Loudness and intensity coding
Abstract and Keywords
This article is concerned with the auditory representation of sound intensity, and its perceptual correlate of loudness. Loudness is the primary perceptual correlate of physical sound intensity or strength, often described as the characteristic of sound ranging from very soft to very loud. It discusses intensity discrimination and the dynamic range of hearing, techniques for measuring loudness, and the complex relation of loudness to the physical characteristics of sounds. Following this, the article explains how loudness is affected by parameters, and how intensity is coded in the auditory system. Furthermore, it provides a historical overview of loudness modelling. It also gives a brief overview of the perception of loudness for hearing-impaired listeners. Finally, it describes important new developments in loudness modelling.
3.1.1 Definition of loudness
Loudness is the primary perceptual correlate of physical sound intensity or strength, often described as the characteristic of sound ranging from very soft to very loud. Loudness is often mistakenly used in non-scientific discussions to refer to the physical volume or the physical intensity of a sound, but proper usage refers only to perception. A number of other physical sound parameters also affect loudness, including frequency, duration, spectrum, bandwidth, and context. The perception of loudness also depends on the individual characteristics of the listener, particularly the integrity of the auditory system. Even within a normal population, loudness can vary from person to person.
Despite the fact that loudness has been studied formally for close to 150 years, there is still substantial scientific discourse and debate regarding not only the complexities of the neural coding of intensity and loudness, but also even the most basic measurements of the perceptions of a simple sound, such as a pure tone. Much like Newtonian laws remained a standard in physics until they were replaced by Einstein’s relativity, the forefathers of loudness put forth simple and elegant models that gave us great insight into perception, but could not explain all the phenomena later discovered. Currently, more and more complex models are being developed to include the loudnesses of complex or mixed sounds, the summation of sounds across the two ears, context effects, and the effects of the many aetiologic causes of hearing impairment.
The purpose of this chapter is to provide a brief, but thorough overview of how loudness is measured, how it is affected by physical parameters, and how intensity is coded in the auditory system. It will also provide a historical overview of loudness modelling and examine how context and impairment can affect our perception of loudness.
3.1.2 Intensity, level, and the dynamic range of hearing
Perhaps the most remarkable characteristic of the auditory system is the dynamic range of hearing. Average, normal-hearing humans are capable of detecting sounds that are presented at 0 dB SPL and are generally able to tolerate sounds, at least brief ones, as high as about 120 dB SPL. That means that the auditory system is capable of handling sound intensities that are 1012 times as large as the minimum detectable intensity. It is an extraordinarily impressive system that can detect the sounds of a mosquito beating its wings 1 m away and still withstand the blast of a nearby thunderclap.
Dynamic ranges are significantly smaller for most common audio systems. Audiotape is capable of approximately a 65-dB dynamic range without significant distortion. This is equivalent to a maximum intensity that is on the order of 106 times the minimum intensity. Compact discs are capable of producing a 96-dB dynamic range or a maximum intensity that is on the order of 109 times the minimum intensity.
(p. 46) 3.1.3 Loudness units
Because loudness is primarily correlated with sound intensity, loudness is most often displayed or described as a function of physical sound intensity or pressure. This typically results in plots of loudness on a logarithmic scale as a function of sound level in dB SPL. Loudness is also sometimes expressed as a function of dB HL (hearing level) or dB SL (sensation level), variations of the dB SPL scale that are shifted by average (dB HL) or individual (dB SL) audibility threshold data as a function of the frequency of the stimuli.
Loudness itself is sometimes mathematically expressed in one of two distinct units, both of which are based on 1-kHz reference tones. The loudness level in phons refers to the level of an equally loud 1-kHz tone in dB SPL presented to both ears of a listener in a free field with frontal incidence (Fletcher and Munson, 1933). For example, if the loudness of a sound were equal to the loudness of a 1-kHz tone at 60 dB SPL, that sound would have a loudness level of 60 phons. By definition, the loudness level of any 1-kHz tone in phons is simply equal to its level in dB SPL.
The sone is a relative loudness unit referenced to a 1-kHz tone at 40 dB SPL, presented to a listener in a free field, by assigning a loudness of 1 sone for that stimulus (Stevens, 1936). If a sound is twice as loud as a 1-kHz tone at 40 dB SPL, then its loudness is 2 sones. If a sound is half as loud, its loudness is 0.5 sones. Therefore, any audible sound must have a loudness in sones that is non-zero and positive.
3.2 Representing and measuring loudness
By definition, there is no objective way to directly measure any psychological phenomenon. Because loudness is a purely perceptual characteristic, its measurement requires subjective reporting of the sensation by a listener. A good loudness measurement technique should provide a repeatable, reliable, and precise description of the perception of loudness while removing possible experimental biases that could affect the outcome. A number of different methodologies have been used, each with its own advantages and disadvantages.
3.2.1 Category scaling
The categorical scaling of loudness is perhaps the most intuitive approach to making subjective judgments of sound. In a category-scaling procedure, listeners are asked to select a category that best describes the loudness of a sound. Category labels may vary somewhat from study-to-study, but typically are similar to: not heard, very soft, soft, comfortable, loud, very loud, and too loud.
Recent advancements in categorical scaling include the use of a greater number of categories, adaptive procedures, and mathematical modelling to generate loudness functions (Keidser et al., 1999; Brand and Hohmann, 2002). Because of the general ease of use of category scaling, it has been used in the clinical assessment of loudness for rapid hearing-aid fitting (Kiessling et al., 1996). However, some studies have indicated that category scaling generally cannot provide valuable information for hearing-aid fitting (Elberling, 1999).
