I need a little help understanding what a particular exercise is trying to teach me. For reference, this is in the Phil DeGreg Jazz Keyboard Harmony book, exercise 2-4A. It's labeled 'Diminished / Tri-tone Sub'.

The first line has the guide tone chords listed for CM7, C#dim7, Dm7, D#dim7, Em7, Eb7, Dm7, Db7, CM7.

While I can see that some of these chords are intended to be diminished substitutions, what are they substituting for?


They are substituting for secondary dominants of the next chord in the progression. The C♯°7 substituting for A7 which is the dominant of D and the D♯°7 substituting for B7 which is the dominant of E. The only diffrence between the diminished 7th chords and the secondary dominants is only one note which is only a half step difference so there is a lot of common ground between the chords being the A and the B♭ in the A7 vs. C♯°7 and the B and the C in the B7 vs. D♯°7.

The tritone subs are also substitutions for the dominants of the next chords in the progressions, however they resolve in an opposite way of typical dominants. So for example the D♭7 to CMaj7 is a substitution for G7 to CMaj7 and the resolutions follow similar patterns like so:

D♭7    CMaj7     G7
D♭ -->   C   <-- D
F  -->   E   <-- F
A♭ -->   G   <-- G  
C♭ -->   B   <-- B  

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