Posts
DOI: 10.4169/amer.math.monthly.124.10.979 Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.124.10.979
 |
| Follow on Twitter as @evelynjlamb |
 |
| Photo: Johns Hopkins University Press |
If someone plays a sine wave with a frequency of 440 Hz, you will most likely perceive an A. (In fact, 440 is probably the most famous musical frequency. 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. The flute probably comes closest to producing a pure sine wave, but even its sound is much more complex. 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 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. But if you start playing with sine waves a little, you will hear some surprises. Frequency is not destiny. 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...
Gareth E. 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. 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. [Read more about the missing fundamental on this blog here and here.] The real explanation, however, belongs to cognitive science, not mathematics. 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 brain assumes it just missed picking up on sine waves with frequency 220 and fills in the gap, perceiving 220 where there is none...
Roberts, a mathematics professor at the College of the Holy Cross, developed the book for an undergraduate course in mathematics and music...
..Roberts makes some suggestions about how to use the text for either a year-long or one-semester class on math and music. I have not taught such a course, but it seems like the text would be appropriate for a freshman seminar or liberal arts math class with a group of students that is diverse with regard to their mathematical and musical backgrounds.
Read more...
Source: Scientific American (blog)