3.2.2 Magnitude estimation and production
In magnitude estimation, proposed for use with auditory stimuli by Stevens (1956), listeners are asked to assign any positive number that represents the loudness of a presented sound. These assignments are allowed to include decimal and fractional numbers. Listeners are typically instructed or assumed to assign numbers within a ratio scale. That is, if a sound were twice as loud as another sound, that sound would be assigned a number with twice the value of the other sound. These estimates expose the relationships among all presented sounds within an experiment (Stevens, 1975).
Magnitude estimation may also be performed with a standard. In such a paradigm, the listener is presented with a reference sound that is assigned to a specific reference number. The listener is then asked to assign numbers to subsequent sounds such that ratios are maintained. If a sound were half as loud as the reference sound, it would be assigned a number that is half of the reference number. If a sound were twice as loud as the reference sound, it would be assigned a number that is twice the reference number.
Some researchers have suggested that there are a number of biases present in magnitude estimation (see Poulton, 1989 for review). To a degree, these biases may be resolved by counterbalancing with the opposing task, i.e. magnitude production. In this task, a listener is given a number and asked to adjust the level of a sound such that the loudness matches that number (Hellman, 1981). Typically, results from magnitude estimation and magnitude production from the same listener might be averaged together to determine a final estimate of loudness.
A number of researchers have also shown that magnitude estimation is a useful tool for assessing group mean loudness, but is less suitable for individual assessments due to variability (Green and Luce, 1974; Epstein and Florentine, 2006). This limitation makes it an uncommon tool for loudness assessment in clinical hearing-aid fitting scenarios.
3.2.3 Cross-modality matching
Cross-modality matching is a general description of a procedure in which two different modalities of perception are matched together. In fact, magnitude estimation is technically a form of (p. 48) cross-modality matching in which the perception of loudness is matched with the perception of the magnitude of a number. Cross-modality matching procedures used for loudness most often use line length (Teghtsoonian and Teghtsoonian, 1983), but other modalities have been used to match features of sounds, including the brightness of a light and tactile vibration. In theory, there is no particular limitation to the selection of a modality for comparison. However, when multiple modalities are compared, a mathematical transformation must be determined in order to equate the measurements with a more direct technique. Cross-modal matches between line length and loudness are most common because they have been found to be consistent with magnitude estimation results in both normal-hearing and hearing-impaired listeners (Hellman and Meiselman, 1988; Hellman, 1999).
3.2.4 Loudness matching
Another approach for determining information about the loudness of sound is to equate the loudness of one sound with the loudness of another. This is particularly useful for comparing the loudnesses of sounds with different physical parameters such as frequency (Fletcher and Steinberg, 1924), duration (Florentine et al., 1996), spectrum (Leibold et al., 2007), or bandwidth (Zwicker et al., 1957), as well as combinations of those parameters (Anweiler and Verhey, 2006). Loudness matching is also the fundamental basis of the phon scale of loudness. Listeners are typically presented with two sounds and are asked to adjust the level of one of the sounds to find the point of subjective equality in which the loudnesses of the two sounds are equal. Most experimenters counterbalance loudness matches by then reversing the roles of the fixed and variable sounds and averaging between these two conditions.
3.2.5 Other correlates of loudness
In addition to measuring loudness subjectively, a number of objective experimental techniques have been examined as possible correlates of loudness. Correlates of loudness that require less active participation or cognitive activity from the listener may be more suitable for rapid, objective clinical assessments of loudness, particularly in young children or cognitively impaired adults.
Measurements of simple reaction time, the speed at which a listener provides a response to a sound, correlate with loudness, with louder sounds yielding faster reaction times. In other words, reaction time is inversely correlated with loudness. In a typical reaction-time task, listeners are presented with sounds and asked to press a trigger immediately upon hearing the sound. In particular, this technique has been used to examine loudness near threshold, which is typically difficult with magnitude estimation (see Wagner et al., 2004 for a review). There may be some limitations to the use of reaction time to derive loudness information as some listeners have shown frequency dependence.
The basilar membrane is a membrane running through the centre of the cochlea and the location on which sound is transduced from physical vibration to neural activity. The relationship between sound intensity and basilar-membrane motion has been measured in animals using laser Doppler velocimetry (Nuttall et al., 1991). Buus and Florentine (2001) plotted human loudness functions modeled from temporal integration data and spectral summation data with basilar-membrane velocity measures made in a chinchilla (Ruggero et al., 1997). They demonstrated that loudness was approximately a linear function of basilar-membrane velocity squared. Other psychoacoustical measurements associated with basilar-membrane compression also correspond well with loudness (p. 49) growth functions (Oxenham and Plack, 1997). This indicates that, at least for pure tones, loudness is closely associated with the intensity of physical vibrations on the basilar membrane (Fig. 3.2). This, of course, is not of direct practical clinical use, but does provide insight into the process of sound transduction and processing within the auditory system.
Auditory brainstem response
The auditory brainstem response (ABR) is an objective neurological measure of the evoked potential that results from auditory stimulation. A number of researchers have examined the relationship between the ABR and loudness (Howe and Decker, 1984; Serpanos et al., 1997). These studies have primarily monitored the most common ABR landmark used for the clinical testing of auditory-system integrity, Wave V latency. Wave V latency decreases with increasing intensity and, much like reaction time, has a strong inverse correlation with loudness. However, Wave V latency asymptotes at a relatively low intensity, while loudness continues to grow with increasing intensity. It is possible to estimate the basilar-membrane response function by measuring the effects of an on- and off-frequency forward masker on Wave V latency (Krishnan and Plack, 2009), and this may provide a better estimate of loudness.
