I'm looking at the chord-scale system and polychords as a shortcut for coming up with nice Jazz voicings.
We start with a scale, for example, C Lydian.
Playing all the notes of this scale at once, we hear a big Jazz chord.
We'll give each note some space to breathe by stacking thirds.
We can decompose this chord into overlapping triads.
Playing Em, G, Bm, or D (possibly in some inversion) over C gives us polychords that correspond to the following chords with a few notes doubled.
In terms of polychords these are much easier to write down and remember.
Of course, each chord may have different voicings: we simply play different inversions and leave out or double notes as we see fit.
To generalize over key we can use Roman numeral analysis. I'm using the traditional notation based on the major scale, so flats and sharps indicate deviations from that scale. For the Lydian mode, the overlapping triads are I (C), iii (Em), V (G), vii (Bm), and II (D). For all seven modes, we get the following table.
| Mode | 1 | 3 | 5 | 7 | 9 | Chord |
|---|---|---|---|---|---|---|
| Ionian (major) | I | iii | V | viio | ii | I▵13 |
| Dorian | i | ♭III | v | ♭VII | ii | i13 |
| Phrygian | i | ♭III | vo | ♭vii | ♭II | i7♭9♭13 |
| Lydian | I | iii | V | vii | II | I▵ ♯11 13 |
| Mixolydian | I | iiio | v | ♭VII | ii | I13 |
| Aeolian (minor) | i | ♭III | v | ♭VII | iio | i7 ♭13 |
| Locrian | io | ♭iii | ♭V | ♭vii | iio | i7♭5♭9♭13 |
As a little exercise, pick a mode and a polychord and move through the cycle of fourths on the piano.