HARMONIC SERIES
While many harpers play harmonics on their harps, many do not really understand them. Since the theory of harmonics is a basic element of music, it seems right to share my understanding.
When a string vibrates, you hear a note. That note is called the Fundamental. Only tuning forks produce solely the Fundamental. This would be called "pure sound".
Most musical instruments do not produce pure sound. We actually hear more than just the fundamental. Take that vibrating string, for example. It is not just vibrating from end to end, which would give us pure sound. It is also vibrating on either side of a "stationary" point in the middle of its length. It is vibrating on either side of other points at 1/3 of its length, at 1/4 of its length, and so on.
Think of waves in a pool of water. You've got the general swell of the body of water, which is analogous to the string vibrating along its entire length. And then you have the waves, and then waves on the waves, and then ripples in the waves on the waves, and so on. Musical sound waves act in much the same way.
Harpers know that if we lightly place our finger half way up the string and pluck, we get a different note than the Fundamental. What we are doing is eliminating the Fundamental (preventing the whole length of the string from vibrating) and reinforcing the 1/2 length vibration. We have played a HARMONIC. This particular harmonic just happens to be the Second harmonic.
(The First harmonic is the Fundamental.)
| The Fundamental is a harmonic, but it is NOT an overtone. SO: The FIRST overtone is really the SECOND harmonic. The words “overtone”, “partial” and “upper partial” are synonymous. |
To recap, we have the Fundamental, also known as the first harmonic, which is produced when we simply pluck our string. The second harmonic is clearly heard when we lightly stop the string half way as we pluck the string. That gives us the second harmonic, which is the octave above the fundamental.
What happens if you stop the string at 1/3 its length? You hear the THIRD harmonic, which is an octave-and-a-fifth above the fundamental.
Take the fourth harmonic (stop the string at 1/4 of its length) and you hear a tone two octaves above the fundamental.
A couple of interesting things have been established here. First, the 4th, 5th and 6th harmonics are a major triad. The major triad EXISTS IN NATURE! It doesn't matter if you are playing guitar, trumpet, harp, flute, the Great Highland Bagpipes, or beating on a tree branch in your backyard: the upper partial series is always the same. There is a mathematical formula underlying the series.
Another interesting observation to be made is that the fifth and octave occur a lot. Naturally!
The harmonic series is generated by principles of acoustical physics and is not bound by our preferences for modern equal temperament tuning. In the series, only the octaves are perfect intervals. Most of the other harmonics are what we call "just intonation".
Three of the Harmonics in the series are not in any of the tuning systems we ordinarily use. Specifically, these are the first two "accidentals", the b flat and f#, and A, (the seventh, eleventh and thirteenth harmonics, respectively). In other words, the b flat, the f# and that A are "approximations" of what we hear. In fact, in the equal tempered tuning system, the f# is actually closer to f than it is to f#.
Below is the harmonic series again, this time built on a different note than C. It is presented here because it is important that it be understood that every note has its own harmonic series. The intervals are all the same, but obviously come out on different notes when the fundamental or first harmonic is different. Look and see the major triad: still on the 4th, 5th and 6th harmonics.
The Harmonic series is important because it determines what your instrument sounds like, providing its timbre or tone.
For example, the flute produces a nearly pure sound. That is to say, it is lacking almost all the harmonics except the first, the fundamental. The oboe, on the other hand, has nearly all the harmonics present in its sound. The clarinet has only the odd numbered harmonics. My wire-strung harp most certainly has a different collection of strong harmonics then does a gut-strung harp. It depends on how the strings, the body, everything: how they vibrate as a whole and in sections of 1/2; 1/3; 1/4; etc.
Now that you know all this, can you explain why your electronic tuner sometimes tells you the C you just plucked is tuned to G?
© Cynthia Cathcart 2001. First published in the 3 January 2001 issue of the Newsletter of the Wire Branch of The Clarsach Society.
