Fig 2.11 The general shape of the average human threshold of hearing and the threshold of pain with approximate conversational speech and musically useful ranges.
The threshold of hearing varies with frequency. The ear is far more sensitive in the middle of its frequency range than at the high and low extremes. The lower curve in is the general shape of the average threshold of hearing curve for sinusoidal stimuli between 20 Hz and 20 kHz. The upper curve in the figure is the general shape of the threshold of pain, which also varies with frequency but not to such a great extent. It can be seen from the figure that the full 130 dB(SPL) range, or 'dynamic range', between the threshold of hearing and the threshold of pain exists at approximately 4 kHz, but that the dynamic range available at lower and higher frequencies is considerably less. For reference, the sound level and frequency range for both average normal conversational speech and music are shown for reference.
The ratio between the loudest and softest is therefore:
Threshold of pain/ Threshold of hearing = 64/10-5 = 6,400,000 = 6.4 X 106
This is a very wide range variation in terms of the numbers involved, and it is not a convenient one to work with. Therefore sound pressure level (SPL), is represented in decibels relative to 20 micro Pa (see Chapter 1), as dB(SPL) as follows:
dB(SPL) = 20 log( Pactual/Preference) where Pactual = measured sound pressure level and Preference = 20 microPa or 2 x 10-5 Pa Example 2.3 Calculate the threshold of hearing and threshold of pain in dB(SPL).
The threshold of hearing at 1 kHz is, in fact, P reference which in dB(SPL) equals: 20 log10 (Preference/Preference) = 20 log10 (2 X 10-5 /2 X 10-5 ) = 20 log10(1) = 20 x 0 = 0 dB(SPL)
and the threshold of pain is 64 Pa which in dB(SPL) equals:
20 log10 (Pactual/Preferencel) = 20 log10 (64/2 X 10-5 ) = 20 log10(6.4 X 106 )= 20 x 6.5 = 130 dB(SPL)
Use of the dB(SPL) scale results in a more convenient range of values (0 to 130) to consider, since values in the range of about 0 to 100 are common in everyday dealings. Also, it is a more appropriate basis for expressing acoustic amplitude values, changes in which are primarily perceived as variations in loudness, since loudness perception turns out to be essentially logarithmic in nature.
Although the perceived loudness of an acoustic sound is related to its amplitude, there is not a simple one-to-one functional relationship. As a psychoacoustic effect it is affected by both the context and nature of the sound. It is also difficult to measure, because it is dependent on the interpretation by listeners of what they hear.
Fig 2.12 Equal loudness countours for the human ear.
The pressure amplitude of a sound wave does not directly relate to its perceived loudness. In fact it is possible for a sound wave with a larger pressure amplitude to sound quieter than a sound wave with a lower pressure amplitude. How can this be so? The answer is that the sounds are at different frequencies and the sensitivity of our hearing varies as the frequency varies. The figure above shows the equal loudness contours for the human ear. These contours, originally measured by Fletcher and Munson (1933) and by others since, represent the relationship between the measured sound pressure level and the perceived loudness of the sound. The curves show how loud a sound must be in terms of the measured sound pressure level to be perceived as being of the same loudness as a 1 kHz tone of a given level.
Frequency response of the ear (Audio Demo)
There are two major features to take note of: The first is that there are some humps and bumps in the contours above 1 kHz. These are due to the resonances of the outer ear. This is a tube about 25 mm long with one open and one closed end. This will have a first resonance at about 3.4 kHz and, due to its non-uniform shape a second resonance at approximately 13 kHz, as shown in the figure. The effect of these resonances is to enhance the sensitivity of the ear around the resonant frequencies. Note that because this enhancement is due to an acoustic effect in the outer ear it is independent of signal level. The second effect is an amplitude dependence of sensitivity which is due to the way the ear transduces and interprets the sound and, as a result, the frequency response is a function of amplitude. This effect is particularly noticeable at low frequencies but there is also an effect at higher frequencies. The net result of these effects is that the sensitivity of the ear is a function of both frequency and amplitude. In other words the frequency response of the ear is not flat and is also dependent on sound level. Therefore two tones of equal sound pressure level will rarely sound equally loud. For example, a sound at a level which is just audible at 20 Hz would sound much louder if it was at 4 kHz. Tones of different frequencies therefore have to be at different sound pressure levels to sound equally loud and their relative loudness will be also a function of their absolute sound pressure levels.
