Do you think the title sounds like theory? You are right! But the following theory is easy to understand and easy to apply.
Let's say you have a 24mm and a 135mm lens and you love them. But there is a huge gap between the two lenses. Therefore you would like to have a third lens between them. But which focal length should you choose?
Of course, some people already have experiences and for them it is clear that the only proper focal length here is for example 35mm. That is absolutely okay.
But you have no idea what your proper focal length between 24 and 135mm could be.
Let's try to find a way to determine a harmonic addition!
How about taking the arithmetic mean:
Some people might like this result, but I would say the gap between 24 and 80mm is greater than the one between 80 and 135mm. If the arithmetic mean would be the solution for your problem, it would be easy to build a harmonic sequence of focal lengths by repeated adding of a constant amount, e.g. like this: 20mm, 40mm, 60mm, 80mm, 100mm, 120mm, 140mm and so forth. But this sequence does not look very consistent to my eyes. The difference between 20 and 40mm is significant, but between 180 and 200mm it is negligible:
More harmonic is a sequence with a constant magnification between one focal length and the next longer one. That is easy, because multiplying the focal length by m just corresponds to an m-times magnification (at infinity).
Here is an example for a sequence with a 2x magnification: 20mm, 40mm, 80mm, 160mm, 320mm and so forth:
With Nikon's real world prime lenses it results in an array like this: 20 - 35 or 45 - 85 - 180 - 300mm.
What do you think is the reason for Nikon to produce PC-E lenses with 24mm, 45mm and 85mm? Yes, the three focal lengths are a harmonic combination - check it out with a factor of 1.875.
Another popular factor is √2, resulting for example in this sequence: 25mm, 35mm, 50mm, 71mm, 100mm, 141mm, 200mm, 283mm and so on. In analogue times I tried such small factors. The drawback is that you have to carry around a lot of lenses. Today I tend to greater factors. My basic setup for nature (including landscape and wildlife) photography is based on a factor of 3.5: 24mm, 84mm and 294mm, in reality 24-85-300mm.
Let's enjoy some mathematics: if f1<f2<f3 are a harmonic array of focal lengths, we can easily conclude:
Thus, we have the solution for your problem - the focal length between 24mm and 135mm that makes the combination harmonic simply is their geometric mean:
The two other cases of the "find-the-third-lens" problem are easier to compute:
You have two prime lenses (for example 50mm and 105mm) and you want to add a shorter or a longer lens. First determine the magnification factor between your lenses, for example m = 105/50 = 2.1. Then just multiply or divide: a longer harmonic addition is 105*2.1 = 221mm (in reality 200mm), a shorter one is 50/2.1 = 24mm.
Again: this was theory for giving you some orientation about harmonic combinations of focal lengths. If you already have preferences for your lens setup, forget about it.
If you don't want to think about these things, just buy Nikon's four professional zoom lenses, giving you the range from 14 to 400mm without any gaps ;)