Just Intonation (JI) represents the pure harmonic structure of sound.
This is the inherent nature of sound.
The first thing to notice is they repeat at some frequencies.
This is indicated by color of the chart.
This explains why an octave, fifth, fourth, M3, m3 and 6th
    are more resonate notes and used harmonically.
The next thing to notice is that some notes are different
   than in equal temperament.
For example the major third (M3 5/4) is 14 cents flat
    and the minor third is 16 cents sharp.
Therefore to have pure resonance, without incoherent beating,
    or to play against a drone,
    one must slightly alter the equal temperament value.
Also of vast significance is the fact that other great notes exist
    outside of the twelve classic notes.
Even if playing music which is basically equal temperament,
   these notes are the sweetest spots to 'bend' to
   or to use as accent or grace notes.
Awareness of the nature of sound can greatly enhance
   any musicians ability to express feelings
    and move their audience.

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Click here for 100 Harmonics in a PDF.
Note that some ratios reduce and represent constructive interference.
This is why the 5th is most promenent, followed by teh 4th, ...

PDF: 100 Harmonics