The loudness of sine wave signals, as a function of frequency and sound pressure levels, is given by the 'phon' scale. The phon scale is a subjective scale of loudness based on the judgements of listeners to match the loudness of tones to reference tones at 1 kHz. The curve for N phons intersects 1 kHz at N dB(SPL) by definition, and it can be seen that the relative shape of the phon curves flattens out at higher sound levels, as shown in Figure 2.12. The relative loudness of different frequencies is not preserved, and therefore the perceived frequency balance of sound varies as the listening level is altered. This is an effect that we have all heard when the volume of a recording is turned down and the bass and treble components appear suppressed relative to the midrange frequencies and the sound becomes 'duller' and 'thinner'. Ideally we should listen to reproduced sound at the level at which it was originally recorded. However, in most cases this would be antisocial, especially as much rock material is mixed at levels in excess of 100 dB(SPL)! In the early 1970s hi-fi manufacturers provided a 'loudness' button which put in bass and treble boost in order to flatten the Fletcher-Munson curves, and so provide a simple compensation for the reduction in hearing sensitivity at low levels. The action of this control was wrong in two important respects:
These effects make it difficult to design a meter which will give a reading which truly relates to the perceived loudness of a sound, and an instrument which gives an approximate result is usually used. This is achieved by using the sound pressure level but frequency weighting it to compensate for the variation of sensitivity as a function of frequency of the ear. Clearly the optimum compensation will depend on the absolute value of the sound pressure level being measured and so some form of compromise is necessary. Figure 2.13 shows two frequency weightings which are commonly used to perform this compen sation: termed 'A' and 'C' weightings. The 'A' weighting is most appropriate for low amplitude sounds as it broadly compensates for the low level sensitivity versus frequency curve of the ear. The 'C' weighting on the other hand is more suited to sound at higher absolute sound pressure levels and because of this is more sensitive to low frequency components than the' A' weighting. The sound levels measured using the 'A' weighting are often given the unit dBA and levels using the 'C' weighting dBC. Despite the fact that it is most appropriate for low sound levels, and is a reasonably good approximation there, the 'A' weighting is now recommended for any sound level, in order to provide a measure of consistency between measurements.
The frequency weighting is not the only factor which must be considered when using a sound level meter. In order to obtain an estimate of the sound pressure level it is necessary to average over at least one cycle, and preferably more, of the sound waveform. Thus most sound level meters have slow and fast time response settings. The slow time response gives an estimate of the average sound level whereas the fast response tracks more rapid variations in the sound pressure level.
Sometimes it is important to be able to calculate an estimate of the equivalent sound level experienced over a period of time. This is especially important when measuring people's noise exposure in order to see if they might suffer noise-induced hearing loss. This cannot be done using the standard fast or slow time responses on a sound level meter; instead a special form of measurement known as the Leq measurement is used. This measure integrates the instantaneous squared pressure over some time interval, such as 15 mins or 8 hrs, and then takes the square root of the result. This provides an estimate of the root mean square level of the signal over the time period of the measurement and so gives the equivalent sound level for the time period. That is the output of the Leq measurement is the constant sound pressure level which is equivalent to the varying sound level over the measurement period. The Leq measurement also provides a means of estimating the total energy in the signal by squaring its output. A series of Leq measurements over short : times can also be easily combined to provide a longer time Leq ; measurement by simply squaring the individual results, adding 1 them together, and then taking the square root of the result.
