Background for the string
One can use a dimensional argument to establish the plausibility of the equation
v = √(T/m)
where T is the tension in a string, m is the density of the material of which it is made, and v is the velocity of a wave propagated along it.
We know from basic wave theory that v=fl, while work on standing waves reminds us that for, the fundamental mode, l=2l, where l is the length of the string. So
v = 2fl
We may also deduce, from basic materials science, that
T = (YA/L)x
where L and A are, respectively, the unstretched length and the cross-sectional area of the string, and x is the extension introduced by the tuning mechanism.
Background for the pitches
Humans have a logarithmic law of perception. This means that when we perceive something to have increased to a set extent, the stimulus producing that perception has in fact gone up by a set multiple. For example, if doubling the output of a loudspeaker from 3 W to 6 W seems to be a particular increase in loudness, then doubling the power again to 12 W will seem like the same increase of loudness, and doubling it yet again to 24 W will only seem like another equal step in loudness.
Every time the pitch rises by a semitone, the frequency increases by a particular factor. We could call it the semitone factor s. It's value turns out to be about 1.05946. Instinctively, we think that this is pretty close to 1, but it actually corresponds to an increase of just under 6%. So if you think of a note of frequency 440 Hz as being A, then an A♯ (or B♭) will have a frequency of
f1 = f0 s = 440 Hz ´ 1.05946 = 466.16 Hz
Each time we go up a semitone, we multiply by another s. By the time we've gone all the way up the chromatic scale, with its twelve semitones, we have arrived at
f12 = f0 s12 = 440 Hz x 1.0594612 = 880 Hz
which is just double what we started with. We have arrived at the octave.
Making different notes
The equations of the first section may readily be combined to produce
Here, we are distinguishing between L, the original length of the whole string (some of which is wrapped up in the tuning mechanism) and l, that portion of it currently being used to set up a standing wave. To make a higher note (increased f ) we may either increase x (during tuning) or decrease l (during playing).
If you make x the subject of this equation, you can use it to calculate the value of x needed to achieve any particular note. For example, if the lowest string is normally tuned to an E, then you set f=frequency of E, make L and l both equal to the length of the guitar string, and give Y, A and m the values appropriate for that string. Then wind the handle. Once you've got the value of x, you can plug it into T = (YA/L)x to find the tension of the string.
You can't check this x value experimentally. BUT . . . . . you can use observation to measure how much extra extension is needed for a semitone rise when tuning the string. Add this to your calculated x value to find the expected new x value. Plug this into the equation at the beginning of Making different notes, and you can calculate the expected new frequency. With luck, this will be 1.05946 times the frequency of E.
[not done yet - but more straighforward]