I know the dominant seventh chords include a 7th harmonic and there are progressions using them. What about utilizing higher harmonics? Maybe an
add6 chord implies the ratio 13/8 for some certain places?
For instance in
i - bIII - IV7 - iv(add6) progression, I suspect the added 6th can serve as the 13th harmonic of the fourth. I lowered the major second of scale to 13/12(at about 138 cents) and it sounds natural to my ears(maybe cause i used to the middle eastern music :) . Also, it seems lowering the minor third degree about 10 cents results a better dominant seventh experience.
Not directly related with the specific progression above but since the topic is in general about the usage of higher harmonics, here is an arrangement of a piece originally written for oud, arranged by professional Turkish classical guitarist Celil Refik Kaya. He says in the description that he transposed the piece to tonic B and the C is bent up about 40 cents. I think that bend is to nail the 13/12 ratio.
I strongly recommend anyone to watch the performance and listen to this beautiful piece.