Fig 2.14 The threshold of hearing as a function of frequency
In Figure 2.11 the two limits of loudness are illustrated: the threshold of hearing and the threshold of pain. As we have already seen, the sensitivity of the ear varies with frequency and therefore so does the threshold of hearing, as shown in Figure 2.14. The peak sensitivities shown in this figure are equivalent to a sound pressure amplitude in the sound wave of 10 μPa or: about -6 dB(SPL). Note that this is for monaural listening to a sound presented at the front of the listener. For sounds presented on the listening side of the head there is a rise in peak sensitivity of about 6 dB due to the increase in pressure caused by reflection from the head. There is also some evidence that the effect of hearing with two ears is to increase the sensitivity by between 3 dB and 6 dB. At 4 kHz, which is about the frequency of the sensitivity peak, the pressure amplitude variations caused by the Brownian motion of air molecules, at room temperature and over a critical bandwidth, corresponds to a sound pressure level of about -23 dB. Thus the human hearing system is close to the theoretical physical limits of sensitivity. In other words there would be little point in being much more sensitive to sound, as all we would hear would be a 'hiss' due to the thermal agitation of the air! Many studio and concert hall designers now try to design the building such that environmental noises are less than the threshold of hearing, and so are inaudible.
The second limit is the just noticeable change in amplitude. This is strongly dependent on the nature of the signal, its frequency, and its amplitude. For broad-band noise the just noticeable difference in amplitude is 0.5 to 1 dB when the sound level lies between 20 dB(SPL) and 100 dB(SPL) relative to a threshold of 0 dB(SPL). Below 20 dB(SPL) the ear is less sensitive to changes in sound level. For pure sine waves however, the sensitivity to change is markedly different and is a strong function of both amplitude and frequency. For example at 1 kHz the just noticeable amplitude change varies from 3 dB at 10 dB to 0.3 dB at 80 dB(SPL). This variation occurs at other frequencies as well but in general the just noticeable difference at other frequencies is greater than the values for 1-4 kHz. These different effects make difficult to judge exactly what difference in amplitude would be noticeable as it is clearly dependent on the precise nature of the sound being listened to. There is some evidence that once more than a few harmonics are present that the just noticeable difference is closer to the broad-band case, of 0.5-1 dB, rather than the pure tone case. As a general rule of thumb the just noticeable difference in sound level is about 1 dB.
The mapping of sound pressure change to loudness variation for larger changes is also dependent on the nature of the sound signal. However, for broad-band noise, or sounds with several harmonics, it generally accepted that a change of about 10 dB in SPL corresponds to a doubling or halving of perceived loudness. However, this scaling factor is dependent on the nature of the sound, and there is some dispute over both its value and its validity.
The total level from combining several uncorrelated sources is given by:
PN uncorrelated = P sqrtN
This can be expressed in terms of the SPL as:
SPLN uncorrelaled = SPLsingle source + 10 log10(N)
In order to double the loudness we need an increase in SPL of 10 dB. Since 10 log10(10) = 10, ten times the number of sources will raise the SPL by 10 dB. Therefore we must increase the number of violinists in the string section by a factor of ten in order to double their volume.
Fig 2.15 The effect of tone duration on loudness
Pitch Salience Tone Duration(Audio Demo)
As well as frequency and amplitude, duration also has an effect In the perception of loudness, as shown in Figure 2.15 for a pure one. Here we can see that once the sound lasts more than about 200 milliseconds then its perceived level does not change. However, when the tone is shorter than this the perceived amplitude reduces. The perceived amplitude is inversely proportional the length of the tone burst. This means that when we listen to sounds which vary in amplitude the loudness level is not perceived significantly by short amplitude peaks, but more by the sound level averaged over 200 milliseconds.
Unlike tones, real sounds occupy more than one frequency. We have already seen that the ear separates sound into frequency bands based on critical bands. The brain seems to treat sounds within a critical band differently to those outside its frequency range and there are consequential effects on the perception of loudness.
