Math’s Gift to Musicians
Is music just a cultural product, or does it also have roots in nature?
I contend that the tonal system underlying most Western music (and arguably other musics as well) is successful because it derives from the natural harmonic series. When a string is plucked or a column of air in a clarinet is placed in motion, the resulting vibration pattern is a complex composite of sine waves of various frequencies, especially the smaller multiples of the fundamental frequency. We refer to these multiples as “harmonics.” Since we are constantly exposed to such patterns in our natural environment, it is not surprising that our ears are naturally attuned to them, perhaps even evolved to hear them. When the key two octaves below middle C is struck on a piano, the most prominent harmonics generated are as shown above.
The most prominent intervals in our music are the ones between adjacent harmonics. The 2:1 interval, which we call the perfect octave, is surely the most important interval in music: If two F-sharps are an octave apart, we often even think of them as being “the same” pitch, in some sense. Next after that is the 3:2 interval, or perfect fifth, which relates the two most important pitches in any key (tonic and dominant) and also underlies our system of key relationships (the famous “circle of fifths”). Historically, the octave was the first interval to be accepted as a consonance, followed by the fifth, and so on.
To be accurate, the octave is the only interval precisely realized in the modern equal-temperament tuning system, which limits us to 12 equally-spaced pitches in order to keep things manageable. Nevertheless, that system gives really good approximations to the most important other harmonics. For example, the perfect fifth in nature has the ratio 3:2 = 1.5. The perfect fifth in our equal-tempered system consists of seven half steps and therefore has a ratio of 2^(7/12) = 1.49831, which is darned tootin’ close, pun intended. We musicians and music lovers are extremely fortunate that the math works out so well for music! 🙂