Before you venture into this ... please study the previous post:
https://fretography.blogspot.com/2023/03/diatonic-symmetry-galore.html
Now ... where were we?
Ok ... we're making connections between modes and keys across the diatonic system. Modes do not only exist in separate keys with walls around them. The cross key relationships we find in so much music have their own logic, grammar and flow.
The character of each mode becomes a pathway from one musical idea to another. We sense when the key has changed, and it is modality that provides the framework of coherence of the harmonic and melodic threads.
Since Dorian mode is diatonically central, lets see how it connects to the seven keys drawn from its traverse of natural tone finals ...
The natural tones across the top, bottom, left and right of each diagram are the finals of seven modes in seven keys. The tonic of each key is found where the blue lines cross.
The blue lines are anchored on each mode with D as the final.
Since all the modes stem from natural tones, it's easy to trace the connections between the keys through their modes. Each mode receiving this treatment will yield a different patter.
Particularly relevant is the connection between the placement of the secondary keys (blue lines) and the common tones (grey lines) and the symmetrical modal relationships.
Look at the palindromic pairs of modes ... those modes who's interval structures mirror one another:
It becomes clear that while there are symmetries to be found in the treatment of the natural tonic (shown in the previous post), the symmetry is far clearer when applying the logic to the natural supertonic ... D.
Notice that all seven converged keys have natural tonics, where using C as the finals of all the modes led to the keys of C, G, F, Db, Eb, Ab and Bb. Also notice the the blue line crossings always converge with a grey line, which does not happen with the yellow lines in the previous post.
The symmetry is clear, which is to be expected with Dorian mode, and why this mode is so essential as a starting place for understanding the Diatonic System
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