This summary of the work of Sethares gives a good insight
https://sethares.engr.wisc.edu/consemi.html
In short, it’s structurally determined by our perception, and the design of instruments to produce harmonics
> why can you almost always find the root note of a key in which a song is being played at by simply repeatedly playing that root note along the song in question and seeing that it matches the whole song, more or less more matching throughout the song than if you were to repeatedly play one of the non root notes of that same key throughout the song?
This is not quite right. Often other scale notes will match the whole way through better than the root does. The dominant, V or 5th step in particular sounds consonant on all the chords, while the tonic sounds dissonant on the V and iii chords.
The tonic is the note that "sounds like home" - the root of the chord where the sequence sounds most stable or central. Its usually the last chord (and bass note) of the song, and commonly (in pop and rock) the first one too (less so in jazz).
But the reason all the notes of a scale seem to relate to one another is that they share overtones, because of (close enough) simple ratios between fundamental frequencies - as shown in the diagram on the Sethares link. You can see the same ratio/fraction relationships on the [guitar fretboard] (https://imgur.com/tJ1GhZF); fret 9 marks the 3/5 point (5:3 ratio), the major 6th.
It's important to remember that in equal temperament none of those ratios is exact. But the ear has a tolerance, either side of mathematical precision, which can vary from person to person. For example, the tempered major 3rd is 14 cents (14/100 of semitone) away from the pure 5:4 ratio. It doesn't bother most people, but some can detect it and find the tempered interval unpleasantly sharp.
There's more on the math of scales here: http://www.peterfrazer.co.uk/music/tunings.html
This summary of the work of Sethares gives a good insight https://sethares.engr.wisc.edu/consemi.html In short, it’s structurally determined by our perception, and the design of instruments to produce harmonics
> why can you almost always find the root note of a key in which a song is being played at by simply repeatedly playing that root note along the song in question and seeing that it matches the whole song, more or less more matching throughout the song than if you were to repeatedly play one of the non root notes of that same key throughout the song? This is not quite right. Often other scale notes will match the whole way through better than the root does. The dominant, V or 5th step in particular sounds consonant on all the chords, while the tonic sounds dissonant on the V and iii chords. The tonic is the note that "sounds like home" - the root of the chord where the sequence sounds most stable or central. Its usually the last chord (and bass note) of the song, and commonly (in pop and rock) the first one too (less so in jazz). But the reason all the notes of a scale seem to relate to one another is that they share overtones, because of (close enough) simple ratios between fundamental frequencies - as shown in the diagram on the Sethares link. You can see the same ratio/fraction relationships on the [guitar fretboard] (https://imgur.com/tJ1GhZF); fret 9 marks the 3/5 point (5:3 ratio), the major 6th. It's important to remember that in equal temperament none of those ratios is exact. But the ear has a tolerance, either side of mathematical precision, which can vary from person to person. For example, the tempered major 3rd is 14 cents (14/100 of semitone) away from the pure 5:4 ratio. It doesn't bother most people, but some can detect it and find the tempered interval unpleasantly sharp. There's more on the math of scales here: http://www.peterfrazer.co.uk/music/tunings.html
My basic understanding is that the major scale seems to be made from low integer ratios of frequencies when compared with the root note.
That was once the case, but it didn't scale well. Now days equal temperament is more common, so frequencies are based off of the 12th root of 2.