Perceptual Coherence : Hearing and Seeing

350 Perceptual Coherence
The Steady State at Maximum Amplitude
In reality, the sound at maximum intensity is not steady at all. There are dynamic
changes in the frequency and amplitude of the vibration modes due
to unstable physical processes as well as the performer’s intentions. We can
derive measures that reflect both the average and dynamic properties in the
steady-state segment.
1. The most commonly used average property of the steady state is the
spectral centroid, a measure of the energy distribution among the frequency
components. A simple way of computing the spectral centroid is the
weighted average of the amplitudes of the components:
Spectral centroid frequency =∑a i f i / ∑ a i
(8.2)
where a i and f i are the amplitudes and frequencies of each harmonic or partial
of the sound.
While the spectral centroid captures the balance point of the amplitudes,
it is incomplete because many different spectral distributions will yield the
same frequency. For example, a sound with equal amplitudes at 400, 500,
and 600 Hz will have the same centroid as a sound with equal amplitudes
at 100, 500, and 900 Hz. Moreover, a sound with equal amplitude at every
frequency component can have the same centroid as another sound with
drastically unequal amplitudes at the component frequencies. For these reasons,
several other measures have been used.
2. The irregularity in amplitude among the spectral components is
another way to characterize the steady state. For example, the relative
strength of odd and even harmonics defines different types of horns, and
an extreme jaggedness of amplitudes indicates a complex resonance structure
such as that found for stringed instruments. In addition, variation in
the frequency ratios among the components can distinguish between
continuously driven instruments (bowed strings, wind instruments, and
voices) and impulsive instruments (piano, plucked strings). J. C. Brown
(1996) found that for continuously driven instruments, the components
were phase-locked and the frequencies occurred at nearly integer ratios.
For impulsive instruments, the components were not related by integer ratios.
After the impulse, each vibration mode decayed independently, and
differences in damping caused the frequencies to deviate out of an integer
relationship.
3. The spectral spread of the amplitudes of the harmonics is yet another
measure. Suppose we have two sounds with components at 100, 500, and
900 Hz. For the first sound the amplitudes are 1, 4, and 1, while for the second
sound the amplitudes are 4, 1, and 4. The energy is spread more widely
for the second sound, and the two sounds, in spite of identical centroids
(500 Hz) and spectral irregularity, will sound different.