I understand that the first note of each octave is twice the frequency of the first note from the octave before, but how are the frequencies for each note chosen subsequently after that. So far, I’ve been able to gather that an equation using geometric progression is used and it has something to do with harmony. I’m having trouble wrapping my head around how it all works. How does geometric progression translate into emotional value in music?
There are probably more versions of this than I am aware of but I'll cite 2 or 3.
In the just tuning system the other intervals were chosen to be a rational fraction of the lowest note in the scale (Do). For example the frequency of a fifth (Sol) is 3/2*(frequency of Do), and a second is 9/8*(Do), etc. You can look up the full chart on wikipedia. The rational for this is that many of these tones are natural harmonics of a linear vibrating system.
The harmonic sequence is
fn = n*f1
where n is an integer and f1 is the lowest note in the sequence. n = 2 is obviously an octave, n = 3 is even higher and not within a single octave but you can divide this frequency by 2 and get 3/2 which is a perfect 5th. It turns out not all of the notes in the major scale are perfectly represented in this sequence but enough are to generate a major triad out of harmonics. If this process is repeated starting a 5th up in the scale and again a 5th below (on the 4th) one will generate all the just intonation scale notes. So one can think of this for of tuning as being based on the physics of vibrating systems.
Later in history people wanted a system that had a fixed ratrio for the half step and devised equal tempered tuning. This is based on fixing 12 half steps in an octave and defines the half step as the 12th root of 2.
f(1/2) = 2^(1/12)*f
this formula defines the frequency of the half step above f. In this manner you can build the major scale completely from half steps. The 5th would be 7 half steps and,
f(5th) = 2^(7/12)*f approximately 1.4983 * f
as compared to 3/2 = 1.5.
This form of tuning uses an irrational number which can never be perfectly represented so there is some fudging in this system.
Beyond that there is a system based entirely on 5ths, and probably a few others.
In fact the human can hear with much better resolution than a half step and frequency truly forms a continuum.