The first effect is that the ear seems to lump all the energy within a critical band together and treat it as one item of sound. So when all the sound energy is concentrated within a critical band the loudness is proportional to the total intensity of the sound within the critical band. That is:
Loudness is proportional to P12 + P22 + ... + Pn2
where P1-n = the pressures of the n individual frequency components
As the ear is sensitive to sound pressures, the sound intensity is proportional to the square of the sound pressures, as discussed in Chapter 1. Because the acoustic intensity of the sound is also proportional to the sum of the squared pressure the loudness of a sound within a critical band is independent of the number of frequency components so long as their total acoustic intensity is constant. When the frequency components of the sound extend beyond a critical band an additional effect occurs due to the presence of components in other critical bands. In this case more than one critical band is contributing to the perception of loudness and the brain appears to add the individual critical band responses together. The effect is to increase the perceived loudness of the sound even though the total acoustic intensity is unchanged.
Fig 2.16 The effect of tone bandwidth on loudness
Figure 2.16 plots the subjective loudness perception of a sound at a constant intensity level as a function of the sound's bandwidth which shows this effect. In the cochlea the critical bands are determined by the place at which the peak of the standing wave occurs, therefore all energy within a critical band will be integrated as one overall effect at that point on the basilar membrane and transduced into nerve impulses as a unit. On the other hand energy which extends beyond a critical band will cause other nerves to fire and it is these extra nerve firings which give rise to an increase in loudness.
The interpretation of complex sounds which cover the whole frequency range is further complicated by psychological effects, in that a listener will attend to particular parts of the sound, such as the soloist, or conversation, and ignore or be less aware of other sounds and will tend to base their perception of loudness on what they have attended to.
Duration also has an effect on the perception of the loudness of complex tones in a similar fashion to that of pure tones. As is the case for pure tones, complex tones have an amplitude which is independent of duration once the sound is longer than about 200 milliseconds and is inversely proportional to duration when duration is less than this.
Although we now have found clear connections between acoustical intensity and subjective perception of loudness, we are not able to determine the subjective loudness for every instant of a piece of music. If we permit a tone of longer duration to be impressed upon the hearing system, the subjective loudness perception for this tone will at first increase, until after about 0.2 sec a gradual decrease in the subjective perception occurs, which results from a gradual fatigue through the constant burdening of the sensory cells and the nerve fibres. The changes in sensitivity in response to the effect of a constant burden of sound affects the following tones in the influence area of the constant tone for several seconds. One then speaks of" fatigue." Bekesy (A) determined the adaptation by supplying to one ear a constant tone and to the other ear a comparison tone, interrupted periodically, by which he measured the decrease in subjective loudness (cf. Fig. 65, top). From this we learn, for example, that one tone following after a loud organ point of 5 sec duration at the same pitch will be heard after 1 ½ sec with a loudness of 5 db less than the original tone, therefore softer than intended by the composer. Important here is the initial loudness of a constant tone of longer duration. In Fig. 66 one sees that the decline in the subjective loudness is greater for the loudest tone, with the result that the end loudness after 2 minutes is the least for the loudest tone. A very loudly played organ point of constant sound pressure would therefore bring about great revaluation of the loudness of
FIG. 65. Decrease of loudness sensation through fatigue when the ear is taxed with a permanent sine wave, measured over 3 min. Hatched: time course of the test signal. (a) Constant sound pressure; (b) doubling of the pressure alter 2 min (c) halving of the pressure after 2 min.
following tones of the same pitch, as well as neighboring tones in the same area of influence. This decline does not occur in beat tones because the ear recovers after each decay, as Helmholtz observed. In musical reproduction we must always keep this in mind.
FIG. 66. Fatigue as a function of the intensity of the fatiguing tone (800 cps) for the three loudness levels 80, 95 and 110 phons (curves 2, 3 and 4, respectively).
Curve I shows the simultancous decrease of the differential thresholds of pitch and loudness level or increase in sensitivity to pitch and loudness level differentiation (according to Békésy).
