Restoration of the Missing Fundamental

acoustic/auditory phenomenon – music, mind and spirit     by Tim McKamey

     I ran across this in Daniel J. Levitin’s wonderful book This Is Your Brain On Music. But the 50 cent interpretation is my own. Sometimes it seems as if the Creator leaves us hidden metaphors within our very being that shout out to be understood on a deeper level. This is an example. It has to do with Overtones and the Harmonic Series.


     As Levitin points out in his book, “It is a natural property of objects in the world that they generally vibrate at several different frequencies at once.” So when we hear any fundamental tone played or sung, there are actually several additional tones occurring simultaneously. Most people don’t hear these, but they are there all the same and sometimes we can train ourselves to pick out a few.

Wave Forms of non-linear harmonic series

These  visualizations above and at right, represent the harmonic series of overtones that occur whenever a single fundamental tone is sounded. These overtones occur at regular intervals of whole integers. So for example, if we play C resonating at a frequency of 256 Hz, (meaning the string is vibrating 256 times per second) the first occurring overtone will be at 512 Hz, which is 256 added to 256. This happens to be the octave of C (C1). Add 256 again to 512 and we get 768 Hz which is G, the perfect fifth of C. Subsequently, each additional overtone in the harmonic series is produced by adding the original frequency (256 in our example) to the tone that precedes it. See the chart below:

Wave images of harmonic series

Vibrational modes of an ideal string, dividing the string length into integer divisions, producing harmonic partials f, 2f, 3f, 4f, etc… where f means fundamental frequency. In the chart above f=N.

Below are 16 harmonic overtones of C. Each interval representing an increase of 256 Hz:

Harmonic Series of C

Below is yet one more view of the same idea represented on a piano keyboard:

harmonics series on keyboard

So you get the picture? Actually sitting down at a grand piano with the lid open is a great way to experience overtones first hand (or first ear). Just press the C note and listen. You should be able to pick out 3 or 4 overtones vibrating along with that fundamental tone of C.

     Before I leap off into the deep end of the pool with all this, I should point out there is quite a bit of historical precedence for attributing mystical properties to this whole idea of overtones. Not surprisingly as often as not this tends to fall on “deaf ears”. Alexander J. Ellis for example points out in his translation of the classic work on acoustics by Hermann Helmholtz how a mistranslation of Helmholtz’s term for overtone may have introduced unnecessary confusion into an already complex subject. In the late 19th century, Helmholtz wrote the book that all subsequent western music theory is based on in terms of the science of acoustics,   On the Sensations of Tone as a Physiological Basis for the Theory of Music. Helmholtz used the term “Obertöne” which was actually a contraction of “Oberpartialtöne”, or in English: “upper partial tones”. Ellis thinks a Professor Tyndall may have mistranslated Helmholtz’s term Obertone to Overtone which in Ellis’ opinion brings mystical connotations into what is otherwise a purely scientific matter.

     Be that as it may, there is a far more ancient science of sound in classical Indian music (and elsewhere) that emphasizes the significance of  sympathetic vibrations and the inaudible tones generated by all audible tones. In the Hindu tradition ahata is the term for audible (or struck) sound, anahata is the term for inaudible (or unstruck) sound. For every ahata either sung or played on an instrument there are corresponding vibrations of anahata which permeate our being and affect change all the way down to the cellular level.

     In This Is Your Brain on Music, Levitin wants the reader to understand that while sound waves vibrate in the physical world, what we perceive as sound, ie, pitch, is a psychological phenomenon, it happens inside our brains. Just as Newton discovered how various frequencies of light pass through the eye and become ‘decoded’ into colors inside the brain, so too sound waves have to be translated by the brain to become the sounds which we identify as various pitches. The brain in both humans and in animals has evolved a quite miraculous process for this. When sound waves enter the ear they excite tiny hair-triggers on the Basilar membrane in a spiral-shaped organ in our inner ear called the cochlea. (See below). Each hair (humans have close to 20,000 individual hair-triggers in there) responds to one frequency. A signal is then transmitted up to the auditory cortex of the brain to corresponding neurons where the sensation of pitch is experienced. We can actually hook up electrodes to people and see which neurons are firing when they hear specific tones. This relationship between the receiving mechanism in the ear and the auditory cortex in the brain is sometimes referred to as tono-topic mapping.

tonotopic map and photo

(Note that this particular spiral shape known as the Golden Spiral is the same as that which occurs frequently in Nature. )

natural examples of golden spiral
basilar membrane wo title

The hairs are laid out in the spiral structure of the cochlea from the smallest to the largest each one responding to a single frequency. Those signals are then generated along the Basilar membrane up to the auditory cortex. 