Pratt and Sohmer (1977) hypothesized that, rather than examining Wave V latency, the first few components, approximately waves I+II of the ABR, are better suited for finding a correlate of loudness. Physiologically, it is likely that the early components of the ABR result from electrical activity within the cochlea itself as well as the first and second order neurons in the auditory nerve.
Otoacoustic emissions, low-level sound by-products of the active mechanism of the auditory system, have been recently examined as a possible correlate of loudness after some suggestion that (p. 50) they might reflect basilar-membrane non-linearity. Acoustic recordings of otoacoustic emissions are typically made by inserting a microphone into the ear canal.
A number of researchers have measured distortion-product otoacoustic emissions, which result from the interaction of two stimuli presented simultaneously, in both normal-hearing and hearing-impaired listeners (Neely et al., 2003; Muller and Janssen, 2004). It has been generally found that distortion-product otoacoustic emissions exhibit many of the characteristics of loudness and basilar-membrane compression. Transient-evoked otoacoustic emissions, acoustic responses of the ear made after short stimuli are presented, have also been measured as a function of level and shown to correlate well with loudness functions estimated using a number of psychoacoustical tasks for normal-hearing listeners (Epstein and Florentine, 2005a).
3.3 Parametric effects
Loudness depends primarily on sound intensity, however two sounds with the same intensity can be perceived with different loudnesses. This results from the effects of other parameters that can influence the loudness of a sound. This section will review the relationships between loudness and several physical parameters of sound.
Intensity is the primary physical parameter that influences loudness. This effect is the most intuitive parametric adjustment of loudness and can easily be experienced by turning the volume knob on a radio up or down. The study of loudness began with an examination of the relationship between intensity and loudness, typically referred to as a ‘loudness function’.
Figure 3.3 reviews a number of experiments measuring the relationship between intensity and loudness using both magnitude estimation and doubling or halving of loudness (Hellman and Zwislocki, 1963). This figure shows the average estimate of loudness on a logarithmic scale as a function of sound level in dB SL. Note that loudness and level have a monotonic relationship (i.e. an increase in level will always be perceived as an increase in loudness). However, this relationship is not linear. If the intensity of a sound is doubled (3-dB increase), its loudness will not be perceived as doubled. The slope of the loudness curve is steepest at low levels and decreases at moderate levels. This implies that the same change in level will be perceived as a greater change in loudness at low levels than at moderate levels.
At moderate levels, the logarithm of loudness is classically modeled as proportional to the level. The slope of this line indicates that it is necessary to raise the level approximately 10 dB in order to double loudness. The shape of the loudness function is approximately independent of the duration (Epstein and Florentine, 2005b) and the presentation mode, binaural vs. monaural (Marozeau et al., 2006), but it does depend somewhat on spectral content (Scharf, 1959) and frequency (Hellman et al., 2001).
3.3.2 Frequency/equal-loudness contours
In addition to being dependent on intensity, loudness is dependent on frequency. In typical experiments to examine this effect, listeners are asked to adjust the level of a tone to match the loudness of a 1-kHz tone. The experiment is repeated many times with the frequency of the adjusted tone varied and the 1-kHz reference tone kept fixed in frequency. Additionally, the level of the 1-kHz tone is changed from condition to condition to obtain a series of functions showing equally loud tones of different frequencies using a particular level 1-kHz tone as reference. Each of these functions is called an equal-loudness contour. Figure 3.4 shows a series of equal-loudness contours for different levels of the 1-kHz reference tone.
Along each contour line, the tones are perceived as equally loud. For example, a 125-Hz tone at 44 dB SPL has the same loudness as a 500-Hz tone at 23 dB SPL. These contours also illustrate the loudness level of a variety of sounds in phons, as the loudness level is equal to the level of the equally loud 1-kHz tone.
Hearing threshold is represented as a dashed line in the equal-loudness contour because some studies have showed that the loudness at threshold varies and that equal detectability does not imply equal loudness level (Buus and Florentine, 2002). However, it seems that there is some relationship between threshold microvariations, relatively large threshold differences for relatively small differences in frequency, and loudness at higher levels (Mauermann et al., 2004).
For low-frequency sounds, the contour lines are closer together as level increases than they are for moderate frequencies. This implies that loudness grows faster with changes in level for low-frequency sounds. For example, the loudness of a 31.5-Hz tone increases from 20 phons to 90 phons with less than a 40-dB increase in level (from 76 to 115 dB SPL). Therefore, for a 31.5-Hz tone, a 40-dB dynamic range will span the same range of loudnesses as a 70-dB dynamic range for a 1-kHz tone. (p. 52)
The duration of a signal also has an effect on its loudness. For sounds shorter than about 500 ms to 1 s, the loudness of the sound will increase with increasing duration. This phenomenon, the summing of sound energy over time, is known as ‘temporal integration’ (see Chapter 5). The effects of duration are typically measured using a loudness-matching procedure similar to that used to measure equal-loudness contours. Listeners are asked to match the loudness of two sounds with different durations by adjusting the level of one until the two sounds are equally loud. Instead of varying frequency, as in the equal-loudness contour measurements, the duration of the sound is varied. Figure 3.5 shows the levels of equally loud 1-kHz tones with different durations for a range of levels from threshold to 95 dB SPL. The figure shows the level of the tone such that it is equally loud to a 640-ms tone at the indicated level as a function of the duration of the tone.
