What happens between two harmonics? I know that standing waves in a simple harmonic system occur when the "echo" of a wave overlaps completely with the original wave at a certain point in time, doubling the amplitude. And then, because both waves have the same speed and frequency but opposite direction, they move away from eachother until they cancel eachother out, and then keep moving until they overlap again to amplify the signal. 
But what happens halfway between two harmonics? Would it be like a half standing wave, or would the signal cancel out?
 A: First of all, to have standing waves you must be talking about a wave carrying system with spatial extent: something like a guitar string. Such a system has a set of possible vibrational modes. The first two modes are shown in the figure. We can describe each mode shape with a function $\phi_n(x)$ where $n$ is a label that indexes the modes, ie. $n$ is an integer going from 0 to $\infty$. Note that for each mode, $n$ is the number of nodes other than the endpoints.

Figure: First two normal modes of a string. The black line indicates the rest position. The blue solid line is the zeroth ("fundamental") mode. The broken red line is the first mode.
Suppose the string is in some arbitrary state, where the deflection from rest at each point $x$ is given by a function $y(x)$. It turns out that you can write any $y(x)$ like this:
$$y(x) = \sum_{n=0}^{\infty}c_n \phi_n(x).$$
That sum is called a Fourier series, and the point is that you can write the state of a string as a linear sum of the normal modes.
What's special about the modes is that if you start the string in exactly one of those modes, it will keep that shape forever; only the deflection amplitude will oscillate. For example, if we start the string in $\phi_0$ the string's deflection for all time is
$$y(x,t) = \phi_0(x)\cos(\omega_0 t)$$
where $\omega_0$ is a frequency associated to the $0^{\text{th}}$ normal mode. The fact that this shape is maintained forever means that the mode is a standing wave, by definition.
If you start the string in the generic shape given in our first equation above, the string's shape for the rest of time will be
$$y(x,t) = \sum_{n=0}^{\infty}c_n\phi_n(x)\cos(\omega_n t).$$
To recap: you can always write the shape of a string as s sum of normal modes (aka standing waves) each with its own shape and vibrational frequency.
Now on to your question...
If you pluck the string in some way such that the initial shape is not one of the normal modes, or in your words, not one of the usual harmonic standing waves, you can still write it as a sum of other modes (sum of other standing waves). The future evolution of the string's shape is then given by our third equation. So, the answer to your question is basically that if you excite the string in a non-standing wave way, it will vibrate as a sum of standing waves, and that sum is just whatever set of standing waves superimpose to match how you're exciting the string.
Note: Everything said here pertains to linear systems. Most things in nature are at least approximately linear as long as the amplitude of the waves isn't too big. There are exceptions to that, however.
If that doesn't make sense, please post a comment :)
