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Trying to find the roots of harmony we can go to the primary phenomena of earth: The ocean. - Hearing to its voice we may hear a deep basic tune - let's say A - offshore and reaching the shore gliding out the waves hearing the fifth-tune - e' -
Supposing to be a sound like c in the middle, the spiritually proportions would be: A (symbolizing God) plus c (the relation of 6/7, symbolizing in "7" - the number of the week and month - the will of God for creation, so the Son Jesus -> As a result of the first overtone-coincidence: The e', a relation of 1/3 (... playing a and e' on an organ and hearing sensible, You may hear the e'') - symbolizing with "3" the Holy Trinty and especially representing the Holy-Spirit-presence of God on earth ...
The link to the greek tetracord-system would be:
from that e' the next, also the bird kukuk as also singing of little children, ist the small third to g'. This small third is 5/6: "6" for the primary-mouth-animals (6 legs, 6 angels of the cradles of the bees, resp. the structure of molecules. - The 5, rising from c to e (Jesus and the Holy-Spitit-presence in creation,) would be the big third, 4/5, in 5 marking the 5 Fingers of many secondary-mouth-animals, like placenta-animals, or some flower-foils....
Especially in voice-accentuated-pronouncing, trying to jump up to the a' as the basic-note A, only octaves higher - but immediately skipping back to the e' as the basic note of earth and origin - God... marking a primary seeking fo own identitiy (3/4, in "4" the double-opposits).
In consequence remains the position of the 4th note of teracord: Near to the g, as f-sharp, the Jonic teracord (feemale), - near to the e, as f, the Doric tetracord (male).
In pro-ancient Greek music, two tetracordes, declining linked as: a'-g'-f'-e' plus e'-d'-c'-b (Doric) respectively: g'-f'-e'-d' plus d'-c'-b-a (Jonic).
The revolution to classic-ancient Greek music was the addition of the 8th cord  (I think, it was the poetrist Terpander, living around the 6th century b.C., when individualization in Greek culture began,) marking the note "a" at the basic. (An example is the song of Seleikos, starting with an upgoing fifth.)
In the Christian era up to the medievals the favorite harmony was also the fifth (in 2/3 marking the Holy Trinity and also the 2 natures of Jesus) - An example we find in the hymn, - some scientists gues, originated at Abrosius of Mailand: "Who You are three in unity ..." (Evang. Hymnbook Nr. 470) - But there the central note is the fifth-note (Trinity - God's Holy Spirit-presence!), not the basic!
According to medieval philosophy the basic has to be leeded by the spiritual Trinity. (A proof are the German note-names with their fifth-notes from (A+e) - (B!!+f) - (c+g) - (d+a) - (e+h!!) - (f+c')) ... Therefore this "under-fifth-parallel-harmonization" makes medieval music sweewing, not standing, but therefore uncompatible with our affinity to accord-hamonization.
Interesting is a similar phenomenon as in pre-ancient Greece, in the "ars-nova-style" of Perutinus de Notre Dame in the 13th century: The jump from the fifth e to a, declining over g is typical: So it marks also a new seeking for Personality, in analogy of the encreasing ability to express feelings e.g. of Jesus on the cross in peintures and carvings. Later on in the 14th century by rising mysticism, Gulliaume de Machaut is prefering the fifth-sound, and Hildegard von Bingen, trying to find her identity between her noble provenience and monastic life, is gently switching from fourth- to fifth-intervall-sounds.
But what happened at the beginnig to the new time of the 16th cenury? - The e',  up to this time, harmonized with a (fifth - 2/3 Trinity, God's Holy-Spirit-presence), was now harmonized with c' (big third - 4/5: 4:opposite of human and God AND human and human - 5: Creation of human body (5 fingers / 5 members (inclusive head)). As a result the new harmoniy was g-c'-e', that is C-Major, basing on creation, or Christ as the HUMAN face of God. -> time of humanism and renaissance).
In the times of Baroque by its bass-melodys of A-B-c-A, this correspondence of God (A) and Jesus Christ c) was perfected.
In the following epoques (Enlightment and Romantic) with their tendence to reduction and unification the basic-note-harmony was preferred (in thinking: Affirmation of the MONarcs by the grace of the ONE God - in matters of belief, there were the "unitarians", who, made up some congregations especially during the 19th century...)
My desire woud be the natural fact, if A and c come togeather, the natural first overtone coincidence of e' could make a harmonization of God's Spirit-presence, as Jesus told in John 4, 23: "... that the true adorers will adore the Father in Spirit and truth ..." - Therefore this over-fifth could be the natural band of a minor-accord (overtone coincidence at fifth and also seventh: At a-minor A-c-e: e'' and g'') or diminuished accord (e.g. at A-c-e(flat): g'') sorry, in this system, major-accords can not be used, therefore the full chromatic 12-notes-scala has to be used for getting the chance to select the most effective of the 3 triads contributable to one melody-note ... Finnally there remains the question, if the fith-overtune makes the harmonic band: If this fifth note ist just the note in the melody, e.g. the e' and contributing the triad a-c'-e': Should the overtune stay in the flow an e''' OR should this e' be the overtnote itsself and exceptionally the two basic nots a and c' should projected in the bass as AA - C to follow the natural overtone-coincidence of the first plus the small third with their overtone-coincidence of the fifth two eigth higher?