1293_Tonotopic mapping
Cortical tonotopy wo title

      Now that we have an idea how the wiring works, let’s return to the Overtones.  The simple whole-integer intervals as described in the charts above of the harmonic series are recognizable by the untrained brain.  We seem to be able to perceive them as harmonic (pleasant) and can also recognize when they are out of pitch, even  without  knowing  very  much  at  all  about  music. One of the most profound aspects of the whole musical experience is how much our brains are able to process. even in those of us who have very limited or no musical training whatsoever. Ask anybody, they can tell what music they like and they know it when they hear it.

      While we know exactly which neurons fire in the brain for any given pitch, we still do not understand very well how the brain processes the relationship between the pitches (the harmonic intervals). But composers know the value of those intervals and use them all the time to indicate whatever dramatic or emotional effect they are after. The arrangement of the intervals in a melody, as in chords, are what give music that feeling of tension or relaxation, the sense that the music is leading us towards or away from something.

     All the musical scales in the world are based on various arrangements of the intervals of tones that appear in that harmonic series of overtones that arise out of a single fundamental tone. This gives the different scales used in different cultures their particular ‘flavor’ which we recognize as different from our own. This is what constitutes the culture-specific aspect to music.

     But there is a more universal aspect which becomes evident in the phenomenon known as the restoration of the missing fundamental. This effect is not even so much a musical effect but is an aspect of the physics of sound itself. Not only is it NOT culture-specific, it is not even species-specific. Each time we hear a tone played or sung, our brain responds by firing the corresponding neurons not only for the fundamental tone, but for its overtones as well.  So when we play a C, our brain also hears the corresponding neurons for the overtones that are present along with that fundamental, even though we may not consciously be aware of them.

Note C

                                                            We may simply play this note:  

But our brain will hear this:

Overtones of C

Now here is where it gets really interesting. If for example we produce the notes of the harmonic series for C at the frequencies of 512, 768, 1024, 1280, and 1536 Hz, but do not include the fundamental tone of C, the brain will just know that this is the harmonic series for C and the neuron corresponding to the fundamental C at 256 Hz will fire!  This phenomenon is called the restoration of the missing fundamental. The brain appears to have an innate natural ability to create harmony. Again, this is not something just found in the brains of trained musicians. This happens in everyone, instinctively, regardless of whether or not they play music at all. When we hook up electrodes to a person and provide the harmonic overtones, naturally we see the neurons firing in the brain that correspond to those overtones but the neurons that correspond to the missing fundamental tone that was NOT played fire as well! Not only that, it is not even restricted to humans.

     Levitin tells about how a graduate student in biology, “Petr Janata, placed electrodes in the inferior colliculus of a barn owl, part of its auditory system. Then, he played the owl a version of Strauss’s Blue Danube Waltz made up of tones from which all the fundamental frequencies had been removed. Petr hypothesized that if the missing fundamental frequency is restored at early levels of auditory processing, neurons in the owl’s inferior colliculus should fire at the rate of the missing fundamental. This was exactly what he found, and because the electrodes put out a small electrical signal with each firing – Petr sent the output of these electrodes to a small amplifier and played back the sound of the owl’s nerurons through a loudspeaker. What he heard was astonishing; the melody of the Blue Danube Waltz sang clearly from the loudspeakers!”

Owls Overtones

So when the owl heard these tones:


Owl Played This

the owl’s brain produced this:


Nature’s Metaphors

     Do recurring patterns and processes in Nature like the Golden Spiral in the cochlea of the inner ear and the other examples pictured above reveal something larger than simply the mechanical processes they describe?  Imagine the universe as a Great Poem into which the Writer has embedded hidden metaphors. This phenomenon that occurs in the auditory cortex of our brain known as the restoration of the fundamental may be such a metaphor.


     Choir Insignia of the Seattle Harmonic Voices

      If there is something missing that is fundamental in our lives, perhaps all we have to do is provide the appropriate Overtones. When the correct Harmonic Series is presented the Fundamental will be restored.


Think about it.

– Tim McKamey, Sept. 6, 2013

For more interesting insights and reflections on Daniel J. Levitin’s book This Is Your Brain on Music, see Elements of Music and If a Tree Falls….


  1. While looking for graphic illustrations of overtones and the harmonic series I happened to run into this interesting design featured in the last two graphics above. It is the Choir Insignia of the Seattle Harmonic Voices, a vocal group dedicated to expanding the possibilities of vocal music that focuses on overtone singing. It uses a geometric design based on the Natural Harmonic Series and appears to be a combination of the Hindu Sri Yantra and the nine-pointed Enneagram from Gurdjieff and Ouspensky’s Fourth Way work.

  2. Thank you so much for your article. I am currently doing more research on the harmonic series and manifestation. The missing tone certainly let me “think about it”, but also deeply feel.

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