Chapter II: What are Harmonics and Overtones?
Harmonics, also called Overtones, are a physical phenomenon that applies to every kind of sound. As we explored in detail in the previous chapter, sound is basically vibration. A single vibratory source has, given the right conditions, the capacity to produce not only its fundamental tone, but also other subtler versions of itself and manifest a series of many different sounds ringing at the same time! Any vibrating wave tends to also vibrate at multiples of its fundamental frequency. These we call harmonics, and they are produced at remarkably accurate mathematical multiples of the base tone.
Figure 1: "Moodswingerscale" by Y. Landman: Depiction of a Fundamental Wave (top) and the Harmonics 2 - 7 (rest)
The vibration multiple has a smaller wavelength and thus proportionally a higher frequency. Because these smaller wavelengths need to fit exactly inside the fundamental frequency to be allowed to resonate with it at all, only a whole-numbered multiple of harmonic waves with respect to the fundamental wave can be produced while the fundamental tone is ringing. Anything not an exact multiple simply does not resonate, and thus never comes into existence. This we can call a quantum property of harmonics, since they are Quanta, or entities occurring only at specified intervals.
We can also talk of a fractal property in the formation of overtones, since all the harmonic waves are kind of replicas of the fundamental in ever decreasing length and amplitude. Fractals occur regularly in nature, but most of them are classified as self-similar. Examples of such fractal structures are leave formations, where each branch of a single leaf resembles the whole leaf. Other examples can be found here. Only in rare instances do we find naturally occurring, self-identical fractals. The intonation of all octaves of a tone is strictly the same in all scales, thus making it a self-identical fractal. You can think of a man and a woman singing the same tune, where the man sings an octave below the woman. Both are singing different tones, but the exact same tune.
Figure 2: An artistic representation of a Fractal - Each branch of the figure resembles the whole
Harmonics can also be seen and measured, using a common digital equalizer. They appear as peaks of energy along the horizontal frequency axis. One can confirm every time, that they occur precisely at the mathematically predicted frequencies. Anyone can learn to produce and experience overtones, here's a simple experiment:
Some traditions, like the Tibetan and the Mongolian, have developed a kind of singing technique, which is very rich in overtones, and which they embrace in their spiritual practices. In India, the tambura is considered the holiest of instruments, because its particular construction enables it to produce enormously rich harmonics. The sound of the tambura and its beautiful overtones is presented in the Sampling Station at the bottom of this page. People all around the world and throughout history have interpreted the musical harmonics as a way not only to raise the pitch, but raise the soul along with it.
The allegory of overtones to explain human spiritual consciousness is quite appropriate, indeed:
You start off with a single tone, which represents the individual consciousness. Most are only aware of this stage, disconnected and blind to a whole sphere of higher harmonics that are undoubtedly present, but not yet in the familiar realm of perception. As you train yourself to listen carefully, you slowly start to become aware of more. In fact, the first time people actually recognize an overtone, they claim that they had always heard it, but didn’t think about it in that way, or didn’t know that is what it actually was. It is a change of perception, not a change of reality. A similar process happens when one makes the practice of meditation or deep prayer a habit, where you “listen carefully” to the subtle voice of God and slowly grow in awareness of your reality, your true human condition.
The similitude continues as you notice that the higher you go in the harmonics, the more harmonics appear. As illustrated in Figure 1 above, this is a very physical property of overtones. In the first octave above the fundamental tone, you only have one overtone, namely the same tone, but an octave higher. This could be seen as the first large leap one has to overcome to understand that there is more to our limited perception of reality; that there is a higher version of yourself. In that sense the first overtone represents your Overself. In the second octave above the fundamental, two overtones appear: the note that bounds the interval of a fifth, followed by another octave of the fundamental. In the third octave more overtones come into being, namely the note that bounds the major third, followed by another fifth, then the minor seventh and finally yet another octave. In the following octave even more notes become perceptible: the major second, the tritone, the sixth and the major seventh, as well as higher versions of the third, the fifth, the minor seventh and the octave. By the time we go to the fifth octave above the fundamental we start perceiving notes that even go beyond the western duodecimal scale, and if we continue the process we will eventually have all possible frequencies manifesting simultaneously. If you were to start with a different fundamental frequency and if you went high enough in the overtones, you would get the same spectrum of frequencies in the highest harmonic spheres for any fundamental tone.
You can see how we literally moved from the individual to the collective to the realm of absolute wholeness, where all individuals share the same potential of becoming the One. You can also see that this reality, which was initially obscured, becomes accessible through the practice of simply listening carefully. You can train yourself to become aware of harmonics. Listen to our monochord recordings and try to become aware of as many overtones as possible!
Here are some more interesting facts about the harmonic series:
- All the necessary intervals to build the 12-semitone western scale are found in the first 4 octaves of the fundamental tone
- The number of harmonics (i) increases exponentially with each octave (n) at the rate of i = 2(n-1)
- This means that by the 21st octave there are over a million harmonics per single fundamental tone, and by the 31st the number of harmonics reached over one billion
- The average person cannot hear the difference in pitch between successive harmonics of any given fundamental tone from the 8th octave on
- Starting with a fundamental tone C = 512 Hz, the complete visible spectrum would correspond to the 21st octave . This means that if you could build an instrument, which produces strong-enough harmonics up until this octave, you would actually see them manifest as rays of light of every possible color
- The harmonic series is inherently "major" in tonality. This is because of the strong presence of the major third (the subdominant) from the 3rd octave on (5th, 10th, 20th harmonic, etc.). The minor third interval does not show until the 5th octave with the 19th harmonic. By this octave the energy present in the harmonics is already very low, which makes these harmonics barely noticeable
Harmonics have had a crucial role in the development of music throughout time and cultures. The notes of a musical scale are all derived originally from the harmonics, Pythagoras being the first one in the western world to have documented the phenomenon. In the next chapter we will discuss how the musical intervals were derived from the observation of this wonderful reality.