It is possibly the missing fundamental created by brain trick.
When out brain is presented with overtones of a missing fundamental it creates the missing fundamental to our perception. For example when 200Hz (2x100Hz), 300Hz (3x100Hz) etc are presented to the ear the brain creates the missing fundamental ie 100Hz.
On the Moonlight Sonata Beethoven plays at some point F#, C#, F#oct at some low register by the left hand. The F# one octave below F# is missing but is possibly created by our brain when we listen as the bass at that point is so deep even for a piano. The right hand plays more on the harmonic series as the chord is F# major.
Note: the 2x, 3x is an oversimplification. A x2 relation in frequency sounds a slightly flat and annoying octave to our brain perception. The same for exactly x3f, ... .Why should it be exactly whole numbers? We are not robots, we are human. Our brain needs slightly more than x2f requency for a beautiful and correct sounding octave. It is how we are made.
This is the difference between an engineering octave and a ear brain octave (euroelectron).
We need enginerring inhamonicity to create human ear brain harmony. It is what came up from hundred of years of evolution of musical instruments so that they sound correct to ear brain and not like an exact x2f, x3f cheap synthesis. For example on a good Steinway piano the strings are so tightly stretched by the correct choice of diameter and strength of the structure (this is called piano string scaling) that every string is stretched to the correct amount for the x2f relation to be increased to the exact amount required by the human brain so that a pleasing octave (the second overone of the same string) is perceived. The the note sounds nice by itself and bit like a sick underirable note. And all this while the same happens for all other overtones when this same string is tuned at the correct pitch. At the same for all strings. This is one reason good pianos sound good. They have the correct engineering inharmonisity for ear brain harmony.
When strings are incorrectly and loose tensioned then the following happens. The overtones become even less than x2 and the strings sound like bells. Even just one note sounds like a bell.
When we say that slightly more than x2 in frequency is needed for a correct ear brain octave do not expect large ratios. For example:
If we have a A440Hz, then 880Hz is the engineering octave and it sounds flat and underirable. Our ear brain may need 881.3Hz (this is just an example, find the right value for yourself) for a harmonically sounding ear brain octave. The actual value in Hz may depends on intensity too. Pitch does not only depend on frequency but on intensity too. [Stevens]
You may find yourself by searching on Google for any online signal generator and just input the frequency numbers you want. (Make sure you keep the volume down to protect your ears. In fact it is possibly at low volume that we hear the correct pitch and therefore can sing correctly too. Pitch depends on intensity as you can very easily verify for yourself with these experiments. When increasing the intensity of 440Hz for example it immediately sounds a flatter A. As intensity is further increased it becomes even more flat.
Frequency and pitch are relating qualities but they are not the same. Frequency is the engineering objective term. It is complete vibration cycles per second. Pitch is the sense of how high is a note to our brain. Why should they be the same?
It is great now that universities and conservatories are teaching this great knowledge that was given to us from authorities such as S.S. Stevens who founded the Harvard Univeristy psycho physics laboratory. His books as sited at the references.
It is also great that the great instruction manual of Bach is being taught at school, university and conservatory throughout the world.
References:
Octave Stretch - Eenst Terhardt
http://euroelectron.blogspot.gr/2018/04/how-beethoven-creates-great-bass-on.html
Engineering octave vs ear brain octave - euroelectron
Sound and Hearing - Stevens, Warshofsky - Time Life Science Series
Musical Acoustics - Donald Hall
Hearing - Stevens, Davis
Bach Bass Rules (the manual of Niedt, Bach to their students):
http://normanschmidt.net/scores/bachjs-general_bass_rules.pdf
Leonard Bernstein Lectures on youtube
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