Page 25–30 describes how the inner ear carries out a frequency analysis of sound due to the mechan-ical properties of the basilar membrane and how this provides the basis behind the “place” theory of hearing. The next important aspect of the place theory to consider is how well the hearing system can discriminate between individual frequency components of an input sound. This will provide the basis for understanding the resolution of the hearing system and it will underpin discussions relating to the psychoacoustics of how we hear music, speech and other sounds.
Each component of an input sound will give rise to a displacement of the basilar membrane at a particular place, as illustrated in Figure 2.5. The displacement due to each individual component is spread to some extent on either side of the peak. Whether or not two components that are of similar amplitude and close together in frequency can be discriminated depends on the extent to which the basilar membrane displacements, due to each of the two components, are clearly separated or not.
Suppose two pure tones, or sine waves, with amplitudes A1 and A2 and frequencies F1 and F2 respec-tively are sounded together. If F1 is fixed and F2 is changed slowly from being equal to or in unison with F1 either upwards or downwards in frequency, the following is generally heard (see Figure 2.6).
When F1 is equal to F2, a single note is heard. As soon as F2 is moved higher or lower than F1 a sound with clearly undulating amplitude variations known as “beats” is heard. The frequency of the beats is equal to (F2 F1), or (F1 F2) if F1 is greater than F2, and the amplitude varies between (A1 A2) and (A1 A2), or (A1 A2) and (A2 A1) if A2 is greater than A1. Note that when the amplitudes are equal (A1 A2) the amplitude of the beats varies between (2 A1) and 0.
For the majority of listeners beats are usually heard when the frequency difference between the tones is less than about 12.5 Hz, and the sensation of beats generally gives way to one of a “fused” tone which sounds “rough” when the frequency difference is increased above 15 Hz. As the frequency difference is increased further there is a point where the fused tone gives way to two separate tones but still with the sensation of roughness, and a further increase in frequency difference is needed for the rough sensation to become smooth. The smooth separate sensation persists while the two tones remain within the fre-quency range of the listener’s hearing.
The changes from fused to separate and from beats to rough to smooth are shown hashed in Figure 2.6 to indicate that there is no exact frequency difference at which these changes in perception occur for every listener. However, the approximate frequencies and order in which they occur is common to all listeners, and, in common with most psychoacoustic effects, average values are quoted which are based on measurements made for a large number of listeners.
The point where the two tones are heard as being separate as opposed to fused when the frequency difference is increased can be thought of as the point where two peak displacements on the basilar membrane begin to emerge from a single maximum displacement on the membrane. However, at this point the underlying motion of the membrane, which gives rise to the two peaks, causes them to interfere with each other giving the rough sensation, and it is only when the rough sensation becomes smooth that the separation of the places on the membrane is sufficient to fully resolve the two tones.
The frequency difference between the pure tones at the point where a listener’s perception changes from rough and separate to smooth and separate is known as the critical bandwidth, and it is therefore
Separate Fused Separate
An illustration of the perceptual changes which occur when a pure tone fixed at frequency F1
is heard combined with a pure tone of variable frequency F2.
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10 000
1000
100
10
1
Bandwidth (Hz)
Center frequency (Hz)
100 1000 10 000
5 semitones 4 semitones 3 semitones 2 semitones 1 semitone ERB (dashed)
fIgure 2.7 The variation of equivalent rectangular bandwidth (ERB) with filter center frequency, and lines indicating where the bandwidth would be equivalent to 1, 2, 3, 4 and 5 semitones.
(Middle C is marked with a spot on the keyboard.)
marked CB in the figure. A more formal definition is given by Scharf (1970), “The critical bandwidth is that bandwidth at which subjective responses rather abruptly change.”
In order to make use of the notion of critical bandwidth practically, an equation relating the effective critical bandwidth to the filter center frequency has been proposed by Glasberg and Moore (1990).
They define a filter with an ideal rectangular frequency response curve which passes the same power as the auditory filter in question, which is known as the equivalent rectangular bandwidth or ERB. The ERB is a direct measurement of the critical bandwidth, and the Glasberg and Moore equation which allows the calculation of the ERB for any filter center frequency is as follows:
ERB{24 7. [(4 37. fc)1]}Hz (2.6)
where
fc is the filter center frequency in kHz
and ERB the equivalent rectangular bandwidth in Hz Equation valid for (100 Hz fc 10 000 Hz) This relationship is plotted in Figure 2.7 and lines representing where the bandwidth is equivalent to 1, 2, 3, 4, and 5 semitones (or a semitone, whole tone, minor third, major third and perfect fourth respectively) are also plotted for comparison purposes. A third octave filter is often used in the studio as an approximation to the critical bandwidth; this is shown in the figure as the 4 semitone line (there are 12 semitones per octave, so a third of an octave is 4 semitones). A keyboard is shown on the filter center frequency axis for convenience, with middle C marked with a spot.
examPle 2.2
Calculate the critical bandwidth at 200 Hz and 2000 Hz to three significant figures.
