I'm 44 years old and learning how to sing. Maths is an old friend but music is new to me so in learning to sing I'm coming across all these new things that many aren't able to explain and you're left with: that's just the way it is or it just sounds right. Your explanation is clear, to the point and all encompassing. You've taken the voodoo out of music and chords and octaves. It's entirely clear to me now why you need to sing spot-on pitch if you're trying to have a nice sounding voice - slightly off is going to throw Xs out of alignment in many different ways. Thank you, sir!