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logarithms also describe many natural phenomena. For example, our perception of pitch is approximately logarithmic. When a musical pitch goes up by successive octaves, from one do to the next, that increase corresponds to successive doublings of the frequency of the associated sound waves. Yet although the waves oscillate twice as fast for every octave increase, we hear the doublings—which are multiplicative changes in frequency—as equal upward steps in pitch, meaning equal additive steps. It’s freaky. Our minds fool us into believing that 1 is as far from 2 as 2 is from 4, and as 4 is from 8,
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