Theory. Modes, Part Two

I don't feel my attempts to understand modes has yet resulted in a coherent model, but here may be another piece.  This follows up on the question of how 7th chords come from modes.

It seems to go something like this.  A7th chord is constructed by assembling everyone other note of a mode.

Working through an example in the key of C...

C major 7 consists of C, E, G, A.  For this mode (remember that major = Ionian mode), the intervals between C and E is a third, between C and G is a perfect fifth, and between C and is a perfect seventh.

Because it is built off the first mode of C it is designed as a I chord (roman numeral one).

A chord build from the second mode of C (the Dorian mode) consists of D, F, G, B.  Because of the place of whole steps and semitones in the second mode, the interval between D and F is a minor third, between D and G is a perfect fifth, and between D and G is a minor seventh.

Because it is build off the second mode of C, it is designed as a II chord.  It is D minor 7.

And so on, but in particular, a chord built from the fifth mode of C (Mixolidian mode), consists of G, B, D, F.  Again, using interval relationships, this chord consists of a minor third, a perfect fifth, and a minor seventh.

Thus, this chord is designed as a V chord, and is G dominant.

To generalize...

Major 7th chords consist of a major 3rd and a major 7th
Minor 7th chords consist of a minor 3rd and a minor 7th
Dominant 7th chords consist of a major 3rd and a minor 7th
(They all contain a perfect 5th).

This is true for all keys.  Here's a test for the reader.

What are the I, II and V 7th chords of F major?
Of F dorian?

Answers to be followed in a later post.

Oh, and why is this important?  The chord sequence II-V-I is the most common chord sequence in jazz.