seems that the octave is an exact doubling of frequency. Traditional
acoustical physics states that "harmonics" occur at
integer frequency ratios, as this supposedly produces zero beating.
I am skeptical about this method of mapping
harmonics, as it seems to me that "harmonics" should
be at the pitches which you can generate on
stringed and other instruments. e.g. by touching the strings
lightly, or blowing and over blowing.
The usual theory is demonstrated using the
image of a sine wave, and how the peaks of waves coincide
at integer ratios.e.g. 440 A and 660 E.
My own belief is that the traditional model
is static and mathematically simplistic.There seems to be
a more subtle (yet also simple) underlying pattern, judging
from my research and experiments into John 'Longitude' Harrison's
Philosophically, you could consider the "sine
wave" as a two dimensional representation of a three
dimensional coiled spring pattern. (Shades of Plato ????;-)
Imagine shining a light through a stretched
spring. The resultant shadow cast by the spring could be seem
as a sine wave, when viewed from the
appropriate angle on a background screen. We know that the
spring is 3 dimensional, "yet we continue to measure
the shadow" ????
I feel that there is much more to this question
that most people realise. Harmonics in the real physical world
tend to also contain vibrato. I suggest that they can be mapped
in a better way which accounts for vibrato, and the resultant
beating. Read between the lines on the site, for I suspect
that the "harmonics" are in reality at the octaves
of the notes generate using intervals derives from pi. It
sounds musically correct, yet I can't yet prove it scientifically.
Hence I am following a "different" paradigm of what
constitutes a "harmonic" in the real world.