In rank-2 interval space (and kind of in meantone tuning systems) you can ignore syntonic commas, and the diatonic modes have a very simple form. Here they are spelled by seconds:
I: [P1, M2, M3, P4, P5, M6, M7], # Ionian
II: [P1, M2, m3, P4, P5, M6, m7], # Dorian
III: [P1, m2, m3, P4, P5, m6, m7], # Phrygian
IV: [P1, M2, M3, A4, P5, M6, M7], # Lydian
V: [P1, M2, M3, P4, P5, M6, m7], # Mixolydian
VI: [P1, M2, m3, P4, P5, m6, m7], # Aeolian
VII: [P1, m2, m3, P4, d5, m6, m7], # Locrian
And here they are spelled by thirds
I: [P1, M3, P5, M7, M9, P11, M13], # Ionian
II: [P1, m3, P5, m7, M9, P11, M13], # Dorian
III: [P1, m3, P5, m7, m9, P11, m13], # Phrygian
IV: [P1, M3, P5, M7, M9, A11, M13], # Lydian
V: [P1, M3, P5, m7, M9, P11, M13], # Mixolydian
VI: [P1, m3, P5, m7, M9, P11, m13], # Aeolian
VII: [P1, m3, d5, m7, m9, P11, m13], # Locrian
I've heard that the "Greek" names for the "church modes", as we learn them in grade school music theory, aren't particularly historically accurate either to ancient Greek music or to early Christian church music, so I'm trying to name the modes by roman numerals.
I think many people will have modes I and VI memorized, and if you do, it's pretty convenient to think of the other modes by how they deviate from one of those:
Mode I (Ionian) # Memorized, all major or perfect
Mode II (Dorian) # Mode VI with #6
Mode III (Phrygian) # Mode VI with b2
Mode IV (Lydian) # Mode I with #4
Mode V (Mixolydian) # Mode I with b7
Mode VI (Aeolian) # Memorized (or Mode I with b3 and b6, and b7)
Mode VII (Locrian) # Mode VI with b2 and b5
In rank-3 interval space, things are a little more complicated. Here are the rank-3 diatonic modes, both Just and Pythagorean, spelled by seconds:
Just I: [P1, M2, M3, P4, P5, M6, M7], # Ionian
Just II: [P1, AcM2, m3, Ac4, P5, AcM6, m7], # Dorian
Just III: [P1, m2, m3, P4, P5, m6, Grm7], # Phrygian
Just IV: [P1, AcM2, M3, AcA4, P5, M6, M7], # Lydian
Just V: [P1, M2, M3, P4, Gr5, M6, Grm7], # Mixolydian
Just VI: [P1, AcM2, m3, P4, P5, m6, m7], # Aeolian
Just VII: [P1, m2, Grm3, P4, Grd5, m6, Grm7], # Locrian
Pythagorean I: [P1, AcM2, AcM3, P4, P5, AcM6, AcM7], # Ionian
Pythagorean II: [P1, AcM2, Grm3, P4, P5, AcM6, Grm7], # Dorian
Pythagorean III: [P1, Grm2, Grm3, P4, P5, Grm6, Grm7], # Phrygian
Pythagorean IV: [P1, AcM2, AcM3, AcAcA4, P5, AcM6, AcM7], # Lydian
Pythagorean V: [P1, AcM2, AcM3, P4, P5, AcM6, Grm7], # Mixolydian
Pythagorean VI: [P1, AcM2, Grm3, P4, P5, Grm6, Grm7], # Aeolian
Pythagorean VII: [P1, Grm2, Grm3, P4, GrGrd5, Grm6, Grm7], # Locrian
The same modes spelled by thirds:
Just I: [P1, M3, P5, M7, M9, P11, M13], # Ionian
Just II: [P1, m3, P5, m7, AcM9, Ac11, AcM13], # Dorian
Just III: [P1, m3, P5, Grm7, m9, P11, m13], # Phrygian
Just IV: [P1, M3, P5, M7, AcM9, AcA11, M13], # Lydian
Just V: [P1, M3, Gr5, Grm7, M9, P11, M13], # Mixolydian
Just VI: [P1, m3, P5, m7, AcM9, P11, m13], # Aeolian
Just VII: [P1, Grm3, Grd5, Grm7, m9, P11, m13], # Locrian
Pythagorean I: [P1, AcM3, P5, AcM7, AcM9, P11, AcM13], # Ionian
Pythagorean II: [P1, Grm3, P5, Grm7, AcM9, P11, AcM13], # Dorian
Pythagorean III: [P1, Grm3, P5, Grm7, Grm9, P11, Grm13], # Phrygian
Pythagorean IV: [P1, AcM3, P5, AcM7, AcM9, AcAcA11, AcM13], # Lydian
Pythagorean V: [P1, AcM3, P5, Grm7, AcM9, P11, AcM13], # Mixolydian
Pythagorean VI: [P1, Grm3, P5, Grm7, AcM9, P11, Grm13], # Aeolian
Pythagorean VII: [P1, Grm3, GrGrd5, Grm7, Grm9, P11, Grm13], # Locrian
What a handy list to have around! Of course the modes spelled by thirds also specify diatonic chords.
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