Each of these lines then represents an equal-loudness contour with respect to duration. For all levels, shorter sounds need to be more intense to be as loud as longer sounds.
It is notable that the amount of temporal integration, that is the difference in level between equally loud short and long sounds, is dependent on level (Florentine et al., 1996). While for the 35-dB 640-ms tone, there is approximately a 22 dB difference between the long tone and an equally loud 5-ms tone; for the 95-dB 640-ms tone, there is approximately a 12 dB difference between the long tone and an equally loud 5-ms tone. Because the loudness function is approximately the same shape for sounds of all durations (Epstein and Florentine, 2005b), this variation in the amount of temporal integration as a function of level also indicates that the slope of the loudness function must be shallower at moderate levels than at low and high levels (Buus et al., 1997). (p. 53)
Loudness also increases with increasing bandwidth. This phenomenon is known as ‘spectral summation’. As with previously discussed experiments to examine parametric effects, spectral summation of loudness is typically measured using loudness matching. Typically, listeners are asked to adjust the level of a 1-kHz tone to match the loudness of a noise geometrically centred at 1 kHz with a constant overall level as a function of the bandwidth of the noise.
Figure 3.6 shows the results of this experiment. The figure shows the level to which the reference tone is set at the point of subjective loudness equality as a function of the bandwidth of the signal. Each curve shows the influence of bandwidth for a specific level of noise. As the bandwidth widens beyond shorter bandwidths, loudness increases, except at 20 phons. The bandwidth at which summation begins is known as the ‘critical bandwidth’. The maximum amount of spectral summation depends on level (Scharf, 1997) and duration (Verhey and Kollmeier, 2002), and the effect can be as large as 18 dB in very specific scenarios (Zwicker and Fastl, 1990).
3.3.5 Binaural summation
A sound presented monaurally, through one headphone, will sound softer than the same sound presented binaurally, through a pair of headphones. Generally, it has been assumed that the loudness ratio between a binaural sound and a monaural sound is equal to two (see Hellman, 1991 for a review). That means that sounds presented to both ears would be twice as loud as the same (p. 54) sounds presented to one ear. However, some studies have suggested a lower binaural-to-monaural loudness ratio, between 1.5 and 1.7, and that the ratio is constant and independent of level (see Marozeau et al., 2006 for a review).
3.4 Intensity coding
The auditory system is capable of coding a tremendous range of physical intensities and identifying relatively small differential intensities. This section will outline the perceptual and physiological effects of intensity variations.
3.4.1 Intensity discrimination and Weber’s law
The smallest intensity change that is detectable is often called the ‘just noticeable difference’, JND, or ‘difference limen’. This can be measured using a number of different experimental paradigms, most of which involve a two (or more) -interval task in which the listener chooses the sound with a greater intensity. These paradigms include direct comparison of two sound bursts to determine which one has a higher intensity, identification of an amplitude modulated sound, and detection of a brief increment in a longer, continuous background sound. Each of these test paradigms results in somewhat different values for the JND, but the trends of the results are the same.
Weber’s law states that the JND is a constant proportion of the intensity of the sound. In other words, some constant K exists such that K is equal to the detectable change in intensity, ΔI, divided by the pedestal intensity of the sound, I. This is known as the Weber fraction:(3.1)
The Weber fraction indicates the proportion of change that is required for detection. The constant proportion for detectability can be experienced in everyday situations. If a jar contains 100 marbles, it would be difficult to distinguish that jar from an identical jar containing one fewer marble. However, if the jar contains five marbles, it would be quite easy to distinguish it from a similar jar containing four marbles. In each case, the same number of marbles has been removed (p. 55) (i.e. the change in intensity, ΔI, is the same, but the ratio of change to pedestal intensity is not). Weber’s law indicates that 20 marbles would need to be removed from the jar with 100 marbles to create the same effect:(3.2)
To determine the level difference in decibels between two just discriminable sounds, ΔL, for a particular pedestal intensity, I:(3.3)Where ΔI = IK
The value of K at higher levels is approximately 0.1 or –10 dB, which corresponds to a ΔL, of about 0.4 dB. That is, listeners can detect changes in level of about 0.4 dB. Figure 3.7 shows the Weber fraction as a function of the level of the pedestal. For wideband sounds (such as a noise with a flat spectrum) Weber’s law holds well from about 30 to 110 dB SPL, but the Weber fraction is greater at low levels (Miller, 1947). For pure tones and other narrowband sounds, however, the Weber fraction decreases with level. This phenomenon is known as the near miss to Weber’s law (McGill and Goldberg, 1968).
The basilar membrane, the vibration of which activates the neural portion of the auditory system, is tonotopically organized. That is, it vibrates in a particular location for a stimulus of a particular frequency. Responses to high-frequency sounds occur near the base of the cochlea and responses to low-frequency sounds occur at the apex of the cochlea. When a tone is presented, vibration peaks at the location specifically tuned to that frequency. Additionally, nearby regions also vibrate. At low levels, the vibration spreads approximately equally toward the apex and the base. At higher levels, the vibrations spread more toward the base than the apex. This is known as the upward spread of excitation. In fact, the upward skirt of the excitation grows linearly with level, while the peak of excitation grows compressively (see Chapter 2).
Figure 3.8 shows the shapes of the excitation of the basilar membrane for 1-kHz pure tones at several levels as a function of frequency. As level increases, there is little change in the amount of excitation that spreads toward low frequencies. In contrast, as level increases, the spread toward higher frequencies increases rapidly.
