Answer: If I continue muttering about musical acoustics people may begin to look askance at me on the street. But that's where this is taking us. I went there in the appendix of Exploring Theory with Practica Musica, but keep getting drawn back by the magnetic force of the subject.

Any time you pluck a vibrating string or blow a note on a pipe, the sound heard is actually not a single pitch but rather is a combination of many pitches heard at once. And due to the inexorable power of math and physics, the frequencies of those pitches have a certain relationship to the frequency produced by the whole string or the whole column of air. Ideally, assuming the string is even and flexible, and the pipe is smooth, the string or pipe will vibrate at various divisions of its length, all even numbers. It will vibrate in halves, in thirds, in fourths, etc., and each of those extra modes of vibration will produce a tone whose frequency is, again ideally, a multiple of the first or fundamental frequency. We call those extra sounds 'partials,' because they are all parts of the full tone we hear. If your guitar string is tuned to E, it will simultaneously sound not only the pitch E, but also a pitch twice that frequency, one three times that, one four times that, and so on. The partials become fainter and higher as their number increases, but the first 6 or so are pretty easy to hear.

Sit in a quiet room and pluck that string, and you can train yourself to hear these partials, whose pitches will be E (the fundamental or first partial), E an octave higher (the 2nd partial), B above that, E above that, then G#, then B again. The next one is going to be the 7th partial, which is an oddity because it agrees with no others except the fundamental, and the 8th is just an octave of E once more, so for practical purposes the first 6 are the most important. To hear them it will help to momentarily isolate the higher pitches by touching your finger to the division points of the string: at the halfway mark, at one third, etc. That's what performers are doing when they play "harmonics."

Did you notice that those first six pitches are all notes of the E major triad? The major triad is heard any time you play a note on an instrument with harmonic tone (i.e. not on a bell or gong, whose partials are distorted). So it is natural that one of the first things that occurred to us humans was to reinforce some of those inner sounds by repeating them on other strings or other instruments. Some critics, like the theorist René Leibowitz, have objected that the acoustical foundations of the triad are irrelevant, because among other things the partials "don't explain the minor triad." But that neglects the fact that there is a noticeable interaction between the partials of two separate tones if they form one of these basic intervals. Play two notes a minor third apart and the 6th partial of one will palpably reinforce the 5th partial of the other, making this a "stopping point" when tuning two notes. Imagine that tightening the second string is like adjusting a radio dial: when you reach a minor third that will be one of the places where you suddenly focus on a station. Two tones a major third apart make a slightly stronger connection: the 5th partial of one against the 4th of the other.

Major and minor, in short, are not arbitrary but both arise from the nature of the material. The fifths, fourths, and thirds have a potential for mutual reinforcement (if they are tuned close to the right proportion, which is another dangerous topic) that is not found in any other intervals. And those intervals are the basis of the major and minor triads, which as mentioned in question 35 would be comprehensible to creatures in a galaxy far, far away.

All other chords in the music we know best are formed as elaborations of the triad, which is why it's called triadic harmony. And here is where I will stop, though many readers might have stopped earlier.