Introduction to Acoustics

a)
1 period
Musical Acoustics 15.1 Vibrational Modes of Instruments 541
0 5 10 (Hz) 0 5
10 (Hz)
b)
1 period
Fig. 15.3a,b Comparison of the envelope, repetitive waveform and spectrum of (a) a synthesised sawtooth and (b) a note
played on an oboe
Range of Hearing and Musical Instruments
A young adult can usually hear musical sounds from
around 20 Hz to 16 kHz, with the high-frequency response
decreasing somewhat with age (typically down to
between 10–12 kHz for 60-year olds). Audio
provides a sequence of 1 s-long sine waves starting from
25 Hz to 12.8 kHz, doubling in frequency each time.
Doubling the frequency of a sinusoidal wave is equivalent
in musical terms of increasing the pitch of the note
by an octave. Audio is a similar sequence of
pure sine waves from 8 kHz to 18 kHz in 2 kHz steps.
Any loss of sound at the low frequencies in
will almost certainly be due to the limitations of the
reproduction system used, which is particularly poor below
≈ 200 Hz on most PC laptops and notebooks, while
the decrease in intensity at high frequencies in
simply reflects the loss of high-frequency sensitivity of
the ear (see Fig. 15.16 and Chap. 13 for more details on
the amplitude and frequency response of the human ear).
The above sounds should be compared with the
much smaller range of notes on a concert grand piano,
typically from the lowest note A0 at 27.5 Hz to the highest
note C9 at 4.18 kHz, as illustrated in Fig. 15.4. The
nomenclature for musical notes is based on octave sequences
of C-major scales with, for example, the note
C1 followed by the white keys D1, E1, F1, G1, A1,
B1, C2, D2, ... . Alternatively, the octave is indicated
2.2 s
by a subscript (e.g. A4 is concert A). Where the white
notes are a tone apart, a black key is inserted to play the
semitone between the adjacent white keys. This is indicated
by the symbol # from the note below or ♭ from
the note above. Figure 15.4 also illustrates the playing
range of many of the instruments to be considered in this
chapter.
Frequency and Pitch
It is important to distinguish between the terms frequency
and pitch. The frequency of a waveform is
strictly only defined in terms of a continuous sinusoidal
waveform. In contrast, the waveforms of real musical
instruments are in general complex, as illustrated by the
oboe waveform in Fig. 15.3. However, despite such complexity,
the repetition frequency and period T can still be
defined provided the waveform does not vary too rapidly
with time. The periodicity of a note (measured in Hz)
can then be defined as the inverse of T. This is generally
the note that the player reads from the written music.
However, as described later, a repetitive waveform does
not necessarily include a sinusoidal component at the
repetition frequency, an effect referred to as the missing
fundamental. Furthermore, depending on the strength of
the various sinusoidal components present, there can often
be an ambiguity in the pitch of a note perceived by
the listener. The subjective pitch, when matched against
Part E 15.1