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The Twenty-First Century
by Don Robertson

2005 Rising World Entertainment

Part Three: The Overtone Series

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"The German musicologist Wilfried Kruger discovered that the eight electrons of the oxygen atom shell and the eight protons of the nucleus of the oxygen atom generate a major scale with the spins of the particles delineating the half tones and whole tones. According to the work of the famous astronomer, Johannes Kepler, certain proportions of the elliptical orbits of the planets correspond beautifully to the simpler relationships naturally found in the overtone series. These same harmonic relationships are found not only throughout nature but also at the basis of the most successful human constructs such as classical Greek architecture."
                                                                      From Paul Stokstad

The Overtone Series

The overtone series, also called the harmonic series, is a naturally occurring phenomenon that reflects the inner laws of creation and is the very essence of music. It is a series of notes that looks like this:

    The first note in overtone series (the lowest one) is called the fundamental. The remaining notes are the partials (also called harmonics, or overtones)
     In this drawing, I assigned a low C note to the fundamental; the series can begin with any note, however. The next note is another C, but this C is one octave higher than the fundamental. Following that is G, then another C that is two octaves above the fundamental. Next in the series is E, then another G, a Bb, and then a scale from C up to the next C with an F# and two Bs: Bb and B natural, but these notes aren't tuned exactly as they would be on a piano.
     The overtone series is the audible example of natural law, the law of our solar system, and the basis for the science of acoustics. Because it defines the very nature of sound, it has always been the basis for all of our scales and chords.

1 - The overtone series is inherent to the nature of the vibrating strings of musical instruments.

2 - The overtone series is the basis of sound that is produced by woodwind and brass musical instruments.

3 - The overtone series is what defines timbre


     A simple instrument called a monochord, said to be invented by Pythagoras somewhere around 550 BC, was used by theorists for centuries to demonstrate the relationship between string length and the notes produced by a vibrating string. The monochord is extremely simple in design. A string, such as a guitar or violin string, is attached at both ends and pulled taunt over a resonating body with a moveable bridge underneath. When the string is plucked, it vibrates at a rate that is directly proportional to the length of the string. However, additionally, it simultaneously vibrates in particular proportional divisions of its length. In other words, an additional vibration occurs at 1/2 the length, another at 1/3 the length, another at 1/4, 1/5, 1/6, and so on: proportional distances. Each of these vibrations produces a harmonic, or overtone that has a frequency that is inversely proportional (2x, 3x, 4x, 5x, 6x, etc.) to each division of the string. 
     These are the members of the overtone series, the partials, harmonics, or overtones. Starting at the fundamental, the loudness of each harmonic gets softer and thus, the higher overtones are so soft that they are difficult to hear. When we pluck the string, our ears hear the resultant sound as a collection of the fundamental and its overtones: this is what makes the sound of that particular string different that, say, a pure sine wave that only contains the fundamental. The order, tuning, and volume of the overtones are what make the notes played by, say, a brass instrument sound different than those of, say, a flute, or a violin.
This is called timbre.
     On stringed instruments, the fundamental can be dampened out when the performer applies a light pressure at one of the proportional locations on the string, thus revealing the rest of the sound that was vibrating as part of the pitch, and thus creating a sound at a higher frequency. To get a sound an octave higher, the musician lightly touches the string at its midpoint. This same light fingering can be applied at 1/3, 1/4, etc. of the string length to create higher and higher overtones. Simply pressing the string to the fingerboard at these positions wouldn't create the same note, nor the same sound, as the overtone, or harmonic (as it is usually called in this case). This technique is called for in 20th century and late romantic orchestral scores for strings and harp, and is also one of the acoustic guitarist's favorite tricks.


     On the monochord, or any string instrument, the fundamental note is determined by the length of the string. For brass and woodwind instruments, the pitch of the fundamental depends upon the length of the tube that comprises the instrument itself. The player overblows (by tightening the embouchure and blowing air faster) to produce overtones. Woodwind instruments have keys that open up holes in the tube, thus changing its length, and hence, the note. Before valves were invented (which mechanically change the length of the tube on brass instruments), trumpet, and horn players could only play the lower overtones (and in the case of the clarion, or clarino trumpets of Torelli, Perti, Cazzati, and the composers at San Petronio in Bologna the full 'upper scale'), and sometimes the fundamental, so these brass parts had to be written that way, playing simple chord notes.
    Trombonists change the length of the tube by means of a slide. A brass instrument with no valves, like the bugle and the post horn, can create only the notes of the harmonic series, and that is why bugle and post horn melodies are use only the three or four notes lower overtone notes.


    Our ears tend to blend the fundamental and its overtones into a single sound that we call pitch. Rather than perceiving the many individual harmonics of a musical tone, we perceive a tone color, or timbre, with a pitch that is of the fundamental. 
     It is the amplitude (loudness) and placement of the overtones that determines what the timbre, or sound quality will be. For example, clarinets sound only the odd numbered overtones, creating notes that have a purer timbre than those produced by a stringed or brass instrument. The strength the brass instrument's higher overtones makes the sound "brassy": a sound that is rich, but slightly dissonant. Additionally, not all musical instruments have overtones that exactly match the pure harmonic partials. The piano's overtones, for example are increasingly sharper than perfect harmonics because stiffness of the metal strings.

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