Mysteries of the Harmonic Series

Scientific and unscientific notes fit different styles (plus, how to annoy everyone by sounding like a mosquito...)

Mother Nature provides us with many ways to listen to musical sounds, some more organized than others. We are able to sing or play with those sounds, and even come up with sounds that can mercilessly annoy our friends and neighbors. It’s all about audio frequencies, which we hear and make use of, whether we understand them or not. Some sounds are pleasant and some are not. At the end of this article is a list of frequencies used by mosquitoes, hornets, musical instruments, and other potentially annoying sounds!

My 7th grade music teacher tantalized us one day with ideas about the connections between music and the frequencies of sounds. After class, I asked him about it, and he gave me a copy of the Instrumentalist magazine explaining the harmonic series.

I was fascinated by this extra dimension of music. Whether listening or struggling physically to play an instrument, we can always try to understand the organized beauty of melody and harmony.

The harmonic series explains why anyone can match two pitches, and why we only named seven notes, A B C D E F G and then cycle back to A again — the eighth note, the octave has the same name as the first note, and is almost as easy to identify as unison, or identical, pitches. The fact that the frequency of any note is double the frequency of the note eight notes below it can help explain to brass, flute, or pennywhistle players why blowing harder can produce a higher octave, and helps string players understand why one string will resonate sympathetically if the octave note above it is played in tune. Even beginners can hear when two notes are the same pitch, or when they’re an octave apart, and they don't have to imagine that this is a skill requiring talent or years of study, because there's a physical, scientific reason for it.

Anything in the world that vibrates at high speed will create a musical pitch — a hummingbird wing, a card buzzing in bicycle spokes — and if we know exactly how fast it's vibrating, we know what pitch it is. For example, the hum in our houses and our sound systems is between an E and an F, because it’s a multiple of the 60-cycle vibration of our electric current.

If I wiggle my 3d finger on my violin A string just a little higher than usually, I can produce a sound that anybody might mistake for a mosquito. That’s because that note vibrates the air around it at about 600 times per second, the same frequency as the beating wings of a mosquito.

Everybody can identify mosquito wings (600 flaps per second) vs those of a housefly (190 flaps) or bumblebee (250 flaps per second). This physical connection suggests that for quite physical reasons, we can all recall and match pitches that we have heard many times.

In This Is Your Brain On Music, Daniel Levitin mentions a study in which average people (i.e. not musicians) tend to sing popular songs in the same key in which they were recorded, as long as different versions in other keys didn’t become popular in the meantime.

There are many who think they have a bad sense of pitch, but I often feel that’s a faulty, if popular, self-diagnosis. I get frustrated hearing chorus teachers allow young kids to sing out of tune, because I know they can do better if people didn’t find singing out of tune to be especially cute when kids do it. I know of a Christmas album that deliberately used an out-of-tune chorus of kids to convey cuteness and hominess and don’t get me started, okay?

People who can’t find a pitch may be looking for the wrong thing, such as focusing on how their vocal cords or finger muscles feel, instead of listening and comparing pitches by focusing on their ears. This is where “ear training” classes can really help people, because, like any good musical instruction, they prioritize what’s helpful rather than what’s easy.

Not everybody can listen to two sounds and tell which one is higher, but just about everyone can tell the moment when two pitches match after they slide toward each other. When the frequencies match (i.e. have same number of beats per second), we sense calm and harmony. If they don’t match, we sense a clash, a buzz, and the more we pay attention to this, the easier it is to identify the difference.

If a string is divided in half and plucked or played, the resulting pitch is an octave higher than the full-length string. If you divide the string in thirds, you hear a note a fifth higher than the octave, because that note has three times the frequency of the original. Examples: one A beats 220 times per second; the A one octave higher beats at 440 times per second; the E above that beats at three times the lower A, or 660 times.

One-half the length of a string or column of air creates a vibration double the frequency of the full length, sounding an octave higher. One-third the length triples the frequency, yielding a note that's a fifth higher than the octave. One-fourth quadruples the frequency of the vibration, 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 doesn’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 in their compositions.

I speak of Western classical music because music of other cultures or of local populations, whose music is categorized as folk or traditional or world music, have developed other sound patterns. For example, the Ottomans went so far as to divide the octave into 57 pitches, and this can be heard in classical Turkish music. Some Indian music apparently divides the scale into 22 pitches.

The most scientifically precise notes in our seven-note scale are the fourths, fifths, and octaves. It’s especially intriguing that the two most ambiguous notes in the harmonic series — the frequencies that don't match our seven-note scale — are the third and the seventh notes of the scale. These are the notes most freely toyed with in most musical cultures.

In classical music, the third note of the scale determines major or minor. Harmonic and melodic minor scales play games with the seventh note of the scale. The blues bends the third and the seventh notes. Many fiddle traditions are ambiguous about how they play the thirds and sevenths (last week’s post laid out patterns for the “correct” pitches but some traditions have a different notion of what they like to hear). These ambiguities are not due to a lack of training; the notes in question can be heard in various keys regardless of the finger spacing, so it has to do with how the players hear the music, not how they were trained physically. Pay attention to the 3ds and 7ths in some of the older Cape Breton or Shetland fiddlers and you’ll hear the ambiguity, regardless of key.

Here’s a recording of the beginning of the reel “The Marquis of Tullybardine” played well in tune by the great Scottish fiddler Charlie McKerron (best known from the band Capercaillie):

Here’s the same tune as played by the great Cape Breton fiddler Joe Cormier, who always had his own, older sense of intonation, especially on those 3d and 7th notes of the scale, that was consistent in his playing:

Fascinating complications come into play when we combine notes, and make them clash or even create new frequencies of their own. The harmonic series of two notes played together will reinforce each other’s frequencies in some places and cancel out each other’s frequencies in other places. On a bowed doublestop, a string player can hear a third frequency if the notes are in tune, or a buzzing clash if they’re not. With some awareness and practice, anybody can hear this phenomenon, since it is a physical fact based on the harmonic series. I wish physics teachers would use music to demonstrate some of this; it’s both practical and fascinating! (My kids’ high school physics teacher, who was otherwise a great teacher, sadly knew nothing about musical frequencies.)

I heard of one music teacher from long ago who went way overboard in his studies and deep-dive thinking on this subject — he actually recommended against taking kids to the symphony because the conflicting vibrations of so many frequencies played at the same time would, he surmised, damage young brain waves!

We often view science as making things better known and more predictable, but learning about the harmonic series and the science of sound only makes music more engaging, mysterious and awe-inspiring.

By the way, if you’re looking for a natural pitchpipe, here are some frequencies you can always make use of —

Frequencies (beats per second)

1046 Highest note humans can sing

988 Highest note in 1st position of violin (B)

600 Mosquito

523 Lowest note on piccolo

440 A for orchestral tuning

261 Middle C

250 Honeybee

196 Lowest violin note (G)

190 Housefly

165 Lowest note on clarinet and guitar

130 Bumblebee and lowest note on viola (C)

100 Hornet

96 Horsefly

85 Moth

65 Lowest cello note (C)

60 (or multiple of 60) buzz from a 60-cycle electric circuit (between Bb and B)

38 Dragonfly

33 Lowest C on piano

20 Fundamental (bottom note of harmonic series) on trombone

12 Butterfly

8 Lowest organ note