Overtones and harmonics

We have discussed the circle of fifths as a way in which the various pitches, major keys, and minor keys can all be arranged in a circle. But certainly there must be something else special about the interval of a perfect fifth to make it a worthy basis for all of this discussion. Sure, the perfect fifth (or its inverted mirror image, the perfect fourth) is the only interval other than the boring half step that you can make a circle out of. But why? What would cause someone to decide to do this? Was the circle of fifths invented by a mathematician-turned-musician-turned-clockmaker whose business was slow and needed some way to pass the time?

Well, there is a sense in which keys (and chords, as it happens) that are near each other in the circle of fifths are closely related. But why? We can begin to guess why the human ear hears such things (or at least ears that are accustomed to Western music) by examining the way that things vibrate.

The vibration of things

If you had a thick rope, or perhaps a long old-fashioned telephone handset cord (its little helical circular squiggles make it sort of rope-like) you could hold one end still (this was easy in the old days when the telephone was generally attached to the wall) and wiggle the other end. If you wiggle it at a certain speed, the whole rope or cord will vibrate up and down together. It’s as if the rope/cord has a natural “resonant” speed that it wants to vibrate at.

Helical telephone handset cord

Helical telephone handset cord

It turns out that the cord wants to vibrate at other speeds as well, in different ways. If you wiggle the end twice as fast as the original resonant speed, it will vibrate in halves. The middle of the cord will stay still, and the two halves will vibrate in opposite directions. You can wiggle the end three times as fast as earlier and get the cord to vibrate in thirds as well. (This was a convenient way to pass the time when stuck in a monotonous telephone conversation.)

In the picture below, imagine the string vibrating between the red and blue positions in the three styles of vibration shown: all at once, in halves, and in thirds.

A string vibrating all at once, in halves, and in thirds

A string vibrating all at once, in halves, and in thirds

Now, it happens that if a shorter string is vibrating (or a smaller thing of any sort), all other things being equal, it will tend to vibrate faster and produce a higher note. (This is the reason for the shape of grand pianos and harps; the strings for the higher notes get gradually shorter.) So when the end of the cord is wiggled twice as fast and the string vibrates in halves, each piece of the string that vibrates is half as long, so this follows the above rule. In fact, things generally “like” to vibrate at some basic speed, or frequency (corresponding to the whole-string vibration) and also at even multiples of that frequency (corresponding to the faster vibration in halves, thirds, quarters, etc.).

Ratios and hearing intervals

It so happens that we hear a certain ratio of frequencies as a certain interval. What does that mean? Well, for example, if you hear something vibrating at a certain speed, and then something else vibrating at twice that speed, the second thing will sound one octave higher than the first. This means that if you sat at the piano and hit all the Cs, starting with C1, and then C2, and then C3, and so on up to the highest note on the piano, C8, each note you play has its strings vibrating twice as fast as the strings did for the previous note.

You may wonder what sorts of intervals you may hear with the other ratios of frequencies. Starting with a general disclaimer that in the usual Western style of tuning used nowadays, the pitches may not be precisely in tune with those written here, the picture below shows you the answer (up to multiplying the frequency by 8):

Harmonic series, multiples of a fundamental frequency

Notes with frequencies multiplied by various numbers

As expected, doubling the frequency gives an interval of an octave. But when you triple the frequency, the interval from the original bottom note is an octave plus our old friend, the perfect fifth! Let’s keep going: quadrupling gives you a note two octaves higher (it would have to, since quadrupling is equivalent to doubling something twice). But quintupling gives us a note two octaves higher plus a major third. Multiplying by six goes up two octaves plus a perfect fifth (same as doubling and then tripling). Multiplying by seven goes up two octaves plus a minor seventh (it happens, though, that it’s a rather out-of-tune minor seventh).

Points of interest

First, it happens that the notes C-E-G are the notes in a C-major chord, and those are the first notes that you see in the picture (although they appear in the order C-G-E, or, if you count duplicates, C-C-G-C-E).

Second, there is something very basic about the fifth for it to appear so early in this series of notes. If you were to look at pieces in the key of C, by far the most common final notes in the bass part are G-C. The last bass note is almost always C, and the chord is almost always C-major, because then you’re “home” in your comfortable resting place for the key of C. The penultimate bass note is usually G because that note suggests being “almost home”; that G wants to go down to a C, and it’s satisfying to the ear when it does so.

Third, it’s common to space the notes in a chord very much like in the series of notes, with bigger intervals near the bottom and close-together notes at the top. If you asked some piano player out of the blue to sit at the piano and play a C-major chord, it’s probable that he’ll just naturally play an octave C in the left hand, and play the other notes in the right hand, as shown in “Example 1” below. Depending on his mood, he might sometimes decide to put a fifth on the bottom as in “Example 2” below. But for him to put a bunch of close-together notes at the bottom as in “Example 3” would be quite unnatural. If this happens, then the jig is up; he’s just messing with you because he knows you’re looking to see how he’s going to play that chord.

Ways to play a C-major chord

Ways to play a C-major chord

Vocabulary

Several words are used to describe this particular series of pitches. The bottom note (in this case, the low C) is called the fundamental. The other notes are called overtones and the whole bunch of them together are called harmonics or partials.

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