A sine wave doesn't have a "timbre", similar to a single point on a paper not having a contour. Sine waves are the "Eigenfunctions" of linear time-invariant systems which are the overwhelming number of components in any sound transmission.
If you put a sine wave into a non-distorting amplifier, you'll get a sine wave out. When you fiddle with the tonal controls, you'll still get a sine wave out. There is no way to distinguish fiddling with the tonal controls with fiddling with the volume control: either way you'll only affect the amplitude of what remains a single sine wave.
If you put a sine wave generator in your throat and form vowels, again there will be nothing except a sine wave of varying volume. There will be no way to actually distinguish the vowels.
Flutes are rather sine-heavy, not having many sound components above the fundamental note being played. The human voice box rather works with a pulse train which has a high content of overtones (for good vocal closure, higher even than a square-shaped wave), so you can use your mouth to form vowels and the shape of the mouth leaves revognizable shapes in the overtone spectrum of the voice rather than just influencing loudness at a single frequency. Reed instruments of various kinds mimic this kind of sound production, making it possible to shape the overtone spectrum into very characteristic timbres by shaping the airways appropriately.
Bowed string instruments start with a saw-tooth kind of tone generator.
With electronic instruments, square-shaped tones can easily be generated (and they are approximated even when starting with clean sounds by overdriving amplifier stages like distortion pedals do) and have even higher overtone content than triangular wave forms but less so than pulse trains (a perfect pulse train has all harmonics with the same strength as the fundamental whereas harmonics of square waveforms taper off inversely with the frequency, and those of triangular waveforms taper off inversely with the square of the frequency).
Square waves (and more so pulse trains) have "too much" of a timbre to be usefully employed, like the color white (which contains actually colors of all the spectrum). You use them more as raw material and shape them with electronic filters (technically related to tone controls on an amplifier or graphical equalizers) into more characteristic timbres.
For example, high-class accordions (which produce their tones by free reeds like a harmonica) may feature a tone chamber for some reed banks, a "cassotto", making for a characteristic mellower sound. It only makes sense with comparatively high-quality reeds since those have the smallest air gaps and consequently the sharpest overtone spectrum. Since those are much more expensive as some additional wood, nowadays you can get cassotto registers even in comparatively cheap instruments where the overall impression is "somewhat more muffled" rather than the sort of vowelizing effect that this was originally built for.
At any rate, starting electronically, squares waves are the simplest reasonably rich timbre building blocks (not a useful finished timbre, mind you, but a good building block). You can turn up the overall harmonic content at the cost of sound energy by playing with the duty-cycle and making the "square waves" asymmetrical (note that your tweeters in particular will not share the impression of a decrease in sound energy, and blowing the same energy through them as through a large bass woofer is not likely going to make them happy once you try for some more solid volume).