In 7th grade, our music teacher tantalized us with ideas about the harmonic series. I remember asking him about it after class, and he gave me a copy of the Instrumentalist magazine with an article and diagrams showing the connection between music and the frequencies of sounds.
I was fascinated, and Iâ€™ve seen students be equally interested, and even taken by surprise, at this meeting of music and science. It adds another dimension to learning about music besides the struggle of trying to coordinate hands on an instrument and understand the organized beauty of melody and harmony.
The harmonic series explains why anyone can match two pitches, and why octaves have notes of the same name, and are almost as easy to identify as unison pitches. The fact that one octave is double the frequency of the lower one helps string playersÂ understand why a string rings sympathetically if the octave note above it is played in tune.Â Even beginners can hear this, and since there’s a physical, scientific reason for it, they don’t have to worry that it’s tied in with talent, or years of study.Â
It’sÂ fun to point out that anything in the world that vibrates at high speed will create a musical pitch–a hummingbird wing, or a card buzzing on bicycle spokes–and if we know how fast it’s vibrating, we know what pitch it is.Â For example, the hum in our houses and our sound systems is between an E and an F, because itâ€™s a multiple of the 60-cycle vibration of our electric current.
Itâ€™s amazing to see that if a string divided in half, the resulting pitch is an octave higher. Even more surprising is to find out that if you divide the string in thirds, youÂ hearÂ a note a fifth higher than the octave. Wind players work with this all the time, and for string players this fact helpsÂ understand why the fifthsÂ our open strings are tuned to are pretty easy to hear–theÂ next easiest thing to identify after the octave.
So the revelations add up: one-half the length of a string (or column of air)Â makes a vibrationÂ double the frequency of the full length, and sounds an octave higher. One-third the length triples the frequency,Â yielding a note that’s a fifth higher than the octave. One-fourth quadruples the frequency of the vibration, adding a fourth (resulting in a note two octaves above the original). One-fifth quintuples the frequency and adds a major third.
On it goes, adding a minor third, a major second and so on. Some say the history of Western classical music follows the harmonic series, with each generation spotlighting the next level of the series as its featured interval. This idea doesnâ€™t apply too well to some of the baroque composers who loved playing with dissonant intervals. But it seems to fit the general historical pattern of succeeding generations of composers moving from unison to fifths to thirds to seconds to twelve-tones to even quarter-tones inÂ their compositions.
I say Western classical music, because music of other cultures or of local populations, whose musicÂ is categorized as folk or traditional or world music, have developed other sound patterns.Â For example, the Ottomans went so far as to divide the octave into 57 pitches, and this can be heard in classical Turkish music.Â Some Indian music apparently divides the scale into 22 pitches.
Itâ€™s especially intriguing, I think, that the two most ambiguous notes in the harmonic series–the multiples of the original frequency that don’t quite match our seven-note scale–are the third and the seventh notes of the scale.Â It’s no coincidence that these are the notes most freely toyed with inÂ most musical cultures. In classical music, these notes determine major and minor. Â Harmonic and melodic minor scalesÂ play aroundÂ with the seventh note of the scale.
The blues likes to bend primarily the third and the seventh notes. Many old fiddle traditions play the thirds and sevenths ambiguously, sort of halfway between the major and minor thirds and sevenths. This can be done easily on a violin by spreading the fingers out evenly:Â place the second finger halfway between the first and third, instead of next to one or the other to form a half step. You might assume a player might do this only because of a lack of training, but in fact, this ambiguous pitch is maintained byÂ some fiddlers even in other keys, so itâ€™s clearly part of a way of hearing the pitches. An example is the playing of Joe Cormier, a great Cape Breton fiddler, or old recordings of traditional Shetland fiddlers.Â A musician unfamiliar with those traditions would simply assume the player isÂ out of tune.
We often view science as making things better known and more predictable, but it seems to me that learning about the harmonic series and the science of soundÂ onlyÂ makes musicmakingÂ more engaging, mysterious and awe-inspiring.