It has to do with overtones and how our brain processes signals.
When a source of a sound has overtones, there is a mathematical relationship between each of the frequencies of the tones. If you understand the relationship, you can simplify your processing by breaking it apart into a single frequency and an encoding of the relationship of the overtones.
As it turns out, the most natural frequency to choose is the lowest frequency in most cases. Why? Well it's in a unique position to describe the sequence the best. Consider a vibrating string example, where I choose to think about the string as a single note and some harmonics -- but I choose different fundamental frequencies. In the first column I pick the lowest frequency, and in the second I pick the third (an arbitrary choice):
Frequencies: 400 800 1200 1600 2000 2400 ...
multiplex of 400 multiples of 1200
400 1 1/3
800 2 2/3
1200 3 1
1600 4 4/3
2000 5 5/3
2400 6 2
As you can see, the math looks a lot simpler if we consider the fundamental frequency of the note to be the lowest frequency! This particular example is for a string, which is different from, say, an open pipe, but the story is the same. The equations are simply more natural if you think about the lowest frequency.
Our brains are smart little processors. They figured this out billions of years ago, before we were even mammals! So when you and I hear things, we naturally hear this decoupled pattern: we hear the "fundamental frequency" defining the note and we hear the "character" of the sound as a separately encoded set of information regarding the overtones.
An interesting breakdown of this model occurs with throat singing. Best known in Tuvan throat singing, this is a way to sing two notes at once. The technique is a brilliant abuse of this processing that our brain does. In virtually all real instruments, the most power (i.e. energy transmitted, in a scientific sense) comes in the lower frequencies, closer to the fundamental. In throat singing, one carefully adjusts the shape of the mouth to highlight one harmonic, making it resonate far more than the others. When the listener processes this, their brain sees a sound that doesn't fit into their usual model, which assumes that for single sound source there will be less and less power in each of the overtones. It then makes the assumption that there is not one sound source but two! Thus, when we hear Tuvan throat singing, it sounds like they are singing both the bass line and the whistling melody at the same time. In fact, they are singing one note, but with a brilliant and tricky precision that causes us to invent a new "lowest note" in our own head to carry the melody with!