Journey in the Ear: Harmony - it's in the wave!
Updated: Nov 2
Some things in our world are fixed. Set in stone, as it were. They can’t be changed. At least not by human intervention. Gravity. The speed of light. The orbit of the moon. Our planet’s spot in the Milky Way.
In fact, most of our world is fixed, in the sense that we can’t create new natural resources that don’t exist, but we can create endless things from those that do. Our world is full of things we have built, created, harnessed, manipulated, replenished or repurposed with the vast resources at our disposal.
So it should come as no surprise that musical harmony is also fixed! Yes, you read that right! There is a scientific law governing harmony. A law no man created and that no man can alter!
Remember that last time we discovered how a music sound wave contains more than one frequency, more than one sound wave. Each music sound wave is actually a complex wave consisting of a fundamental frequency, the primary sound we hear and that our ear identifies as a particular pitch, plus numerous harmonic overtones, secondary frequencies that occur above the fundamental at higher frequencies or pitches but with varied intensities. And these overtones are all integers of the fundamental. This is what separates music from the haphazard, irritating frequencies of noise. There’s a pleasing, logical, repetitive mathematical pattern perpetuated in the sound wave created by a musical instrument. We also discovered that the first overtone of a music sound wave is always one octave above the original note played.
But it gets even more fascinating!
Let’s examine the first few harmonics produced by striking middle C on the piano:
Harmonic 1 = 261.62 Hz (1 x 261.62) = C4 (middle C)
Harmonic 2 = 523.24 Hz (2 x 261.62) = C5 (octave above middle C)
Harmonic 3 = 784.86 Hz (3 x 261.62) = G5 (next G)
Harmonic 4 = 1046.48 Hz (4 x 261.62) = C6 (next C octave)
Harmonic 5 = 1308.10 Hz (5 x 261.62) = E6 (next E)
Harmonic 6 = 1569.72 Hz (6 x 261.62) = G6 (next G)
The fundamental frequency, which is what our ear identifies as middle C, is 261.62 Hz. As you can see, the other naturally occurring frequencies, or overtones, are exact whole number multiples of the fundamental.
Notice that harmonics 4, 5 & 6 spell the C major chord: C E G! All the notes of the C major triad appear in that same sound wave, even occurring in order on the 4th, 5th and 6th harmonics! Furthermore, the chord actually occurs in an inverted order first (in this case, 2nd inversion) starting at the 3rd harmonic, G, to form G C E , then occurs in what is called root position (C being the root), as in C E G.
Every key on a piano produces this same pattern of multiples of the fundamental. We call this pattern the harmonic series. If you strike a G (let’s assume the G5 frequency shown above), the harmonic series would look like this:
Harmonic 1 = 784.86 Hz (1 x 784.86) = G5 (2 G’s above middle C)
Harmonic 2 = 1569.72 Hz (2 x 784.86) = G6 (next G)
Harmonic 3 = 2354.58 Hz (3 x 784.86) = D6 (next D)
Harmonic 4 = 3139.44Hz (4 x 784.86) = G7 (next G octave)
Harmonic 5 = 3924.30 Hz (5 x 784.86) = B7 (next B)
Harmonic 6 = 4709.16 Hz (6 x 784.86) = D7 (next D)
Again, the major chord corresponding to the note played, in this case, G, is present in the sound wave: G B D! That’s the G major triad! And the 2nd inversion form, D G B, is present as well, just as in the harmonic series for C.
If that weren’t fascinating enough, every pitched musical instrument (as opposed to a non-pitched instrument such as a snare drum) produces harmonics based on this same harmonic series since they’re all producing integer multiples of the fundamental of any note played.
What differs from instrument to instrument is the amplitude or intensity of the overtones. For example, a flute has a very strong 2nd harmonic, while a recorder’s 2nd harmonic is so subdued, it’s almost missing. Consider how very different these two wind instruments sound. The materials used, the mouthpiece, the length of the tube etc all impact the sound wave produced and the harmonics therein. These and other variants are what create the unique timbre (sound colour) of every instrument.
While most musical instruments produce the complete harmonic series, some sound only the odd or even-numbered harmonics. The clarinet, for example, in its lower registers essentially sounds only the odd-numbered harmonics, but sounds both odd and even in higher registers. Nevertheless, the 1st, 5th and 3rd notes (also known as the tonic, dominant and mediant, respectively) of the major key corresponding to the fundamental note are all present within the sound waves of all musical instruments because these fall on both odd and even harmonics early in the series. These three tones comprise your major triads!
Let's take a look the notes corresponding to the first 10 harmonics in the harmonic series. We'll use a C as our fundamental note:
C Major Scale: C D E F G A B C
tonic mediant dominant
Harmonic: 1 2 3 4 5 6 7 8 9 10
Corresp. Note: C C G C E G B Flat C D E
There's simply no missing the C major triad, C E G! It falls on both odd and even-numbered harmonics.
So virtually every time you play a note on an instrument, you are also playing a major chord! We hear it as one note, one complex sound, but it’s there in the background, adding depth and colour to the note. And it’s not alone! We’ve only covered the first 10 harmonics. The 7th harmonic, as you can see, is a whole other story in itself with the twists and turns and potential that it injects into music! It teams up with the major triad to form another chord, a dominant 7th chord: C E G B Flat, also known as C7, which belongs to the key of F Major, a key closely related to C Major. That's only a simple peak into the 7th harmonic, which is a far more complex issue, due, in part, to the constraints of instrument tuning methods.
And the story doesn’t end there! There are potentially an infinite number of harmonics in the series, but they diminish in volume and perceptibility as they increase in frequency. This is why some of the lower notes on the piano, for example, have a richer, fuller sound than some of the higher notes, because their harmonics are more pronounced due to their lower frequency.
Modern science affirms what our ears, for millennia, have intuitively and progressively detected as sounding harmonious together! The science is simply exposing why: because in every note of every instrument the foundation of harmony has been laid in the sound wave itself by the presence of the overtones. A major chord sounds harmonious because it already exists in the sound wave and powerful resonances and tonal relationships are occurring. In fact, you could say that music is all about relationships. We'll introduce some of those key relationships next time!
Harmony really is all in the wave! As for who put it there? Well ... the same One who hung the stars in the sky, I suppose.
Leonard Bernstein, The Unanswered Question - Six Talks at Harvard, 1973, Lecture 1: Musical Phonology
Michigan Technological University, Physics of Music -Notes, https://pages.mtu.edu/~suits/Physicsofmusic.html
Glenn Elert, The Physics Hypertextbook; Chapter: Music and Noise, https://physics.info/music/