Does a harmonic become a fundamental of its own harmonic series? Simple question, hopefully there's a simple answer. I'm about half a piano tuner, not a physicist.

A musical tone has a fundamental frequency, say $220\,\text{Hz}$. Its second harmonic is $440\,\text{Hz}$, its third harmonic is $660\,\text{Hz}$, etc.

My question is: 
Does a harmonic have its own harmonic series with itself as the fundamental? For example, does a $220\,\text{Hz}$ vibration, a 2nd harmonic which exists only because someone banged on a piano string that sounded a $110\,\text{Hz}$ fundamental, have its own second harmonic that is $440\,\text{Hz}$, a third harmonic, etc.
If not why not? It seems to me that if harmonics are real which I know they are because I learned to hear them, they must have their own harmonic series too.
If so, do these other harmonics have a name? I couldn't find them on google or wikipedia. I would think that if they exist, they must have a relatively low amplitude.
 A: If I denote by $f$ the fundamental frequency, then the $n$-th harmonics has a frequency $n\times f$. So the $m$-th harmonics of that $n$-th harmonics would have a frequency $m\times n\times f$: this is the $(m\times n)$-th harmonics of the fundamental, which does exist. So that nomenclature you devised is consistent. We don't use it in Physics afaik.
A: A periodic waveform has a fundamental period $T$ (the length in time of the repeating pattern).  By Fourier's theorem, such a periodic waveform can be decomposed into a fundamental component, with (fundamental) frequency $f_1 \equiv \frac{1}{T}$, plus components with frequencies that are integer multiples of $f_1$ which are called harmonics.
Note that it's not the fundamental component that 'has' harmonics (it doesn't, it's a pure tone), it's the waveform itself that has (contains) harmonics.
So I don't really grok the notion of a "harmonic [having] its own harmonic series with itself as a fundamental".  It is the (periodic) waveform itself that has a harmonic series, not the components (which are pure tones) of the waveform.
A: If you play the A below middle C, you get 220Hz and it has harmonics at 440, 660, 880, etc.  440 should match the A above middle C.  660 will be the just tempered E above that which is nearly but not quite the well tempered E.  880 will the next A etc.
If you play the A above middle C then you get 440Hz and it will have harmonics at 880, 1320, etc.  These will be a subset of the first set of harmonics, the nth harmonic of this series will be the 2nth of the previous set.  The odd harmonics of the lower note, e.g. 660 and 1100, will not appear in this set.  
Similarly, if you play the the E near the top of the treble clef then you will get nearly 660Hz (not exact with well temperament) and its harmonics will be nearly every third of the original A at 220.  
I am not aware of any special name for this relationship.  I guess that no one has ever felt the need.  
