I've been thinking of ways to harmonize melodies. One of my ways is to not think in terms of the seven diatonic chords, but to just think of in terms of "odd chords" and "even chords".
Let's say we take the C major scale in two octaves:
C D E F G A B C D E F G A B C
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
So we have two types of chords, where their notes can be:
Even: 2 4 6 8 10 12 14 16
Odd: 1 3 5 7 9 11 13 15
Where to generate a chord you'd pick a subset of three or more notes. So example of even chords would be: [2,4,6] [4,6,8] [8,12,16] And example of odd chords would be: [1,3,5] [3,5,7] [5,11,15].
So to harmonize a melody, if the melody note falls on an even number, harmonize it with an even chord, and if the melody note is odd, harmonize it with an odd chord.
For example in the melody of Mary had a little lamb:
 2 1 2 3 3 3  2 2 3 3 3 ...
I harmonize the  (an odd note) with [1,3,5] (an odd chord)
I harmonize the  (an even note) with [2,4,6] (an even chord)
This Odd/Even idea is also compatible with the idea that people often substitute the ii,IV,vi with eachother. And the iii,V,vii with each other.
But I was wondering if others think this holds any truth or maybe there is an actual name for this in music theory?
Note: after writing this I came across a cool concept. If you take the odd scale degrees on a piano: 1,3,5,7,9,11,13,15. and have the left hand in charge of the first four notes (1,3,5,7), and the right hand in charge of the last four notes (9,11,13,15). then without moving your fingers you can get to all the diatonic chords in the scale. For example: I would be [1,3,5] and ii would be (using right hand): [9,11,13] .. iii would be [3,5,7] and so forth.