The 7 modal formulas will show us how the other scales relate to ionian. We will start off by assuming that ionian is :

 
1 2 3 4 5 6 7 1


One letter for each of the seven notes. By assuming this information, we can now label the other scales and have something to relate it to. I will explain further below.

The 7 modal formulas are :

 

IONIAN - 0 sharps, 0 flats 1 2 3 4 5 6 7 1
DORIAN - 2 flats 1 2 b3 4 5 6 b7 1
PHRYGIAN - 4 flats 1 b2 b3 4 5 b6 b7 1
LYDIAN - 1 sharp 1 2 3 #4 5 6 7 1
MIXOLYDIAN - 1 flat 1 2 3 4 5 6 b7 1
AEOLIAN - 3 flats 1 2 b3 4 5 b6 b7 1
LOCRIAN - 5 flats 1 b2 b3 4 b5 b6 b7 1


What this means is that to get dorian (which has two flats) you would take ionian and just flat the 3rd and 7th notes. This is because it will be all of the exact same notes as ionian EXCEPT for the 3rd and the 7th, as the chart above shows.

 

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