Ok, so I am very curious about why the major scale 4th and 5th are deemed "perfect" while other intervals are "major." This may just be an arbitrary thing based on sounds we find pleasing, but I've been at least trying to find something more rigorous. I did notice one thing.
If you take the C major scale, C D E F G A B C, we know there are 12 semitones from C to C. The 4th, F, is five semitones from C, and the 5th, G is at 7 semitones from C. 5 + 7 = 12. In other words, if you choose either one of those notes and find its distance from C in semitones, then it turns out the remaining number of semitones to reach 12 is the position of a note that is IN THE SCALE AS WELL.
If you check D, the 2nd, it's at 2 semitones, and you'd need 10 to get to 12, but the scale doesn't have a note 10 semitones above C. Similarly for E, the 3rd; it's 4 semitones from C, and there is nothing in key at 8. F and G, the 4th and 5th, are unique by this criterion.
Ok, so far so good. I then said to myself, "Ok, the major scale is the Ionian mode. What about the other modes?" So I proceeded to walk through all seven of them, and the answer varies from mode to mode. Specifically, you get this:
Dorian: EVERY SINGLE NOTE
Lydian: 4th only (it's 6 semitones above the root, so it's its own partner)
Mixolydian: 4th+5th and 2nd+7th
Aeolian: 4th+5th and 2nd+7th
Locrian: 5th only (same story - it's 6 semitones above the root)
So this is all very interesting. But really I'm just a mathematically-inclined engineer with a strong curiosity about music theory. I'm primarily self-taught using the internet. So my question is simple:
Does what I laid out above actually mean anything? Or is it just a curious coincidence? I don't really believe in coincidences - usually when something this intriguing shows up there's a reason for it.
As a possible motivator for insights, the Aeolian mode is the natural minor scale, and in addition to 4th+5th that one has 2nd+7th. So is there any existing knowledge out there about the 2nd and 7th intervals taking on any new prominence in the natural minor scale?
One more interesting thing. If you use the circle of fifths to "order" the modes, you get this order:
Lydian: 4th only
Mixolydian: 4th+5th, 2nd+7th
Dorian: Every note
Aeolian: 4th+5th, 2nd+7th
Locrian: 5th only
There's structure and pattern there...