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 $n$.

When you push down on the fret 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 $n$.

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.

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 $n$.