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The Harmonic Overtone Series

The harmonic overtone series is a mathematical property of a vibrating string - also of other vibrating objects such as bells, columns of air contained by wind instruments such as trumpets and flutes, and even drums - but being guitarists we're going to talk about vibrating strings.

Pythagoras, the great Greek mathematician and music theorist, did the basic work on the properties of vibrating strings in about 500 BCE, so there is nothing new here. It's even possible that Pythagoras was only passing on something he had learned in his early studies in Egypt, and that these ideas are much older than his own.

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A vibrating string, as well as vibrating along its full length from one end to the other, has an infinite series of smaller wave patterns contained in the fundamental pattern. These smaller patterns all divide the string into smaller and smaller sections. The sections are all equal to fractions - with whole numbers as numerators and denominators - of the total length of the string. The largest divisions are the strongest contributors, by and large, to the overall sound, but the variations in the relative strength of the various overtones contribute to the unique tone of different instruments.

In the case of a vibrating string, the string must be of completely even density, thickness and roundness in order to vibrate with a pure tone and with pure harmonic overtones. A string which is uneven in its density or dimensions will of course vibrate unevenly, and the mathematical purity of the overtone series will be disturbed. When you leave your strings on your guitar for months or years on end, they become deformed or worn where they have been pressed against the frets, even when there is no obvious breakage in the surface, and they vibrate more unevenly and are harder to tune, and to play in tune, as a result.

The Series

We will take the A string of the guitar for an example; since it vibrates at 110 hz, this is an easy frequency to multiply, and it's easy to understand the resulting mathematical patterns. You can get your guitar out and play the harmonics as you read.

If you touch the A string very lightly with a fingertip at exactly the midpoint of the string, and then pluck the string with your fingernail near the bridge, you will hear a sound usually called a "harmonic" (the full technical name being "harmonic overtone".) The midpoint of the string is exactly above the 12th fret, and the harmonic will produce the same sound as stopping the note at the 12th fret. When you touch the string lightly at that point, the string continues to vibrate on both sides of your finger, but the vibrating pattern which goes the entire length of the string (the sound of which we call the "fundamental tone") is interrupted, and now we hear only the sound of those vibrations which are 1/2 or less the length of the fundamental pattern. The pattern which vibrates for 1/2 the length of the string is the strongest of these, and is the one which is now heard. The frequency of this sound is 220 hz, twice the frequency of the fundamental tone, and it is exactly one octave higher. The ratio of the harmonic to the fundamental tone is 2 to 1. All multiples of the original frequency which belong to the series 2, 4, 8, 16, 32 etc., will produce octave multiples of the fundamental tone, and they will all have the name "A" - i.e., A-110, A-220, A-440, A-880, A-1760, etc., and their ratios to the fundamental tone will be 4-1, 8-1, 16-1, etc.

Next we will touch the string lightly at the one-third point. This point is directly above the 7th fret. (There is also a 2-thirds point at exactly the 19th fret, which is identical from the point of view of the string's vibrations.) If we pluck the string with a fingernail close to the bridge while touching the string at this "node", we hear a harmonic tone which is 3 times the frequency of the fundamental tone. Its pitch is "E", its frequency is 330 hz, and it is one octave and a fifth above the fundamental tone.

Now the interesting property of this second harmonic is not so much its relationship with the fundamental tone, but its relationship with the first harmonic which we obtained by touching the string at the halfway point. These two harmonic overtones have respectively the frequencies 330 hz and 220 hz, and therefore a ratio of 3:2. This ratio defines a musical interval call the "perfect fifth" which is universally recognized in music around the world, along with the octave (ratio 2:1). It was this 3:2 ratio that Pythagoras used in his system of tuning the notes of the scale which is now called Pythagorean Tuning (but which has fallen out of use in favor of a newer system called "Tempered Tuning".)

(Tempered tuning uses a slightly smaller ratio to tune the fifths, about 2.996:2 instead of 3:2 - this is one of the peculiarities of modern western music.)

The next several harmonic overtones give us our most basic musical intervals. These are intervals which are recognized by many musical cultures around the world, precisely because they are derived from these very basic mathematical and musical properties of vibrating strings.

To obtain the 3rd harmonic, we touch the string lightly just exactly above the 5th fret, while plucking with a fingernail close to the bridge. This harmonic is 4 times the original frequency, or "A-440". Its ratio to the previous harmonic is 4:3, and the musical interval between the two is called a "perfect fourth". This is another very basic musical interval recognized by all musical cultures.

The 4th harmonic (which will be 5 times the frequency of the fundamental tone) is NOT directly over a fret. (The frets are placed according to a different mathematical formula, and the resulting pattern intersects the pattern of the harmonic overtones ONLY at the octave, 4th and 5th above the fundamental tone, which is to say, at the 5th, 7th and 12th frets.) We will find the node for the 4th harmonic near, but not directly above, the 4th fret. Since the vibrating pattern which we are intercepting at this node divides the vibrating string into 5 equal parts, there are then at total of 4 nodes for this harmonic. You will find the 2nd one near the 9th fret, and the 3rd one near the 16th fret; the last you will find somewhere near the side of the soundhole closest to the bridge. The pitch of the 4th harmonic is C#. The ratio of the 5th and 4th harmonics is 5:4, and this interval is called a "major third". This major third is a pure result of the harmonic overtone series, and it is interesting to observe that it is somewhat smaller (or flatter) than the interval actually used in either Pythagorean tuning or modern tempered tuning.