The near miss to Weber’s law can be explained by the additional information provided by the skirt excitation at high levels. In an experiment performed by Moore and Raab (1974), a masking noise with a band notch, designed to only mask skirt excitation, presented during a JND experiment resulted in only a small decrease in performance that only occurred at high levels. This change in performance eliminated the near miss to Weber’s law. This is also consistent with the findings of Florentine et al. (1987). They found that high-frequency tones do not exhibit the near miss, probably because there is no room in the cochlea for excitation to spread toward higher frequencies.
3.4.3 Neural intensity coding (dynamic range problem)
Auditory nerve fibres increase their firing rate as the intensity of a sound increases. Rate-level functions show the number of times a nerve fibre activates, on average, per second for a given stimulus. These functions show that the vast majority of auditory nerve fibres tend to saturate (i.e. reach maximum firing rate) within only a 30–60 dB dynamic range. Because the auditory system as a whole is capable of processing a dynamic range of 120 dB, some mechanism other than general firing rate of the neurons must account for this ability. It is not yet well understood how this dynamic-range problem is overcome by the system.
One mechanism that may contribute to the wide dynamic range of the system is the spread of excitation, which increases the number of neurons involved in coding the sound and includes neurons from the skirts of the excitation pattern that receive less excitation and are therefore not saturated. Another is phase locking (see Volume 1, Chapter 9), for which the synchrony of firing to the waveform of the stimulus becomes greater as level increases. Despite these contributions, (p. 57) when both of these cues are eliminated by masking and using high frequencies to eliminate phase-locking cues (Carlyon and Moore, 1984), the auditory system is still capable of quite good intensity discrimination, so these cues cannot explain psychophysical performance on their own. It is also possible that the small proportion of fibres with wide dynamic ranges are sufficient to account for the wide dynamic range and acute intensity discrimination seen in the auditory system (for reviews, see Viemeister, 1988; Plack, 2005).
In general, the relationship between neural coding and intensity discrimination is not well understood. One example is the expectation that rapidity of onset of an increment in a continuous sound would improve the detection of that increment. This would be expected because rapid onsets result in strong transient responses in the auditory nerve and elsewhere in the auditory system (see Volume 2, Chapter 6). However, Plack et al. (2006) found that increment detection was not altered by changing the slope of the onset ramp on the increment. All these findings indicate that there may be variations in the function of specific fibres, and that the information in these fibres is integrated in a form that is far more complex than simply energy summation.
3.4.4 Profile analysis
Profile analysis experiments demonstrate further that the complexities of neural coding go far beyond the use of firing rate or firing population. Profile analysis occurs in the auditory system when comparisons are made between two sounds with different spectral envelopes. Typically, experimental comparisons are made between multitone complexes in two presentation intervals. In one interval, all the tones in the multitone complex are presented at the same level (flat spectrum). In the other interval, one of the tones (target tone) is presented at a higher level than all the other tones (bumped spectrum). Listeners are asked to identify the interval containing the bumped-spectrum sound. In order to prevent listeners from simply comparing the loudnesses of the target tone in the two intervals, the level of the entire complex is varied randomly in each interval. Therefore, the bumped-spectrum sound is higher in level than the flat-spectrum sound only 50% of the time. In the experiment, the level of the target tone is varied in the bumped spectrum to find the smallest detectable bump. Even when the overall levels of the sounds are varied within a 40-dB range, listeners are able to identify bumps of only a few dB. It has been hypothesized that this ability is closely related to the spectral analysis used in speech perception, and demonstrates that auditory intensity analysis does not occur solely on a frequency-by-frequency basis (for a review see Green, 1988).
3.5 Models of loudness
Loudness is assessed by asking listeners to make judgments of their perception of sound using psychoacoustic methods. Although individual variations exist, there has been a continuing search to develop a mathematical model of loudness such that, given the physical parameters of a particular sound, one could predict the loudness of that sound for a particular listener and listening condition. Originally, loudness models predicted loudness from only intensity. Newer models grow more and more complex and attempt to incorporate many physical parameters as well as the interactions between those parameters.
3.5.1 Pure-tone loudness model
The simplest loudness model predicts the loudness of a pure tone from the intensity of that tone. However, even these simple models, which originated almost 150 years ago, are still being revised to improve accuracy today.
(p. 58) Logarithmic function
Fechner (1860) was the first to propose a loudness model based on a logarithmic relationship between intensity and loudness. This relationship is a natural evolution of Weber’s law, which, as described in Section 3.4.1, states that the minimum amount of intensity increase ΔI needed to induce a perceptible loudness difference is a constant proportion of intensity:(3.5)
According to Fechner, every ΔI can be associated with an increase on the perceptual (or sensation) scale, ΔS, corresponding to the smallest perceptible sensation increase, and that each sensation increase would be identical in magnitude. That is, for each detectable intensity increase, sensation would change the same amount. Figure 3.9, which shows the sensation of the sound as a function of intensity, illustrates this relationship.
