Overtones

It is perhaps counterintuitive that when we hear musical notes played on instruments, we don't just hear those notes. Each note is actually a compound sonority, almost a chord, consisting of the note that you perceive and an infinite amount of overtones, or harmonics.

Overtones are not immediately apparent, but they're not elusive, and we can definitely tell when they're not there. Overtones are what separates music and noise. Sounds with indeterminate pitch have unevenly recurring harmonics. Musical notes have evenly-spaced harmonics. Sounds with no harmonics at all are called sinus tones. When the doctor checks your hearing, he plays sinus tones in those headphones.

The Overtone Series

According to the laws of physics, a string [1] does not just vibrate along its entire length, it also vibrates in halves, thirds, fourths and so on. These partial vibrations manifest themselves as overtones, a.k.a. harmonics.

All other things being equal, if you cut a string in half, the resulting note will be exactly one octave above the original string. That is, we will perceive it as the same note but in a different register.

If you cut the string in three equal parts, these will be an octave and a fifth above the original string.

What happens physically when you do this is that cutting the string in half doubles the vibrating frequency. Cutting it in three parts trebles the frequency. Put in other words, the frequencies of the overtones are in inverse proportion to divisions of the string. Half the string—double the frequency, and so on.

The notes derived from these partial vibrations can be arranged in a sequence, the harmonic overtone series. The following diagram illustrates the overtone series for the note 1C (corresponding to the lowest note on a cello, a major third below the low E on a guitar):

So, if you play 1C, you will actually hear all those other notes at the same time as well. And these are just the first 16 harmonics; the series continues indefinitely, but with progressively lower amplitude, and obviously, at 20 KHz, our ears no longer perceive any vibrations.

Note that some of the above notes are approximations. The seventh harmonic (B flat) is actually noticeably flatter than indicated, and the 11th harmonic (F sharp) is noticeably sharper.

Some sources separate the fundamental note from the harmonics, but it is more logical to look at all these notes as parts of a whole, where the fundamental note is considered the first harmonic, or first partial.

Frequency Ratios

Since we know the frequency ratio between the first harmonic and all the others (1:n), we can also quite easily find out the ratios between other harmonics as well. If you put numbers under each note, 1 through 16, you will know exactly how to divide a string and what frequency that division will yield compared to the fundamental note.

If you locate all the C:s on the harmonic series, you will find that they are found in slots 1, 2, 4, 8 and 16. If you convert this to ratios, you will get 1:2, 2:4, 4:8 and 8:16. All these ratios can be simplified into 1:2. It is the same case with the G:s, which are found in slots 3, 6 and 12; 3:6 and 6:12 can also be simplified into 1:2.

Consequently, there are perfect fifths between harmonics 2 and 3, between 4 and 6, between 6 and 9 and between 8 and 12. These are all equivalent to the ratio 2:3.

So, if you want to find out the frequency ratio of a major sixth, find the interval on the harmonic series. I can find two occurrences: G (harmonic #3)/E (harmonic #5) and G (harmonic #6)/E (harmonic #10). The ratio 6:10 can be simplified into 3:5.

The simplest frequency ratios yield the most consonant intervals: unison (1:1), octave (1:2), fifth (2:3) and fourth (3:4). This is why these intervals are referred to as the perfect intervals. The more complicated frequency ratios yield thirds, sixths, seconds and sevenths, which are the intervals that appear in two basic qualities: major and minor. This will be elaborated upon in the essay on interval theory.

Finding Overtones

As hinted at above, harmonics are not mysterious entities that only affect us subconsciously. They are very real and can be isolated pretty easily on a guitar or even a piano.

Natural Harmonics

Since you play a guitar by shortening the strings against the frets, the harmonics can be easily located by finding the appropriate frets. The 12th fret is in the exact middle of the string, so there is where you will find the second harmonic. Lightly touch the string (do not press down!) directly over the fretwire and play the note.

This method can be used for all other harmonics, except that the higher partials are difficult to bring out clearly because of their low volume and because they are pretty close together on the string.

For instance, the third harmonic is an octave and a fifth above the open string, i.e. the 19th fret (2/3 of the way from the nut to the bridge). But—since the third harmonic is a tripartite division of the string, there is a second node somewhere on the string. This node corresponds to the 7th fret (which is 2/3 of the way from the bridge to the nut).

There are three nodes for the fourth harmonic. These can be found at the 5th and 24th frets. The third would theoretically also exist at the 12-fret node, but this is cancelled by the much stronger second harmonic.

To summarize, with the 12th fret as a starting point, the higher harmonics can be found in either direction—towards both the bridge and the nut.

Artificial Harmonics

Artificial harmonics go by many names: false harmonics, pick squeals, you name it. It's the same general principle as natural harmonics, except that the technique is different. Instead of touching open strings at nodal points, you "pinch" fretted strings at certain nodes to generate overtones. This "pinch" is executed by striking the string with both the pick and the finger. To do this, you must hold the pick closer to the tip to make sure the fingertip (thumb or whichever other finger you hold the pick with) strikes the string almost simultaneously with the pick. The pick sets the string in motion, the finger immediately afterwards stops the entire string from vibrating, generating a harmonic.

Difference Tones

A difference tone is a psychoacoustic phenomenon where a third note is perceived when playing a two-note interval. The oscillation, or "beating" heard when an instrument is ever so slightly out of tune is also an aspect of this phenomenon.

To determine the pitch of the difference tone, simply subtract the frequency of the lower note from that of the higher note. In the case of the untuned guitar or piano, the beating stems from that the notes are close enough in pitch that you can actually hear the individual oscillations (one oscillation per second = 1 Hz). When playing actual musical intervals, under the right conditions, it can turn into a perceptible interval.

A unison has a frequency ratio of 1:1. This obviously means that no difference tone is generated, because 100 Hz minus 100 Hz equals 0 Hz.
The octave has a frequency ratio of 1:2. If the lower note is at 100 Hz, the higher note will be at 200 Hz. The difference tone will be at 200 Hz - 100 = 100 Hz, which is a perceived doubling of the lower note.
The perfect fifth, ratio 2:3, also has a consonant difference tone. If the lower note is at 200 Hz, the perfect fifth will be at 300 Hz. 300 - 200 = 100 Hz. The difference tone will be exactly one octave below the lower note.
The perfect fourth, ratio 3:4, is actually subtly different. If you subtract 300 Hz from 400 Hz, you will get a difference tone at 100 Hz. This note will be two octaves below the higher note. It will be an octave and a fifth below the lower note. [2]
Continuing from major thirds down to minor seconds, the smaller the pitch difference, the lower the difference tone.

1. Because of my guitar-centric musical world, I will consistently use the string as my example of choice. However, there are other vibrating bodies that have the same properties, e.g. the air pillar of a wind or brass instrument or the wooden blocks of the xylophone. [Go back]

2. This reinforces the notion that the fourth is an inverted fifth. Some music theories actually consider the fourth to be a dissonant interval, because it is so easily perceived as a suspended interval that requires resolution to a major or minor third. [Go back]