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Overtone Series: Harmonicity and Anharmonicity as the Foundation of Harmony


WHAT IS THE OVERTONE SERIES?



The beginning of our Harmonic journey starts with Physics.


The first notion of this acoustic phenomenon started in Ancient Greek with the famous mathematician, Pythagoras. He discovered the mathematical ratio when a string was "cut" into two parts, two thirds, etc. A Mathematical proportion exists in every string. Pythagoras used a "Monochord" that can be seen below:




https://en.wikipedia.org/wiki/Monochord



Let us compare with something more visual: the sound acts in the same manner than light. Imagine a white light, refracted by a prism. What happens? You will be able to see all the colors that conform to this white light. The perfect example of this phenomenon is the rainbow.


We can see the refraction of a beam of like through a prism in the picture below:





https://en.wikipedia.org/wiki/Prism




We can compare this refraction of the light to a sound being "refracted" in many smaller sounds, called "partials" or "overtones". These sounds coexist with the main sound we perceive as being the main one, we can says on a "vertical plane", but we can't hear them as separated from the main tone as they are much softer than the lowest one.



Now, the example is through a string or "Monochord" through a technique called "harmonics". This is very common in any string instrument, either the guitar, Violin, Viola, Cello or Double-Bass.



The graph below shows the mathematical proportion of overtones. On the bottom one, we have the fundamental or first harmonic, then the second appears which is half of the first one and the frequency is one octave higher than the bottom one. The first one divides the string into three parts and the frequency that corresponds with that sound is a perfect fifth higher than the second partial:



From the book "The Elements of Music: Melody, Rhythm and Harmony" By Jason Martineau





PARTIAL, HARMONIC, FUNDAMENTAL, ANHARMONICITY And OVERTONE



- A Complex Tone:


The sound of a note with a timbre particular to the instrument playing the note, "can be described as a combination of many simple periodic waves (i.e., sine waves) or partials, each with its frequency of vibration, amplitude, and phase." (See also, Fourier analysis.)




- A Partial:


It is any of the sine waves (or "simple tones", as Ellis calls them when translating Helmholtz) of which a complex tone is composed, not necessarily with an integer multiple of the lowest harmonic.




- An Overtone


It is any partial above the lowest partial. The term overtone does not imply harmonicity or anharmonicity and has no other special meaning other than to exclude the fundamental. It is mostly the relative strength of the different overtones that gives an instrument its particular timbre, tone color, or character.


When writing or speaking of overtones and partials numerically, care must be taken to designate each correctly to avoid any confusion of one for the other, so the second overtone may not be the third partial, because it is the second sound in a series.



https://en.wikipedia.org/wiki/Harmonic_series_(music) (Source)



If you would like to know more about Harmonic series, you can visit this website;



https://www.oberton.org/en/overtone-singing/harmonic-series/




A pianist can make an experiment if they have an acoustic instrument:



The pupil can partially demonstrate the phenomenon of overtones for himself by silently depressing the keys c', a', g' and then forcefully striking C once, staccato, or the lower octave contra C (all without pedal).

Then he can hear the tones c', e', g', the overtones, whose sound is similar to that of the harmonics of stringed instruments (Arnold Schonberg, Treatise of Harmony. page 20).



You can find more information at the beginning of this documentary from minute 5' 30'':



Howard Goodall Big Bangs 2 Equal Temperament


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OVERTONE SERIES GRAPH





We can see above the full extension upon a low C note or C1.

The higher harmonics (above number 20) are less than a semitone away from each other, creating quarter tones and octaves of a tone.


In the overtone series, which is one of the most remarkable properties of the tone, there appear after some stronger sounding overtones several weaker-sounding ones. Without a doubt, the former is more familiar to the ear, while the latter, hardly perceptible, is rather strange. In other words: the overtones loser to the fundamental seem to contribute more or more perceptibly to the total phenomenon of the tone -tone accepted as euphonious, suitable for art - while the more distant seem to contribute less or less perceptibly. But it is quite certain that they all do contribute more or less, that of the acoustical emanations of the tone nothing is lost.

And it is just as certain that the world of feeling somehow takes into account the entire complex, hence the more distant overtones as well. Even if the analyzing ear does not become conscious of them, they are still heard as tone "color."


That is to say, here the musical ear does indeed abandon the attempt at exact analysis, but it still takes note of the impression. The more remote overtones are recorded by the subconscious, and when they ascend into the conscious they are analyzed, and their relation to the total sound is determined. But this relation is, to repeat, as follows: the more immediate overtones contribute more, the more remote contribute less.


Hence, the distinction between them is only a matter of degree, not of kind.


They are no more opposites than two and ten are opposites, as the frequency numbers indeed show; and the expressions 'consonance' and 'dissonance', which signify an antithesis, are false. It all simply depends on the growing ability of the analyzing ear to familiarize itself with the remote overtones, thereby expanding the conception of what is euphonious, suitable for art, so that it embraces the whole natural phenomenon. (Treatise of Harmony, Page 20).


Related to the previous paragraph in which Arnold Schoenberg in his book states that the consonance and the dissonance is not a matter of polarity, like white or black , or good or bad; but as a matter of degree, as the higher harmonics are also part of the main sound, just into a lesser degree, but they are still the same kind, the same thing.


This late statement lead us again to the beginning of the page, which is the refraction of the light, in which we clearly see that all the colors ARE the white light, so the difference between, let's say, the color blue and red is not a matter of kind, as all the colors are the same kind, all are light, but of degree as the colors, as the musical notes, have something in common; they are classified in frequencies as well, so no color is considered to be opposite to any other color, just in the same manner the notes of a chord should be considered as part of a whole, which in music is the overtones of a sound.


He says: "I will define consonances as the closer, simple relations to the fundamental tone, dissonances as those that are more remote, more complicated. The consonances are accordingly the 6th overtones, and they are the more nearly perfect, the closer they are to the fundamental. That means, the closer they lie to the fundamental, the more easily we can grasp their similarity to it, the more easily the ear can fit them into the total sound and assimilate them, and the more easily we can determine that the sound of these overtones together with the fundamental is 'restful' and euphonious, needing no resolution. The same should hold for the dissonances as well. If it does not, if the ability to assimilate the dissonances in use cannot be judged by the same method, if the distance from the fundamental is no measure of the degree of dissonance, this is, even so, no evidence against the view presented here. For it is harder to gauge these differences precisely since they are relatively small."




HARMONICITY

Overtone Series Through History Timeline



Below we can see how the awareness of the harmonic series was "accepted" aurally through history:




This graph shows how upon a fundamental or first harmonic or partial, humanity introduced the partials contained within a sound.


Now, what this has to do specifically with chords? If you see the first five harmonics or partials, you will notice a triad.

In the example above, The C is the root, the G the fifth of the chord and the last one that appears is the fifth partial, which is the third of the trial. The order in which the partials appears, and subsequently on a chord, tells us about the "Spacing" of the chord that we will cover in the next lesson.

If we follow the natural order of these notes, we will know how to space them through an instrument, for example, the Piano or the Guitar, or different ensembles, such as a string quartet, a Woodwind quintet and even a whole orchestra. The same concept applies.


As a rule of thumb, space more the lower notes of any chord, and allow smaller spaces as you go high in pitch, following the pattern of the overtone series.




WKMT Lesson Video




*Do not miss my previous post, Historical framing of the XV Century music.

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© 2012 by Juan J. Rezzuto. All the tracks, scores and articles you can find in here are copyright.