If you accept a physical basis for fifths (by virtue of the harmonic series) then surely it's not a major leap (no pun intended) from that to thirds?
As you get further along the series you're moving from simple to complex ratios. Thirds are next in line after 4ths/5ths and they have a function that solves some obvious harmonic problems.
A lot of this has echoes in the historical development of music and the commonalities between harmony in different parts of the world. There's a reason cultures independently discover similar constructs i.e. the pentatonic scale
> If you accept a physical basis for fifths (by virtue of the harmonic series) then surely it's not a major leap (no pun intended) from that to thirds?
It’s not a leap to go from the fifth (as an interval, two notes) to the major triad, three notes. However, if you consider major triads, you’re just dipping your toe in the water as far as chords are considered.
The question is not, “Why does the major third above the root appear in chords?” but rather “Why do we build chords out of stacked thirds, which can be minor, major, or even diminished or augmented?” At this point, the part about the major third appearing in the harmonic series lacks explicatory power.
Consider, for example, that a chord “built out of thirds” can contain a major or minor seventh, a minor second, or an augmented fourth. Just as examples.
> At this point, the part about the major third appearing in the harmonic series lacks explicatory power.
I really don't think it does. Fifths (and their inverted form fourths) are the most consonant interval (ignoring octaves as the trivial case).
Most cultures develop musical processes that involve fifths very early on in their history.
Where do you go from fifths if you want to add another voice? If you have three notes and you want the most consonant result - you put a note that forms a third between the root and the fifth.
Given a few basic assumptions ("prefer simpler integer ratios") there's no better way to play 3 non-octave notes together.
> Consider, for example, that a chord “built out of thirds” can contain a major or minor seventh, a minor second, or an augmented fourth.
According to modern harmonic rules those are valid. For a long time, these were not harmonies you'd hear in music. Really only since counterpoint fell out of style, since it forbids augmented intervals.