Each vertical dashed line represents one ΔI and each horizontal dashed line represents one ΔS. For convenience, a stimulus with an intensity of one has been set to a sensation of one. Starting with that first reference point, one can add a new point with an intensity of I+KI (for which I is equal to the present value of one) and a sensation of S+ΔS, for which S = 1. The next point is added at intensity (1+(K×1))+K×(1+(K×1)), which is equal to the previous intensity, (1+(K×1)), plus ΔI, which equals K times the previous intensity. This simplifies to 1+2K+K2 or (1+K)2. The sensation for this intensity is equal to 1+2×ΔS. The next point would have an intensity of I+KI = (1+K)2 + K(1+K)2 = 1+2K+K2 + K+2K2+K3 = K3+3K2+K+1 = (1+K)3 and a sensation of 1+3×ΔS. Therefore, the nth point would be equal to [(1+K)n, 1+n×ΔS]. The function is logarithmic and the sensation S due to a stimulus with an intensity I can be predicted by Mlog10(I/Io); where M and Io are constants. Based on this type of logarithmic relationship, sound level in dB SPL [10log10(I/Io)] is commonly used as the model parameter for which to evaluate the loudness of a sound. This model does not predict loudness correctly as it has been shown that equal JNDs do not correspond to equal loudness increments (Harper and Stevens, 1948). (p. 59)
Another loudness model based on Weber’s law is the power function, which can be derived by assuming that Weber’s law also holds true for sensation. That is, for a constant, T:(3.6)as proposed by Bretano (Stevens, 1961). Figure 3.10 illustrates this relationship. Again, an intensity of 1 is defined to evoke a sensation of magnitude 1. The next noticeable intensity will be equal to 1+K, corresponding to a sensation 1+T. Then an intensity of (1+ K)2 will evoke a sensation of (1+T)2. Therefore, an intensity equal to (1+K)n will evoke a sensation equal to (1+T)n. The general form of this relationship can be described by a power function:(3.7) where M and a are constants.
Referring back to Fig. 3.3, intensity and loudness are both plotted on logarithmic scales (dB SL is proportional to the logarithm of intensity). The results show that, for levels above 40 dB SPL, the logarithm of loudness is approximately proportional to the logarithm of the intensity (dB SPL). This relationship is predicted by the power function: log(S) = alog(I)+alog(M). The slope of the function in this study (Hellman and Zwislocki, 1963), which is the exponent a, is equal to 0.26 (0.54 when measured as a function of pressure rather than intensity, as pressure is proportional to intensity squared). Most early examinations of the power function determined a slope close to 0.3 (Hellman, 1991).
Modern views on classical models
Recently, studies have suggested that the power law is a fairly good descriptor for loudness growth, but that it does not accommodate all local variations. Florentine and Epstein (2006) summarizes these variations and they have synthesized them into the form of a modified function, known as an ‘inflected exponential function’ (INEX). These variations result from two primary deviations from the power function. At low levels, the slope is close to 1, and then the slope gradually declines as level increases until it reaches a minimum slope at moderate levels, in the 40–60 dB (p. 60) SPL range. At these levels, the slope of the loudness function must be shallower than the standard power function slope of 0.3 in order to explain temporal integration and loudness summation data. At higher levels, the slope gradually increases, but does not reach 1. To account for these variations in the loudness function, the INEX function replaces the constant slope of the power function model with a continuous polynomial such that the slope is allowed to vary slowly with level.
Figure 3.11 shows a model INEX function for an average of normal-hearing listeners. For this data set, the INEX is modelled as:(3.8)where S is the sensation and L is the level in dB SPL. In fact, these parameters typically vary somewhat from listener to listener, but the general form is roughly the same.
Although the differences between the INEX and the power function are relatively subtle, the INEX’s modifications to the power function provide a better description and predictor for loudness data gathered across a wide range of levels than the original power function.
3.5.2 Models of complex sounds
The basic loudness functions are sufficient for modelling the relationship between loudness and intensity for pure tones, but do not account for variations in loudness caused by other physical parameters, listening conditions, or listener impairments. As a result, more advanced models of loudness have been developed for a variety of predictive and informative purposes.
Sound pressure level weighting
The loudness of sounds varies with changes in frequency, so it is not surprising that for wideband signals that contain energy at many frequencies, overall loudness is not equally dependent on (p. 61) each frequency. Perhaps the most common field examining this problem is hearing conservation. Conservationists want to predict the potential hazard of a particular sound. In order to do this, they typically weight sound such that the frequencies that result in the most auditory system activity are given more weight than those that create less activity in the system. The typical weighting systems are A-, B-, and C-weighting. Each weighting is based on sound sensitivity at a particular sound pressure level determined using the equal-loudness contours described in Section 3.3.2. A-weighting uses a 40 dB SPL reference, B-weighting uses a 70 dB SPL reference, and C-weighting uses a 100 dB SPL reference. Sounds weighted using these references are given levels in units of dB A, dB B and dB C, respectively. Effectively, the sound is filtered using the inverse of the loudness contour, so that frequency regions that contribute less to loudness are attenuated. These are, however, rough approximations and do not guarantee that loudness is properly balanced across frequency. It is possible to generate sounds with the same level in dB A that have distinctly different loudnesses (Zwicker and Fastl, 1990). The way loudness increases with bandwidth, particularly the critical bandwidth, is not taken into consideration in any of these weightings.
Complex loudness model
The current ANSI standard (ANSI, 2007) is a five-stage predictor of loudness for a variety of sounds. The model accepts the power spectrum of the signal as input and outputs a prediction of loudness. The five stages of the model are:
1 Transmission through the outer ear: The transmission of a sound based on its power spectrum depends on how the sound is delivered to the ear. Primarily, sounds are presented via sound fields or headphones. Each presentation mode modifies the sound by attenuating or amplifying certain frequencies. Regardless of presentation mode, the ear canal and tympanic membrane resonances are included in this power spectrum alteration. Provided that the sound is not presented via headphones, the resonances of the room, head, body, and pinna shape will also alter the characteristics of the sound. The effects of all these external processes are modelled in this stage by applying a filter to the power spectrum to characterize the listening condition.