The ratio of the 4th overtone to the second overtone is 5:3, and this interval is called a "major sixth". Again, the interval is not exactly the same as the interval used in either Pythagorean or Tempered tuning. This is true of all the remaing overtone-generated intervals that we will explore in this paper: although they are the true basis and origin of harmony, we use approximated versions of them in our modern tuning system which sound somewhat harsher than the true intervals, and we do so in order to make all the semitones have equal ratios each to the next, which would not be possible if the intervals were true to the overtone series. (You can do the math on this for yourself, referring to the chart of tempered pitches.) This is as true on the guitar as on any other instrument. The frets are laid out according to a ratio of approximately 18:17 each to the next, without consideration for the true intervals generated by the overtone series. This is necessary for the guitar to be able to play all chords more or less equally in tune.

The 5th harmonic is near the 3rd fret. Its pitch is E-660. Its relationship to the fundamental tone is 6:1, and since this is twice the ratio of the 2nd harmonic (3:1) the pitch is one octave higher than that of the 2nd harmonic. Its ratio to the 4th harmonic is 6:5, and this interval is called a "minor third". Its ratio to the 3rd harmonic is 6:4, and this is a perfect fifth. Note that the ratio is double that of the ratio 3:2 which is also a perfect fifth. Its ratio to the 2nd harmonic is 6:3, and this is an octave.

The sixth harmonic is between the 2nd and 3rd fret, but closer to the 3rd. Its pitch is G, but quite flat when compared to the tempered G. It has a ratio to the fundamental tone of 7:1. Its ratio to the previous harmonic is 7:6, which is also a minor third, but it is a somewhat smaller minor third than the one created by the ratio 6:5. Although the ratios are proceeding by exact proportions, it's one of the curiousities of our musical system that we do not make a distinction between the "minor thirds" of the ratios 7:6 and 6:5, although the difference is quite audible. In the "Procrustean Bed" of the tempered tuning system, there is no room for subtleties like the difference between these two intervals.

The ratio of the 6th harmonic to the 4th is 7:5, and this interval is a "diminished fifth". This is a pure and beautiful interval in the overtone series, and yet the corresponding interval used in tempered tuning is quite harsh. In western music, this is a very, very important interval, and yet, the interval that we actually use is not the same as the pure original produced by the harmonic overtones.

The ratio of the 6th harmonic to the 3rd is 7:4. This is a minor seventh.

Now we can play the 3rd, 4th, 5th and 6th harmonics all in turn. Interestingly, this sequence outlines a chord which we call the "dominant seventh chord", one of the most important chords in western music, with a very interesting history which we won't recapitulate here.

The seventh harmonic is between the 2nd and 3rd fret, and its pitch is A-880. Its ratio to the 6th harmonic is 8:7, and this interval is a major second. The ratio 8:6 is a perfect fourth; 8:5 is a minor 6th (close to the ratio 5:3 which gave us the major sixth.) 8:4 is an octave.

The 8th harmonic is near the 2nd fret. Its ratio to the fundamental tone is 9:1, and its pitch is "B". The ratio 9:8 gives us another major second. 9:7 is a major third; 9:6 is a perfect fifth; 9:5 is a minor 7th (close to the minor 7th of the ratio 7:4).

The 9th harmonic, with the ratio 10:1 to the fundamental, gives us another C#. 10:9 is a major second. Although the proportions of the overtone series continue exactly, note that now we have had three different intervals in a row: 10:9, 9:8, 8:7, for all of which we have no better definition than the single one of "major second". This shows how our tempered tuning system has become biased toward its own ends and away from the overtone series.

The 10th harmonic, 11:1, gives us a D#. The interesting interval here is the ratio 11:8, which is an augmented 4th. It's fairly close to the ratio 7:5 which gave us the diminished fifth. In the tempered system these two intervals are the same, but in the overtone series you can see and hear that they are different.

The 11th harmonic, 12:1, is an E. 12:11 gives us a half step. This is the smallest interval recognized in our tempered system. Every succeeding overtone after this will generate an even smaller interval. However, we are reaching diminishing returns trying to play them on the guitar, because the A string that we are playing on gets very stiff this close to the end, and the harmonics are very hard to hear and to play. However, you have seen and heard that all of the important intervals of the tempered scale are approximated by the relationships between the various overtones, and it is safe to say that the overtone series, being the natural property of a vibrating string (or an other regularly proportioned vibrating object) is the system that came first. You might say that it has always existed. Indeed, very similar sets of relationships are found in the color frequencies of light, in the periodic table of the elements, and in the relationships of the planets of the solar system.

Although the system of melody and harmony that we have created in the west, built from the tempered scale of 12 equal-ratio'd semitones, is enough to keep any musician studying and practicing for many years, we shouldn't pretend that this is the only valid or the only possible approach to music. A musical system which pays close attention to the subtleties of difference between consecutive intervals generated by the harmonic overtone series has the potential for expressive beauty of another level. So far, however, the complexity of such a system has discouraged the widespread use of any new system based on western harmony. We can, however, consider the Indian Raga system from this point of view - they speak of the "ten thousand" modes, and recognize subtleties of difference in tuning intervals that are very fine, and also critical to the expressive potential of the different ragas.

Last page update 03-16-08