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Sound and Music 2

February 1, 2015               Archives

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When a sound is produced it is not just the single sine wave discussed in the previous article, but rather a rich plethora of ancillary tones musicians refer to as overtones. Those overtones a musician is interested in are called harmonics. The harmonics are a proportional set of sounds relating to the lowest tone generated by the sound producing agent. This lowest tone is called the fundamental tone of the pitch. This series of proportional tones is known as the harmonic series. Whatever the fundamental frequency of the musical tone, (since this is about music, I'll refer to tone and pitch as a musically generated thing) there are multiples of this fundamental generated.

If a musical fundamental is 100Hz (this was chosen for arithmetic ease), then there will be overtones generated on increasing integer (1, 2, 3,...) multiples. So there will be, as part of the tone, vibrations not only at the fundamental 100Hz, but also at 200Hz, 300 Hz, 400Hz and so on.

One more concept needs to be introduced - the octave. The octave of a note is simply the double of that note. From our above example, the octave of the 100Hz pitch is 200Hz. The octave of the 200Hz pitch is 400Hz. This doubling continues at 800Hz, 1600Hz (1.6kHz), 3.2kHz, 6.4kHz, 12.8kHz, and 25.6kHz. Now this continues, but each successive overtone has less amplitude than the last, and the 25.6kHz is at the upper end threshold of what the human ear is capable of hearing, leaving the continuation of this process with little practical value.

Back to our example. The graph below has both the overtones of the 100Hz fundamental pitch along with the first four octaves:

overtone series

Notice that the first overtone at 200Hz is the octave of the fundamental pitch of 100Hz. As the octaves double in size notice that there are an ever increasing number of overtones between the octaves. The 300Hz overtone is between the 1st and second octave, the 500Hz, 600Hz and 700Hz overtones are between the 2nd and 3rd octaves and so on. (This is actually a geometric progression of 1, 2, 4, 8, 16.....-not essential to this discussion, but none the less interesting!). As stated earlier, some of the overtones are not musical and are not included in the musical harmonic series. The reason for this is really how our ears perceive these overtones as pleasing and therefore musical. This has historical significance in the study of music and how intervals are considered either consonant (at rest) or dissonant (feels as if another harmony should follow - again very subjective). Below is another graph that shows the relationship of the harmonic series and how the various musical intervals used in Western music are derived.

musical intervals from harmonic series

This actually follows the many vibrating patterns found on a musical instrument string. It actually vibrates as a whole unit, and then in two equal portions, and then three and so on. Visually, plucking an open string on a guitar, the string would vibrate on the fundamental pitch of the string plucked and its harmonics. The fundamental and first three harmonics (to save space - the other harmonics from above are also active) as they vibrate on the string are shown in the diagram below.

guitar harmonics

Next: The Harmonic Series on the Grand Staff



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