Measurements by Bekesy (A) show how with increasing beating rate the fatigue effect declines, and completely disappears for a beat of 15/sec (Fig. 67). Still, this behavior varies with individuals. It is further interesting to observe that a constant tone, which, for example, swells in 2 minutes to double the sound pressure, finally attains scarcely half of its initial subjective loudness (Fig. 65, center). On the other hand, the reduction to half of the sound pressure results in the same subjective loudness as in the previous experiment (Fig. 65, bottom). One can further assume that an acoustically fatigued ear becomes more sensitive, and might overvalue a sudden change in loudness, as Bekesy mentioned in 1929 (A) and Lerche (51) established more recently. These experiments
FIG. 67. Effect of beating rate on fatigue. Duration of the fatigue 2 min.
demonstrate that a musician has no absolute measure of objective loudness, just as he has no absolute measure of time. According to the musical context, a piano can unconsciously be played forte, since the reference level is relative. After a pp a mf seems like ff, and vice versa. The influence area of the adaptation frequency extends also to the neighboring frequencies in an area of an octave above and below. In the case of a biasing stimulation of the ear by means of an organ point it will happen that a tone a certain interval away will sound out of tune if it lies partly in the influence area of the constant tone, as a result of the localized biasing stimulation of the basilar membrane in conjunction with the groups of sensory cells connected to it. According to measurements (Fig. 68), for a constant tone at 800 cps (approximately g2), a tone a fifth higher sounds about 7 per cent out of tune, that is a little bit more than a half tone, and a lower tone at 500 cps sounds about 6 per cent out of tune, which is exactly a half tone (A). This can have the effect for an organ point at g2 of the following octaves c2 and C3 sounding a whole tone out of tune. Therefore, in the case of pieces with octaves following organ points, there is not only change in loudness, but also considerable distuning. It can occur pathologically that one ear loses its sensitivity in limited frequency areas, so that this ear perceives pitches differently
FIG. 68. Effects of fatigue on pitch by distuning in the area of an octave above and below. Constant tone of 800 cps (according to Bekesy).
from the other ear; thus in binaural hearing tones will be heard doubly (diplacusis). C. Stumpf observed as early as 1883 that a minor diplacusis can occur in normal hearing as a result of unequal response in the two ears (99). The above mentioned fatigue phenomena resulting from the acoustical overstimulation of the hearing system permit one to recognize the significance of a general rest, especially after fortissimo passages, as a time span necessary for the restoration of the normal hearing response. One must remember that during a part or the whole of the rest the sound energy decays according to the time constant of the reverberation, and the musical effect of the general rest can lie in this decay.
Figure 66 demonstrates further that as a result of adaptation the relative difference thresholds for intensity and frequency decay in the same way, that is, with increasing adaptation the ear becomes more sensitive to intensity and pitch differentiation (A). This means that in connection with our organ point example slight modulatory changes in the sense of an amplitude and frequency modulation will be more clearly heard for the following tones in a certain influence area on both sides of the adapted tone than without the previous biasing stimulation. One can see from Figure 66 that the highest sensitivity for frequency differentiation is reached only after one minute, while the loudness decline is reached immediately. Because of this sharpened sensitivity, sudden changes in loudness are quantitatively overvalued.
A further effect of an organ point consists in the following: When there is sufficient loudness, higher tones even above an octave will be masked, but not lower tones. A low-lying organ point used as a bass foundation is therefore more significant with respect to masking than is a higher one.
In summarizing, we can say that an organ point which lasts long enough and has sufficient loudness exerts a considerable influence on the tones which follow, with respect to the entire sound structure, because of the combined effect of the reduction in loudness, because of the distuning of neighboring tones, the masking of higher tones on the one hand and the increase in response for the perception of the smallest pitch and loudness fluctuations (inner modulation of sounds) on the other.
Before next class please read Sections
pages 79 to 108 of Acoustics and Psychoacoustics. We will have a brief quiz on these sections at the beginning of the next class.