I recently noticed a pattern between the Primary (I IV and V) Chords and Secondary (II III VI VII) chords as they are further harmonized in 3rds all the way to 13ths.
Starting with the basic triads and arranging them into a chart, I broke down note counts in the two chord categories and got this chart:
Then I noticed that the secondary chords taken together, are opposite to the primary chords, in that any note that only appears once in the primary chords, appeared twice in the secondary chords, and any note that shows up twice in the primary chords taken together, shows up only once in the secondary chords, taken together.
As the harmonization continues in 3rds, the notes in each chord category that increase with each degree of harmonization are opposite in the Primary category as opposed to the Secondary category:
I noticed how under the primary chord category, the counts for the notes A, D, and E were added to, and in the secondary chord category the counts for the notes C, F, G, and B were increased. This pattern was present through each step of harmonization in 3rds: 7ths, 9ths, 11ths and finally in the 13ths pictured above.
Is this pattern important in understanding something to do with why the I, IV and V chords ARE primary chords, and the II, III, VI, and VII chords are secondary?
P.S: I'm not sure if "Primary" and "Secondary" are the right terms to describe the 'I IV V' chords versus 'II III VI VII' chords. Sorry for any misunderstandings.
P.P.S: I know it isn't appropriate to upload pictures if the post wouldn't be clear without them, but posting these photos of the charts I made was the only way I could get the formatting right.