While the average RMS voltage represents the average volume, what we hear is our brain's interpretation of the spectral content of the signal. What is that? Let's take a look.
I fired up Audacity, and recorded myself playing an F major triad on a recorder. What do we see when we do that?
Right away, we can see 4 notes F4, A4, C5, F5, of varying volume. The last note has a larger height, as it required a bit more air and so it was louder. So the thickness of those blobs represents the volume, and if I was pumping this signal out to a speaker, would be proportional to the average voltage. But let's zoom in a bit more on that last note and see what's happening:
We can see that F5 isn't just a solid blob, but the signal actually cycles back and forth the whole time. Let's zoom in just a little bit more:
This graph essentially shows the movement of air molecules over time. Back and forth, vibrating. It looks a lot like a sine wave. If we measure how many times that peak to peak repetition occurs per second, we'll find out the fundamental frequency. That fundamental frequency is the pitch we hear. In the case of this recording, it's about 697 Hz. F5 for an A440 tuning is 698.46 Hz, so I'm playing just a hair flat.
But, there is more to what we hear than just 697 Hz, after all, we can tell the difference between different instruments. Earlier I said it looks like a sin wave, but it in fact isn't. Or at least, it isn't just a single sine wave. To illustrate, let's see what happens if I pluck a guitar string instead of playing the recorder:
What we can see here, is a dramatically different shape to the repeating signal. We can hear that difference in the waveform's shape, because it has a different spectral content.
Mathematically, every repeating signal can be represented by the sum of an infinite number of perfect sine waves, with varying amplitude at each frequency. This is the spectral content that I mentioned at the beginning of my answer. When we play a note on an instrument, the fundamental frequency will have the largest amplitude, and we'll see large amplitudes at several harmonics, or multiples of the fundamental, and we'll usually see a bit of non-harmonic content as well. The distribution and proportion of that spectral content is what we hear as timbre*. Your brain does this when it interprets the signals from your cochlea. When a sound has a lot of high-frequency components, we hear that as tinny or metallic sounding. Drum sounds often have a much flatter harmonic response with significant non-harmonic components. Electrified instruments can have bizarre spectral profiles which makes distorted guitars and other unique sounds possible that aren't possible with purely acoustic, physical devices. Let's look at some of these in particular.
First the recorder:
And the guitar:
Looking at the guitar spectral analysis, we can see significantly more high-frequency components than the recorder, reflective of the metallic sound of its steel strings. The recorder has a much different overtone profile of odd-numbered harmonics, which leads to its "pure" sound. Every instrument will have a different profile on these frequency analysis graphs. Some instruments, the profile will also change from low notes to high notes, in ways beyond just sliding left or right on the horizontal scale, while others will stay relatively consistent.
Lastly, just for fun, let's look at what happens when I play a C Maj chord on the guitar:
What is that? That looks pretty bad. How are we supposed to make sense out of that, where is the the fundamental? Well, the isn't really one, as I'm playing a chord. We have 3 notes speaking, and so we have three of those waveforms piled on top of each other. When some notes are tying to move up, others are moving down, and back and forth, and different frequencies, making that chaotic shape. However, our brains are able to make sense of that and can parse out the frequency components. Let's look at the spectral content:
We can see three fundamentals at 264, 330, and 393 for just about C4, E4, and G4 respectively. The overall profile is pretty similar to the single note as well, but we have groups of three for each harmonic, because of each string vibrating.
*So, timber represents the qualitative aspect of the sounds, some of which is shown here by its volume, its pitch, and the spectral content. But there is more to what makes a sound a sound. I was looking at these signals from a steady state perspective, during a sustained note. But the full experience includes the beginning, middle, and end. The sound you hear varies with time after the note is played, (e.g. the sound of the guitar pluck vs the sustain), how quickly the sound dies off, sometimes it changes pitch while sounding out, such as vibrato. There are a lot of aspects that can be modeled, but I think my answer here covers the main aspects.