I have seen a diagram on sites such as hyperphysics.com that show that there is a missing bit every time so that it makes every harmonic odd. I was hoping I could get a more intuitive explanation. We recently got introduced to harmonics and standing waves in class and I am a bit lost :)
Musical notes are made by strings and by pipes. Standard discussion of harmonics and standing waves focusses on strings, but pipes are just a little more subtle.
For a wave on a string you plot the displacement and get sine waves and that's obvious as the plot looks like the excited string.
For a wave in a pipe the displacement is still described by sine waves but it is longitudinal instead of transverse so the graph is no longer a picture. You could also graph the pressure in the wave - it's basically the same as the density - and if you do that you get the equivalent cosines: in the peaks of displacement the nearby molecules have all moved by the same amount so the density is unchanged, whereas around the region of zero displacement the gradient means that some molecules are displaced further than others leading to variations in density. Displacement nodes are pressure antinodes, and vice versa.
The end of a string is simple. It's clamped. the displacement is zero.
The end of a pipe is ambiguous. If it is closed then the longitudinal displacement is zero: a displacement node. But if it is open then it must always be at atmospheric pressure: a pressure node and thus a displacement antinode.
So instruments involving pipes fall into two types: similar at both ends (like a flute, which is open-open) for which the harmonics are similar to a string, or closed at one end and open at the other (like a trumpet: the lips and mouthpiece form a closed end) which have a displacement node at one end and a pressure node (and thus displacement antinode) at the other. This requires $1 \over 4$ of a wavelengt to fit along the length, or $3 \over 4$, or $5 \over 4$... the odd harmonics.