Introduction to Acoustics

682 Part E Music, Speech, Electroacoustics
Part E 16.3
shows how accurately MFDR could be predicted from
Ps and F0 for previously published data for untrained
male and female singers and for professional baritone
singers [16.45, 46]. Both Ps and F0 are linearly related
to MFDR. However, the singers showed a much greater
variation with F0 than the untrained voices. This difference
reflected the fact that unlike the untrained subjects
16.3 The Vocal Tract Filter
The source-filter theory, schematically illustrated in
Fig. 16.19, describes vocal sound production as a threestep
process: (1) generation of a steady flow of air from
Level
Radiated spectrum
Frequency
Level
Vocal tract frequency
curve formants
Level
Frequency
Glottal voice source
Spectrum
Frequency
Transplottal airflow
Waveform
Time
Lungs
Velum
Vocal
tract
Vocal
folds
Trachea
Fig. 16.19 Schematic illustration of the generation of voice
sounds. The vocal fold vibrations result in a sequence of
voice pulses (bottom) corresponding to a series of harmonic
overtones, the amplitudes of which decrease monotonically
with frequency (second from bottom). This spectrum is filtered
according to the sound transfer characteristics of the
vocal tract with its peaks, the formants, and the valleys between
them. In the spectrum radiated from the lip opening,
the formants are depicted in terms of peaks, because the partials
closest to a formant frequency reach higher amplitudes
than neighboring partials
the singers could sing a high F0 much more softly than
the untrained voices. The ability to sing high notes also
softly would belong to the essential expressive skills of
a singer. Recalling that an increase of Ps increases F0
by a few Hz/cm H2O, we realize that singing high tones
softly requires more forceful contraction of the pitchraising
laryngeal muscles than singing such tones loudly.
the lungs (DC component); (2) conversion of this airflow
into a pseudo-periodically pulsating transglottal airflow
(DC-to-AC conversion), referred to as the voice source;
and (3) response of the vocal tract to this excitation signal
(modulation of AC signal) which is characterized by the
frequency curve or transfer function of the vocal tract.
So far the first two stages, respiration and phonation,
have been considered.
In this section we will discuss the third step, viz. how
the vocal tract filter, i. e. the resonance characteristics of
the vocal tract, modifies, and to some extent interacts
with, the glottal source and shapes the final sound output
radiated from the talker’s/singer’s lips.
Resonance is a key feature of the filter response.
The oral, pharyngeal and nasal cavities of the vocal tract
form a system of resonators. During each glottal cycle
the air enclosed by these cavities is set in motion by the
glottal pulse, the main moment of excitation occurring
during the closing of the vocal folds, more precisely at
the time of the MFDR, the maximum flow declination
rate (cf. the previous section on source).
The behavior of a vocal tract resonance, or formant,
is specified both in the time and the frequency domains.
For any transient excitation, the time response is an
exponentially decaying cosine [16.27, p. 46]. The frequency
response is a continuous amplitude-frequency
spectrum with a single peak. The shape of either function
is uniquely determined by two numbers (in Hz): the
formant frequency F and the bandwidth B. The bandwidth
quantifies the degree of damping, i. e., how fast the
formant oscillation decays. Expressed as sound pressure
variations, the time response is
p(t) = A e −π Bt cos (2π Ft) . (16.1)
For a single formant curve, the amplitude variations
as a function of frequency f is given (in dB) by
[F
L(f)=20 log
2 + � �2 B
2 ]
�
( f − F) 2 + � �2 B
2
�
( f + F) 2 + � �2 B
2
.
(16.2)