Question

The only variable we found that affects the speed of a wave on a string was the tension of the string. How does this relate to how a musician tunes a stringed instrument?

Answer 1

We know that in a string with two fixed points (like the one you will find in a guitar, where the string is "fixed" at the bridge and at the nut) the only thing that defines the speed (the frequency) at which the string vibrates is the tension (the length is also important, but for the 6 strings in the guitar all of them have the same length, but, as you know, when you press in a given fret the note changes, this happens because you are changing the length of the string, finally, the mass of the string is also important, but all the strings have almost the same mass, so we can ignore this).

Then for the open notes (the notes that you play when you don't fret any note) the only thing that defines the note that will sound is the tension of the string.

And a musician can tune the stringed instrument by changing the tension of each string using a tuner, which is the mechanism in the headstock of the instrument). As more tense is the string, higher will be the open note.