and
trumpet are shown in Fig. 15.15.
The starting transient provides an immediate clue
to the ear enabling the listener to recognise the instrument
being played quickly. However, the characteristic
fluctuations in frequency and amplitude within the overall
envelope and noise associated with the method of
excitation (e.g. bowing and blowing) are just as important
in the recognition of specific instruments. This can
easily be shown by removing the starting transient from
a musical sound altogether, as illustrated in .In
this example comparisons are made between the sounds
of a violin, flute, trombone and oboe played first with
the initial 200 ms transient removed and then with the
transient reinserted. In each case the instrument can immediately
be identified even in the absence of the starting
transient. The audio example ends with a constant amplitude
sawtooth waveform having an unvarying sound
quite unlike the sound of any real musical instrument.
Nevertheless, the starting transient and subsequent
decay of sound are extremely important in the identification
of the sounds of plucked or hammered strings and all
percussion instruments, where the waveform and spectral
content changes very rapidly with time after the start
of the note. This is illustrated by the dramatic difference
in the unrecognisable sound of a piano when played
backwards and then replayed in the normal direction
( ).
Noise
There are several potential sources of fluctuations in the
envelope of musical instruments, which help to characterise
their characteristic sounds, such as the breathiness
induced by the noise of turbulent air passing over the
sound hole in a recorder, flute or organ pipe (Verge and
Hirschberg [15.15]) and irregularities in the sound of any
bowed instrument due to inherent noise in the slip–stick
bowing mechanism (McIntyre et al. [15.16]).
Amplitude and Frequency Modulation
Another important source of fluctuations is vibrato,
which involves periodic changes in the amplitude, frequency,
or spectral content of a note and often all
three (Meyer [15.17], Gough [15.18]). Vibrato is pro-
Musical Acoustics 15.1 Vibrational Modes of Instruments 551
duced on a stringed instrument by periodically changing
the length of the bowed string by rocking the finger
stopping the string backwards and forwards. In
singing (Prame [15.19]) and wind instruments (Gilbert
et al. [15.20]) vibrato is produced by periodic modulations
of the pressure exerted by the lungs or mouth on
the exciting reed or air passage.
Amplitude modulation of a sinusoidal frequency
component can be expressed as
y(t) = (1 + am cos Ωt) sin ωt
= sin ωt + am
[sin(ω + Ω)t + sin(ω − Ω)t] ,
2
(15.27)
where Ω is the modulation frequency and a the modulation
parameter. Amplitude modulation introduces two
“side-bands” with amplitude am/2 at frequencies Ω
above and below that of the principal central component.
The side-bands have a net resultant that remains
in phase with the central component giving a fractional
change in amplitude [1 + am cos Ωt].
Frequency modulation should more strictly be referred
to as phase modulation, with the phase varying
as
φ(t) = ωt + af cos Ωt . (15.28)
where af is frequency-modulation index. The timevarying
frequency can then be defined by the rate of
change of phase, such that
dφ
dt = ω − afΩ sin Ωt , (15.29)
with a fractional shift in frequency varying as
∆ω(t)
ω =−af
Ω
sin Ωt . (15.30)
ω
For small modulation index, a phase-modulated wave
can be written as
y(t) = sin ωt + af
[cos(ω + Ω)t + cos(ω − Ω)t] ,
2
(15.31)
which again results in equally spaced side-bands about
the central frequency with amplitude af/2, but with a resultant
now in phase-quadrature with that of the central
frequency giving the above phase modulation.
Because of the multi-resonant frequency response
of all musical instruments, changes in driving frequency
also induce significant fluctuations in amplitude. Such
fluctuations are particularly important for the strongly
peaked multi-resonant instruments of the violin family,
as illustrated in Fig. 15.15.
Part E 15.1