Using Equation 2.6 and substituting 200 Hz and 2000 Hz for fc (noting that fc should be expressed in kHz in this equation as 0.2 kHz and 2 kHz respectively) gives the critical bandwidth (ERB) as:
ERB at200Hz{24 7. [(4 37. 0 2. )1]}46 3. Hz ERB at2000Hz{24 7. [(4 37. 2)1]}241Hz
37 The change in critical bandwidth with frequency can be demonstrated if the fixed
frequency F1 in Figure 2.6 is altered to a new value and the new position of CB is found. In practice, critical bandwidth is usually measured by an effect known as masking , in which the “rather abrupt change” is more clearly perceived by listeners.
The response characteristic of an individual filter is illustrated in the bottom curve in Figure 2.8, the vertical axis of which is marked “filter response” (notice that increasing frequency is plotted from right to left in this figure in keeping with Figure 2.5 relating to basilar membrane displacement). The other curves in the fig-ure are idealized envelopes of basilar membrane displacement for pfig-ure tone inputs spaced by f Hz, where f is the distance between each vertical line as marked. The filter center frequency Fc Hz is indicated with an unbroken vertical line, which also represents the place on the basilar membrane corresponding to a frequency Fc Hz.
The filter response curve is plotted by observing the basilar membrane displace-ment at the place corresponding to Fc Hz for each input pure tone and plotting this as the filter response at the frequency of the pure tone. This results in the response curve shape illustrated as follows.
As the input pure tone is raised to Fc Hz, the membrane displacement gradually increases with the less steep side of the displacement curve. As the frequency is increased above Fc Hz, the membrane displacement falls rapidly with the steeper side of the displacement curve. This results in the filter response curve as shown, which is an exact mirror image about Fc Hz of the basilar membrane displace-ment curve. Figure 2.9(a) shows the filter response curve plotted with increasing frequency and plotted more conventionally from left to right.
The action of the basilar membrane can be thought of as being equivalent to a large number of overlapping band-pass filters, or a “bank” of band-pass filters, each responding to a particular band of frequencies. Based on the idealized filter response curve shape in Figure 2.9(a), an illustration of the nature of this bank of
filters is given in Figure 2.9(b). Each filter has an asymmetric shape to its response with a steeper roll-off on the high-frequency side than on the low-frequency side; and the bandwidth of a particular filter is given by the critical bandwidth (see Figure 2.7) for any particular center frequency. It is not possible to be particularly exact with regard to the extent to which the filters overlap. A common practical com-promise, for example, in studio third octave graphic equalizer filter banks, is to overlap adjacent filters at the 3 dB points on their response curves.
In terms of the perception of two pure tones illustrated in Figure 2.6, the “critical bandwidth” can be thought of as the bandwidth of the band-pass filter in the bank of filters, the center frequencies of which are exactly halfway between the frequencies of the two tones. This ignores the asymmetry of the basilar membrane response (see Figure 2.5) and the consequent asymmetry in the individual filter response curve (see Figure 2.9(a)), but it provides a good working approximation for calculations. Such a filter (and others close to it in center frequency) would capture both tones while they are perceived as
“beats,” “rough fused” or “rough separate,” and at the point where rough changes to smooth, the two tones are too far apart to be both captured by this or any other filter. At this point there is no single filter which captures both tones, but there are filters which capture each of the tones individually and they are therefore resolved and the two tones are perceived as being “separate and smooth.”
A musical sound can be described by the frequency components which make it up, and an understand-ing of the application of the critical band mechanism in human hearunderstand-ing in terms of the analysis of the components of musical sounds gives the basis for the study of psychoacoustics. The resolution with which the hearing system can analyze the individual components or sine waves in a sound is impor-tant for understanding psychoacoustic discussions relating to, for example, how we perceive:
n melody
Displacement envelopes for input pure tones
Fc � (4 * f )
fIgure 2.8
Derivation of response of an auditory filter with center frequency Fc Hz based on idealized envelope of basilar membrane movement to pure tones with frequencies local to the center frequency of the filter.
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n musical dynamics
n the sounds of different instruments
n blend
n ensemble
n interactions between sounds produced simultaneously by different instruments.