So let's say we have two vectors $a \in \mathbb R^n$ and $b \in \mathbb R^m$. Let $L=\text{lcm}(n,m)$.
Consider the natural inclusions $a,b \mapsto \mathbb R^L$, which simply continually concatenate $a$ or $b$ to itself until the dimension matches $L$, which of course is possible due to $L$ being the least common multiple of the dimensions of $a$ and $b$ themselves.
Now, take the fourier transform of the inclusions. For $a$, you end up with every $L/n$ frequencies being identical.
It seems like it would make more sense to "spread out" the vector being included, so that the amplitude of the wave relative to how far percentage-wise you are into the vector should be identical? So that for example $(1\ 0\ 1) \mapsto (1\ 0\ 0\ 0\ 1\ 0)$ would make sense. If you expand in this way, what you will find is that the fourier coefficients themselves will begin to repeat across the period $L$. That is, if you go from $n \mapsto L$ by multiplying by $k$, then, the fourier expansion of $a$ will be $k$-periodic, again, only if you're putting 0's in between. i.e. $111$ maps to $100010001000$ if $k=4$. Trying to make the cases clear here.
So either way you take it, you can expand a or b in terms of a L dimensional repeated, spaced out, vector. However, both of those lead to artifacts being present in the fourier transform coefficients. In the former case, you have spaced out fourier coefficients, and in the latter case, you have periodic fourier coefficients. So there doesn't seem to be a natural way to take the dot in fourier space of two vectors of different lengths, since if you try to include upwards into the LCM dimension space, it is not clear how meaningful it would be to at that point dot together these two vectors. Because, you end up recording artifacts of the dimensions $n$ and $m$, rather than simply elucidating structure within $a$ and $b$. Perhaps, you can modulo out this lcm dimensional structure and be left with a lower dimensional fourier object you can then compute a dot product for, but it doesn't seem likely. My question is essentially, how can we compute a meaningful dot product between $a$ and $b$ in the fourier space, so that, let's say a major chord $a$ and a major seven chord $b$ (I'm referencing the fourier coefficients of vectors $a$ and $b$ of different lengths, meaning 4,5,6, but not 7, is highly present in $a$ and 4,5,6,7 is highly present in $b$), could dot together and get a higher number than let's say $a$ dotted in the fourier space with a vector $b$ whose fourier coefficients (in $m$ dimensions, by the way) is like a half diminished seventh chord. Seems like it would be natural that $a$ dot $b$ in fourier space (whatever way you define it, which I don't know how to yet because of what I wrote above, and that's where I need your help) should be larger in magnitude than $a$ dot $b'$ where $b'$.
You may assume the norms $||a||=||b||=1$ and $a_i,b_i \in \mathbb C$.