When you push down on the free you are essentially changing the length of the string. This means you *are* chaining the fundamental frequency. 

This means that, while the $220\ \rm{Hz}$ is a harmonic of the string with the $110\ \rm{Hz}$ fundamental frequency, when you half the length of the string the fundamental frequency is now $220\ \rm{Hz}$, and the $110\ \rm{Hz}$ is no longer a harmonic of the shortened string.

Mathematically, the modes of the string are achieved when the string of length $L$ is broken up into parts such that 
$$\lambda_n=\frac{2L}{n}$$
Where $\lambda_n$ is the wavelength of the standing wave, and $n$ is a positive integer. Since $v=f\lambda$ is true for the waves, where $v$ is the wave velocity that depends on the string properties, we have 
$$f_n=\frac{nv}{2L}$$

Since the fundamental frequency is when $n=1$, if $f_1=220\ \rm{Hz}$, then this is the lowest $f_n$ can be for larger $n$.