6.3 Loudness

Probably the most familiar result of psychoacoustic testing is the chart shown in Fig. 1. This shows contours of equal perceived loudness, for listening to sine waves. The vertical axis shows the amplitude of the sound pressure, plotted on a decibel scale. The zero level is defined by an international convention, and is roughly the level of the quietest sine wave you can hear. Now notice the numbers on the vertical scale: it covers a range of 140 dB, and if you remember that 20 dB corresponds to a change in amplitude by a factor of 10, this means that your full range of hearing (for sine waves, remember) covers 7 factors of 10, so a total range of a factor of 10,000,000.

Figure 1. Contours of equal loudness for sinusoidal waveforms. Image: Lindosland, Public domain, via Wikimedia Commons

The lowest curve in the chart has a simple interpretation. It is labelled “threshold”, which is probably self-explanatory: it indicates the quietest sine wave that can be detected at each frequency, by a typical healthy young adult with normal hearing. “Young” because as we get older our hearing deteriorates, most obviously at higher frequencies. At age 70, I can no longer detect a sine wave above about 10 kHz at any amplitude. Your personal version of this threshold curve is one of the things that is commonly measured if you are given an audiometry test to assess the state of your hearing.

The highest curve in the plot is only “estimated” at frequencies above 1 kHz: this is because exposure to sounds that loud can cause damage to the listener’s ears, so that it is not ethically acceptable to collect the data to complete the measured curve or to map out curves at higher levels. Somewhere around a sound pressure level of 120–140 dB, we would reach the “threshold of pain”: the level at which a sine wave becomes physically painful to the listener. This does not mean that sounds this loud are never heard, of course. Notoriously, loud sounds such as those from aircraft engines, firearms or loud amplified music commonly cause hearing damage unless precautions are taken. “Ear defenders” are now routinely worn by people working in noisy environments — but not usually by party-goers or members of rock bands.

The curves are labelled in terms of a unit of subjective loudness called the phon. The numerical value associated with each curve is defined to be the physical sound pressure level in decibels where the curve passes through 1 kHz. The curious wiggles in the curves are caused by resonances in the ear canal and the middle ear: as a result of these resonances, sine waves at some frequencies are amplified before they reach the cochlea, while at other frequencies they are somewhat suppressed.

The chart in Fig. 1 is often interpreted as saying that the human ear is most sensitive for frequencies around 1–5 kHz. Well, this is true in one sense, but it is the opposite of the truth in another sense. If you are thinking about detecting very quiet sounds, then it is true that your ear is at its most sensitive in this frequency range. But notice that this is also the range where the curves are widest apart: so in this range your ear is at its least sensitive to changes in loudness. In the context of listening to music, sensitivity to changes can be just as important as absolute sensitivity to whether there is a sound at all. For example, where the curves are wide apart a performer would need to make a bigger physical change in order that the listener hears a crescendo.

It is useful to have an idea of how the charts in Fig. 1 were determined. In a typical experiment, a test subject would be sitting in a quiet place, with headphones on. They would be presented with a “reference” tone, perhaps at 1 kHz, at a particular level. They would then hear a second tone at a different frequency, which would start at a random level. Their task is adjust the level of this second tone until the loudness is judged to match the reference tone. This procedure is then repeated many, many times. The frequency of the test tone will be varied to cover the entire audible range, including repeat runs with the same frequency to test for consistency of judgement. The level, and perhaps the frequency, of the reference tone will be varied. Many different test subjects will take the same test — but will be presented with the individual loudness-matching tasks in a different random sequence. All the results are stirred together in the computer, and some statistical tests performed to tell the experimenter when they have enough results that the answer is reliable within a specified limit.

This kind of testing is subject to many traps and pitfalls. We can see a strong hint of this in Fig. 1. The red curves show the current international standard, but the blue curve shows an alternative version of one of these curves, which used to be the international standard before a new set of trials were carried out. They differ by a remarkably large amount, up to about 20 dB. Both sets of experimenters knew what they were doing, and followed a very similar procedure, so how come they reached an answer that was so different?

There are many factors that can disturb the results of an experiment like this. We will give a few examples, but for a full discussion see Moore [1]. Presenting the sounds via headphones is not the same as presenting them with loudspeakers. Even different kinds of headphones will give different results. The reasons are to do with diffraction of the sound field around the subject’s head and external ears. The position at which the reference sound pressure level is measured can make a significant difference: just moving from a few millimetres outside the ear canal to a few millimetres inside it is important. Sensitivity to low-frequency sounds presented through headphones can be affected by sounds generated by your own body, especially your heartbeat and blood flow.

Even less obviously, there are subtle sources of bias that can arise from details of the testing procedure: here is an example. For a given level of the reference tone, the test tone is initially presented at a random level, chosen from within a certain range using a random number generator. But it has been discovered that if this range of random amplitudes is not symmetrical above and below the eventual “correct” level which matches the loudness of the reference tone, there will be a bias in the results of the test. Of course, you don’t know where the matching level will be when you start the experiment, so the testing has to be done in an iterative way. Choose a range, find out what the listeners judge to be the matching level, then if this turns out not to lie at the centre of the random range, you have to repeat the whole experiment with a new range. Then keep repeating this until you have results based on a range which is in fact symmetric about the final chosen level.

All of this complication, and we are still only talking about hearing sine waves! Normal sounds, and real music, involve far more complicated sounds. They will involve multiple frequency components, and they will be varying in time in a complicated way. We will start to investigate how to deal with this in the next section.

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[1] Brian C. J. Moore; “An Introduction to the Psychology of Hearing”, Academic Press (6th edition 2013).