2 Transmission through the middle ear: The principal role of the middle ear is to match the impedance between the air of the outside world and ear canal and the lymphic fluid of the inner ear. This system does not have a flat frequency response and, therefore, the amount of acoustic energy transmitted to the inner ear depends on the frequency of the signal. Again, a filter is used to model the middle ear response.
3 Transformation of the spectrum to excitation pattern: Once transmitted to the inner ear, sound vibrations travel along the basilar membrane, which is organized tonotopically. That is, each part of the basilar membrane responds best to a different frequency. A bank of band-pass filters models this stage, with each filter representing the frequency selectivity at a specific place on the basilar membrane. The shapes of these filters were selected using data gathered from notched-noise experiments, a paradigm designed to examine frequency-specific listening (see Chapter 2).
4 Transformation of excitation pattern to specific loudness: Specific loudness is roughly like a loudness density. Psychoacoustical experiments have shown that the magnitude of the excitation on the basilar membrane is not linearly related to the acoustic power. Specifically, changes in basilar-membrane vibration do not occur proportionally to level changes, as the system is compressive at moderate levels. The behaviour is modelled in this stage by applying a non-linear, compressive, monotonic function to the excitation pattern to transform it into specific loudness.
5 Transformation from specific loudness to overall loudness. Finally, the overall loudness of the sound is obtained by integrating the specific loudnesses across the whole basilar membrane.
(p. 62) The ANSI standard also calls for a doubling of the loudness if the sound is presented binaurally. As discussed in Section 3.3.5 and specifically suggested for the ANSI loudness model by Moore and Glasberg (2007), less than perfect summation serves as a better predictor of binaural loudness.
This standard is also only valid for steady sounds heard by normal-hearing listeners. Variations of the model have been presented to predict loudnesses for time-varying sound (Glasberg and Moore, 2002) and for listeners with hearing impairment (Moore and Glasberg, 2004).
3.6 Context effects
Although all the presented loudness models compute loudness a priori, the loudness of a sound can be substantially affected by the sounds that precede it, both immediately and from relatively long periods prior.
The loudness of sound can be reduced if other competing sounds are presented simultaneously. This phenomenon is known as ‘partial masking’ (Scharf, 1964, 1971; Pavel and Inverson, 1981). Partial masking is primarily believed to result from overlapping excitation patterns. The loudness of a sound is characterized not only by the activity it causes on the basilar membrane at the place tuned to the frequency of the sound, but also the regions activated by the spread of excitation. Therefore, if two sounds are presented simultaneously, it is possible that the sounds may have some overlap in excitation. If the excitation of one sound is so strong that the excitation of the second sound is completely covered by it, the first sound becomes inaudible. This is known simply as ‘masking’. If only some portion of the excitation of the second sound is covered by the excitation of the first sound, the second sound may be audible, but its loudness is reduced. This is known as ‘partial masking’.
In fact, this masking effect can also occur when the masking sound is not presented at the same time as the masked sound. This is known as ‘non-simultaneous masking’ (see Chapter 2). When the masker precedes the target sound by a small amount of time, typically less than 100 ms, the detectability of the target sound may decrease. This is known as ‘forward masking’. This effect increases as the masker and target move closer together in time. The amount of forward masking depends on the temporal proximity of the masker and target as well as the relationship between their levels. Backward masking, the opposite effect in which a masker is presented after a target, results in a very weak effect and only occurs under specific conditions.
3.6.2 Loudness adaptation
Loudness adaptation is the decrease in loudness of a sound presented continuously over an extended period. Loudness adaptation occurs only for sounds presented below about 30 dB SL, with high-frequency tones adapting more than low-frequency tones. In fact, it is possible for continuous tones at high frequencies and low levels to become completely inaudible over time. However, it is important to note that adaptation occurs only for continuous, constant sounds and if the sound is amplitude-modulated sufficiently, the effect disappears (see Scharf, 1983 for a review).
3.6.3 Loudness fatigue/temporary threshold shift
Exposure to very high-level sounds, particularly for an extended period, may cause a reduction in sensitivity to intensity. This is known as ‘loudness fatigue’ or specifically, in the case of threshold, ‘temporary threshold shift’. This effect can last from minutes to days and is often a result of noise (p. 63) exposures like gunfire (Bapat and Tolley, 2007) or loud music (Sadhra et al., 2002). Temporary threshold shift is known to increase rapidly with increases in exposure level and exposure time. Hirsh and Ward (1952) observed that after just 3 minutes of exposure to a 120 dB SPL 500-Hz tone, a maximum temporary threshold shift of about 20 dB occurred 2 minutes after the tone ended. The maximum shift also tends to occur for sounds around 4 Hz, even when exposure occurs at lower frequencies. Recovery from a 3-minute exposure took somewhat over 10 minutes. Sometimes, sound exposure can result in longer term temporary effects. Rabinowitz (2000) discusses a case of a young girl who attended a rock concert the night before auditory evaluation. She showed a temporary 30-dB hearing loss at 4 kHz, which returned to normal after several days.
It is not clear that loudness fatigues by the same amount as threshold shifts. In fact, Botte et al. (1993) hypothesized that loudness fatigue and temporary threshold shift result from two different, but correlated mechanisms. Regardless when a temporary threshold shift occurs, loudness fatigue is typically also present.
3.6.4 Loudness enhancement/decrement
Loudness enhancement is a context effect in which the loudness of a sound (target) is increased by a higher level sound that immediately precedes it (much like a forward masker). The forward masker and target must be presented at similar frequencies and the forward masker must be presented less than 500 ms prior to the target in order for the effect to occur. This is different than the partial masking effect discussed in Section 3.6.1, as that effect primarily results from maskers that are lower in frequency than the target sounds.
