Although the term
perception is difficult to define, we all have an
intuitive notion of what is meant by it. For the sake of this writeup we can take
perception to mean the
human interpretation of and reaction to stimuli. While certainly each of us responds to stimuli differently, experiments have shown that, to a large extent, we all have
logarithmic perception. Logarithmic perception is the topic of this writeup.
Before I scare those who aren't mathematically inclined, let me give a simple description of the logarithm. The most important effect of the logarithm is to take numbers that are spread out over many orders of magnitude and bring them closer to eachother. Consider the following table:
Number Log(number)
10 1
100 2
1,000 3
10,000 4
100,000 5
1,000,000 6
The chart above is for the "log base 10." It is defined such that log10x is the number y with the property that 10y = x.
Usually we consider the difference between one million and ten to be quite large. On a linear scale the ratio between one million and ten is 100,000. Notice, however, that on a logarithmic scale one million is only "six times" ten.
So what does "logarithmic perception" mean? Well, suppose the numbers in the lefthand column of the chart above are measures of some stimulus. Let's suppose they are the amplitudes of sound waves in some arbitrary unit system. Psychological studies have repeatedly shown that our perception of the "loudness" of the sound is proportional to the logarithm of the amplitude. For example, we perceive the difference in loudness between sound waves with amplitudes 100 and 10 to be equal to the difference between sound waves with amplitudes 1,000,000 and 100,000. Our perception of loudness (which is purely mental) is a logarithmic function of sound wave amplitude (which is a real physical entity). It is for this reason that the most common unit for measuring loudness is the decibel, which is proportional to the logarithm of sound wave amplitude. A difference of one decibel (dB) corresponds to the minimum change in loudness that the typical human ear can discern.
We just discussed the fact that constant ratios of sound wave amplitudes correspond to constant differences in perception of loudness. One way to define what we mean by logarithmic perception could be just that--constant ratios of the magnitude of physical stimuli produce constant differences in our perception of them.
Let's examine another perceived property of sound--pitch. We're all familiar with musical scales. In particular, we know that a note sounds "the same" as the note an octave below it and an octave above it. Although we perceive a slight difference between middle C and the C one octave higher, they're "the same note." The pitch we hear (a purely mental entity) is related to the frequency of oscillation of the sound waves (a real physical entity). It turns out that a rise of one octave corresponds to a doubling of the frequency of the sound wave. As an example, sound waves with frequencies of 220Hz, 440Hz, and 880Hz (1Hz means one oscillation per second) are the note A at progressively higher octaves. Notice that the ratio of frequencies between octaves remains constant at 2. Constant ratios correspond to constant differences on the logarithmic scale. Like our perception of loudness, our perception of pitch is logarithmic. Why some chords and melodies are beautiful, sad, or triumphant is a fascinating question that I don't think has been completely answered! Actually human perception of sound is rather fascinating and I plan to add a writeup on that subject.
The law of logarithmic perception holds quite generally (within sensible constraints--obviously a sound wave with amplitude so great that it results in death is not "15 times as strong" as the sound wave that results from a man rudely slurping his coffee). In optics, studies have shown that perceived brightness is a logarithmic function of light's luminance. A website listing animation principles suggests that the eye's perception of motion is logarithmic, not linear (see http://www.writer2001.com/animprin.htm). Studies involving the perception of the weight of blocks in one's hand have shown that the weight is perceived logarithmically. Many researchers have even suggested that human perception of time is a logarithmic function of age, which would explain why older people perceive that years are getting shorter. I personally think this is a misuse of the word "perception" but it speaks to the general acceptance and validity of the law of logarithmic perception.
What I've been calling the "law of logarithmic perception" has a more common name--the Weber-Fechner Law. Ernst Weber and Gustav Fechner independently arrived at the law in the middle of the nineteenth century. Today, researchers in psychophysics still use the Weber-Fechner Law as an approximation, but have developed more detailed models of perception.
Why is it that our brains perceive stimuli logarithmically? It seems to me that our perception evolved to allow us to perceive a wide range of stimuli using a finite number of neurons. Our brains can make sense of both whispers and ambulance sirens. While we have difficulty discerning subtle changes in stimuli amplitudes, our abilities to make sense of wide ranges of stimuli amplitudes more than offset that limitation.