![]() Unlike some textbooks in this area, such as David J. Roberts, a mathematics professor at the College of the Holy Cross, developed the book for an undergraduate course in mathematics and music. Roberts does not oversell mathematics as the explanation for the missing fundamental, but shows how mathematics can be used as a tool to predict the perceived pitch based on the spectrum of pitches produced. The effect is also found in some pipe organs, which do not have room to make pipes large enough for the lowest notes on the organ, and instead cleverly trick the listener with precisely-calibrated smaller pipes. On a practical level, this effect is exploited by telephones, which do not pick up frequencies as low as most humans’ speaking voices, but nonetheless manage to transmit normal-sounding messages. The brain assumes it just missed picking up on sine waves with frequency 220 and fills in the gap, perceiving 220 where there is none. Our pattern-recognizing brains, which must often make snap judgments with incomplete information, notice that 440 and 660 are among the expected frequencies that an instrument or voice would create when producing a note with fundamental frequency 220. The real explanation, however, belongs to cognitive science, not mathematics. There is a mathematical explanation for the auditory illusion in my example, sometimes called the missing fundamental: The perceived pitch is the greatest common divisor of the frequencies of the sine waves present. Roberts’s textbook From Music to Mathematics: Exploring the Connections does a good job of separating the objective from the subjective in its discussion of pitch and frequency. The fact that pitch is a perception, rather than an objective, measurable aspect of sound waves, is one of the challenges in trying to use mathematics to describe music. The combination of the frequencies 440 and 660 creates a perceived pitch of 220. Even if you know that nothing is actually playing a pitch with frequency 220 Hz, that pitch is what you will hear. So, you will perceive the same pitch as a sine wave with a frequency of 220 Hz. We hear pitch logarithmically, and an octave corresponds to a frequency ratio of 2 : 1. Instead, you will hear a pitch an octave below the A 440. (A perfect fifth is seven half-steps, the interval between an A and the E above it.) If you play the sine waves with frequencies 440 and 660 together, you will not hear a perfect fifth. For example, if you play a sine wave with a frequency of 440 Hz by itself and one with a frequency of 660 Hz a few seconds later, you will hear two distinct pitches, one a perfect fifth higher than the other. But if you start playing with sine waves a little, you will hear some surprises. The process works in reverse as well: By adding sine waves of various frequencies, computers and electric keyboards can create decent imitations of the sounds of these instruments. If a harmonic analyzer took the sound of an instrument or human voice as its input, it would break the input into sine waves of many different frequencies, generally all integral multiples of one lowest frequency. The flute probably comes closest to producing a pure sine wave, but even its sound is much more complex. It is the tuning standard for most modern orchestras and instruments.) However, that A will not sound like an A played by any instrument in the orchestra. (In fact, 440 is probably the most famous musical frequency. If someone plays a sine wave with a frequency of 440 Hz, you will most likely perceive an A. This review first appeared in the December 2017 issue of the American Mathematical Monthly.
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