Conversely, loudness decrement is a context effect in which the loudness of the target is decreased by a lower level forward masker that immediately precedes it. The relationship between the levels of the forward masker and the target affects the quantity of enhancement and decrement with maximal changes in loudness around 20 phons (see Oberfeld, 2007 for a review). Additionally, there is some debate regarding the degree to which loudness enhancement and decrement are intertwined with induced loudness reduction (Scharf et al., 2002; Oberfeld, 2007).
3.6.5 Induced loudness reduction
Induced loudness reduction is a phenomenon by which the loudness of a sound is reduced when it is preceded by one or more higher intensity sounds presented at a nearby frequency, anywhere from several hundred milliseconds to several minutes prior. While loudness adaptation and fatigue may affect the loudness of all levels of sound equally, induced loudness reduction tends to primarily reduce the loudnesses of sounds at moderate levels. This phenomenon is one likely cause of discrepancies between estimates of loudness performed by listeners at the beginning of an experiment, when a listener has little prior sound exposure, and at the end of an experiment, after the listener is exposed to a wide range of sound levels (see Epstein, 2007 for a review). In fact, because only moderate levels are affected, induced loudness reduction can alter the entire shape of a loudness curve, when plotted as a function of level.
3.7 Hearing-impaired listeners
Chapter 14 examines the details of many aspects of auditory perception for hearing-impaired listeners. Here, a brief review of loudness for these listeners is presented.
3.7.1 Impaired loudness models
Hearing-impaired listeners (HILs) with sensorineural hearing have a reduced dynamic range. That is, the difference between the highest tolerable sound level and the lowest detectable sound (p. 64) level is less than for normal-hearing listeners (NHLs). This primarily results from the fact that HILs have elevated thresholds and cannot detect low-level sounds. Despite threshold elevation, most HILs experience the same tolerability at high levels as NHLs. Therefore, the loudness function must be different for HILs than for NHLs. Two models have been proposed that can explain the reduction of dynamic range.
The primary assumption for the recruitment model is that loudness at threshold is the same for both HILs and NHLs. If loudness at threshold for a HIL is the same as for a NHL, then loudness functions must grow more steeply for HILs than NHLs in order for loudness to catch up at high levels.
The primary assumption for the softness imperception model is that loudness at threshold is not the same for the HILs and NHLs. Softness imperception is based on the ideas that the loudness at threshold is higher for a HIL than for a NHL and the loudness function shows little-to-no compression at moderate levels, but otherwise is not steeper than the loudness function seen for NHLs.
The recruitment model has traditionally been accepted as the foundation for algorithms used in hearing aids to adapt gain as function of level. Although softness imperception is a more recent concept and is less commonly implemented in hearing aid fits, many fitting procedures include gating such that low-level sounds are not amplified into the audible range. This fitting is compatible with the idea that low-level sounds amplified to audibility may be disconcertingly loud for HILs.
There is still significant debate regarding these two models (Florentine et al., 2004; Moore, 2004) and some evidence supports the idea that some HILs exhibit recruitment, some exhibit softness imperception, and some exhibit an intermediate behaviour (see Marozeau and Florentine, 2007 for a review).
Figure 3.12 shows a variety of loudness models including: the power function (dashed) and INEX (star) for NHLs, and the recruitment (circle) and softness imperception (square) models for HILs. Note that the loudness function for HILs begins at a higher level because the listener has an elevated threshold. In the case of the recruitment HIL, loudness at threshold is the same as it is for a NHL and the function is steeper so that it can catch up in loudness. For the HIL with softness imperception, loudness at threshold is elevated. Although the function is never steeper than the maximum INEX steepness, this listener does not exhibit compression at moderate levels, so the function approaches the INEX loudness function at high levels. Around 80 dB SPL, all the functions show close to the same loudness and slope. If a listener exhibited a loudness function that exceeded normal loudness at high levels, this listener would be said to have hyperacusis.
Hyperacusis is an abnormal sensitivity to loudness. While it is typical for impaired listeners to have a reduced dynamic range of hearing resulting from elevations of their threshold, these impaired listeners typically have approximately the same sensitivity to high-level sounds as normal-hearing listeners. That is, their maximum tolerable level remains about 120 dB SPL. Those who experience hyperacusis have a reduction in tolerable sound level. Some listeners may experience discomfort for sounds as low in level as 65 dB SPL. Additionally, many of these (p. 65) listeners also cope with elevated thresholds, so they may experience a substantially truncated dynamic range of hearing. Relatively little is know about the causes of hyperacusis, but it is often associated with traumatic noise exposure, tinnitus, drug reaction, neurological disease, or head injury (Katzenell and Segal, 2001; Nelson and Chen, 2004).
Perhaps the greatest limitation present in hearing aids and auditory prostheses is the inability to replicate the functions of the auditory system responsible for helping to provide the tremendous dynamic range of hearing as well as remarkable acuity within that range. Despite many years of scientific study, there is still much to ascertain about the processes used to encode the intensities of basic and complex sounds into both loudness and envelope-shape information. Models of loudness and intensity coding have gone from simple predictors relying on a single sound parameter to multistage processes based on many physical parameters of sound as well as the anatomy and physiology of the auditory system. Future mathematical models will perhaps incorporate more about the experience of listening: the context of sound, the syntax of sound, and the meaning of sound, as well as the condition of the listener: awareness, attention, and impairment aetiology. Despite nearly 150 years of study, the field of loudness still holds many mysteries.
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