equated the loudness of a 1000‐Hz pure tone with bands of noise of increasing bandwidth. The level at which the pure tone was judged to be equal in loudness to the band of noise is shown as the ordinate. Thus the curves do not represent equal loudness contours, but rather they show how the loudness of the band of noise centered at 1000 Hz changes as a function of bandwidth. The loudness of a sound does not change until its bandwidth exceeds the so‐called critical bandwidth. The critical bandwidth at 1000 Hz is about 160 Hz. (Notice that, except for sounds of very low level of about 20 phons, for which loudness is almost independent of bandwidth, the critical bandwidth is almost independent of level and that the slopes of the loudness curves are very similar for sounds of different levels.) Critical bands are discussed further in Section 4.3.6 of this chapter.
The solid line in Figure 4.17 shows that sounds of very short duration are judged to be very quiet and to become louder as their duration is increased. However, once the duration has reached about 100–200 ms, then the loudness level reaches an asymptotic value. Also shown by broken lines in Figure 4.17 are A‐weighted sound pressure levels recorded by a sound level meter using the “impulse,” “fast,” and “slow” settings. It is observed that the A‐weighted sound pressure level measured by the fast setting on the sound level meter is closest of the three settings to the loudness level of the sounds.
Further methods of rating loudness, noisiness, and annoyance of noise are discussed in Chapter 6.
4.3.6 Critical Bands
Another important factor is the way that the ear analyzes the frequency of sounds. The critical band concept already discussed in Section 4.3.5 is important here as well. It appears that the human hearing mechanism analyzes sound like a group of parallel frequency filters. Figure 4.18 shows the bandwidth of these filters as a function of their center frequency. These filters are often called critical bands and the bandwidth each possesses is known as its critical bandwidth. As a band of noise of a constant sound pressure level increases in bandwidth, its loudness does not increase until the critical bandwidth is exceeded, after which the loudness continually increases. Thus the critical band may be considered to be that bandwidth at which subjective responses abruptly change [28]. It is observed that up to 500 Hz the critical bandwidth is about 100 Hz and is independent of frequency, while above that frequency it is about 21% of the center frequency and is thus almost the same as one‐third octave band filters, which have a bandwidth of 23% of the center frequency shown by the solid line. This fact is also of practical importance in sound quality considerations since it explains some of the masking phenomena observed.
Figure 4.18 Critical bandwidth, critical ratio, and equivalent rectangular bandwidth as a function of frequency. (○) Critical bandwidth from Zwicker [41] is compared to the equivalent rectangular bandwidths according to a cochlear map function (‐ ‐ ‐) (dashed line).
(Source: Based in part on Ref. [42].)
The critical ratio shown in Figure 4.18 originates from the early work of Fletcher and Munson in 1937 [28]. They conducted studies on the masking effects of wide‐band noise on pure tones at different frequencies. They concluded that a pure tone is only masked by a narrow critical band of frequencies surrounding the tone, and that the power (mean‐square sound pressure) in this band is equal to the power (mean‐square sound pressure) in the tone [28]. The critical band can easily be calculated. From these assumptions, the critical bandwidth (in hertz) is defined to be the ratio of the sound pressure level of the tone to sound pressure level in a 1‐Hz band (i.e. the spectral density) of the masking noise. This ratio is called the critical ratio to distinguish it from the directly measured critical band [28]. A good correspondence can be obtained between the critical band and the critical ratio by multiplying the critical ratio by a factor of 2.5. The critical ratio is given in decibels in Figure 4.18 and is shown by the broken line.
4.3.7 Frequency (Bark)
It is well known from music that humans do not hear the frequency of sound on a linear scale. A piano keyboard is a good example. For each doubling of frequency (known as an octave), the same distance is moved along the keyboard in a logarithmic fashion. If the critical bands are placed next to each other, the bark scale is produced. The unit bark was chosen to honor the German physicist Heinrich Barkhausen. Figure 4.19 illustrates the relationship between the bark (Z) as the ordinate and the frequency as the abscissa; on the left (Figure 4.19a), frequency is given using a linear scale, and on the right (Figure 4.19b) the frequency is given with a logarithmic scale. Also shown in Figure 4.19 are useful fits for calculating bark from frequency. At low frequency a linear fit is useful (Figure 4.19a), while at high frequency a logarithmic fit is more suitable (Figure 4.19b). One advantage of the bark scale is that, when the masking patterns of narrow‐band noises are plotted against the bark scale, their shapes are largely independent of the center frequency, except at very low frequency. (See Figure 4.20.)
Figure 4.19 Relations between bark scale and frequency scale [17, 18].
Figure 4.20 Masking patterns [17, 18] of narrow‐band noises centered at different frequencies fm.
An approximate analytical expression for auditory frequency in barks as a function of auditory frequency in hertz is given by [43]
Care should be taken to note that the ear does not hear sounds at a fixed number of fixed center frequencies as might be suspected from Figure 4.20. Rather, at any frequency fm considered, the average ear has a given bandwidth.
Example
