The Human Ear
2.4 SUBJ ECTIVE RESPONSE TO SOUND PRESSURE LEVEL
2.4.4 Relative Loudness and the Sone
In the discussion of the previous section, the comparative loudness of either a tone or an octave band of noise, in both cases of variable centre frequency, compared to a reference tone, was considered and a system of equal loudness contours was established. However, the labelling of the equal loudness contours was arbitrarily chosen so that at 1 kHz the loudness in phons was the same as the sound pressure level of the reference tone at 1 kHz. This labelling provides no information about relative loudness; that is, how much louder is one sound than another.
In this section the relative loudness of two sounds, as judged subjectively, will be considered. Reference to Table 2.1 suggests that an increase in sound pressure level of 10 decibels will result in a subjectively judged increase in loudness of a factor of
120
100 100
80 80
60 60
40 40
20 20 MA F
MA F 0
0
20 20
40 40
60 60
80 80
100 100
120 120
31 .5 31 .5
6363
125 125
250 250
500 500
1k 1k
2k 2k
4k4k 8k8k O ctave band centre frequency (Hz)
T onal frequency (Hz)
Sound pressure level (dB re 20Pa)µSound pressure level (dB re 20Pa)µ
14 0
(a)
(b)
Figure 2.9 Equal loudness free-field frontal incidence contours in phons: (a) tonal noise;
(b) octave band noise. MAF is the mean of the minimum audible field.
S ' 2(P&40)/10 (2.32)
Table 2.2 Sound pressure level differences between tones and octave bands of noise of equal loudness based on data in Figure 2.9
Loudness level differences (phons) Band centre
frequency (Hz) 20 40 60 80 100 110
31.5 3 2 0 3 3 6
63 3 1 -2 -1 0 3
125 3 0 -2 0 1 2
250 0 -2 -1 -2 0 1
500 1 1 1 -1 2 3
1000 1 4 4 3 5 6
2000 4 6 4 4 3 3
4000 3 4 3 0 -1 -4
8000 14 16 14 11 7 9
2. To take account of the information in the latter table, yet another unit, called the sone, has been introduced. The 40-phon contour of Figure 2.9 has arbitrarily been labelled 1 sone. Then the 50-phon contour of the figure, which, according to Table 2.1, would be judged twice as loud, has been labelled two sones, etc. The relation between the sone, S, and the phon, P, is summarised as follows:
At levels of 40 phons and above, the preceding equation fairly well approximates subjective judgment of loudness. However, at levels of 100 phons and higher, the physiological mechanism of the ear begins to saturate, and subjective loudness will increase less rapidly than predicted by Equation (2.32). On the other hand, at levels below 40 phons the subjective perception of increasing loudness will increase more rapidly than predicted by the equation. The definition of the sone is thus a compromise that works best in the mid-level range of ordinary experience, between extremely quiet (40 phons) and extremely loud (100 phons).
In Section 2.2.7, the question was raised “What bandwidth is equivalent to a single frequency?” A possible answer was discussed in terms of the known mechanical properties of the ear but no quantitative answer could be given. Fortunately, it is not necessary to bother with the narrow band filter properties of the ear, which are unknown. The practical solution to the question of how one compares tones with narrow bands of noise is to carry out the implied experiment with a large number of healthy young people, and determine the comparisons empirically.
The experiment has been carried out and an appropriate scheme has been devised for estimating loudness of bands of noise, which may be directly related to loudness of tones (see Moore, 1982 for discussion). The method will be illustrated here for octave bands by making reference to Figure 2.10(a). To begin, sound pressure levels in bands are first determined (see Sections 1.10.1 and 3.2). As Figure 2.10(a) shows, nine octave bands may be considered.
L ' Smax % Bj)
i
Si (sones) (2.33)
The band loudness index for each of the octave bands is read for each band from Figure 2.10(a) and recorded. For example, according to the figure, a 250 Hz octave band level of 50 dB has an index S4 of 1.8. The band with the highest index Smax is determined, and the loudness in sones is then calculated by adding to it the weighted sum of the indices of the remaining bands. The following equation is used for the calculation, where the weighting B is equal to 0.3 for octave band and 0.15 for one-third octave band analysis, and the prime on the sum is a reminder that the highest-level band is omitted from the sum (Stevens, 1961):
When the composite loudness level, L (sones), has been determined, it may be converted back to phons and to the equivalent sound pressure level of a 1 kHz tone.
For example, the composite loudness number computed according to Equation (2.33) is used to enter the scale on the left and read across to the scale on the right of Figure 2.10(a). The corresponding sound level in phons is then read from the scale on the right. The latter number, however, is also the sound pressure level for a 1 kHz tone.
Figure 2.10(b) is a more accurate, alternative representation of Figure 2.10(a), which makes it easier to read off the sone value for a given sound pressure level value.
Ex ample 2.1
Given the octave band sound pressure levels shown in the example table in row 1, determine the loudness index for each band, the composite loudness in sones and in phons, and rank order the various bands in order of descending loudness.
Ex ample 2.1 Table
Octave band centre frequencies (Hz)
Row 31.5 63 125 250 500 1000 2000 4000 8000
1. Band level 57 58 60 65 75 80 75 70 65
(dB re 20 µPa)
2. Band loudness 0.8 1.3 2.5 4.6 10 17 14 13 11
index (sones)
3. Ranking 9 8 7 6 5 1 2 3 4
4. Adjustment 0 3 6 9 12 15 18 21 24
5. Ranking level 57 61 66 74 87 95 93 91 89
Solution
1. Enter the band levels in row 1 of the example table, calculated using Figure 2.10(b), read the loudness indices Si and record them in row 2 of the example table.
2. Rank the indices as shown in row 3.
3. Enter the indices of row 2 in Equation (2.33):
L = 17 + 0.3 x 57.2 = 34 sones
4. Enter the computed loudness, 34, in the scale on the left and reading across of Figure 2.10(a), read the corresponding loudness on the right in the figure as 91 phons.
Ex ample 2.2
For the purpose of noise control, a rank ordering of loudness may be sufficient. Given such a rank ordering, the effect of concentrated control on the important bands may be determined. A comparison of the cost of control and the effectiveness of loudness reduction may then be possible. In such a case, a short-cut method of rank ordering band levels, which always gives results similar to the more exact method discussed above is illustrated here. Note that reducing the sound level in dB(A) does not necessarily mean that the perceived loudness will be reduced, especially for sound levels exceeding 70 dB(A). Referring to the table of the previous example and given the data of row 1 of the example table, use a short-cut method to rank order the various bands.
Solution
1. Enter adjustment levels shown in row 4 of the Example 2.1 table.
2. Add the adjustment levels to the band levels of row 1.
3. Enter adjusted levels in row 5.
4. Note that the rank ordering is exactly as shown previously in row 3.