Constructing modes is as simple as using a chart with a Major Scale. To be a little more specific, we're going to look at how this chart is formed.
Ionian Modes are the easiest. They are the same as the Major Scale. But lets say we want to know what D Dorian scale looked like. D Dorian is a mode of the C Major Scale.
C Major Scale -> C D E F G A B C
D Dorian Mode -> D E F G A B C D
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When we compare the D Dorian Mode with the D Major Scale. We find that the F and C are sharp in D Major but not in D Dorian.
D Major Scale -> D E F# G A B C# D
D Dorian Mode -> D E F G A B C D
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So in order to create the D Dorian Mode from the D Major Scale we would flatten the 3rd and 7th Notes. We can then create a rule for creating a Dorian Mode from any Major Scale.
1 2 b3 4 5 6 b7 1
(this represents making the 3rd and 7th notes flat)
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We can now apply other scales to this rule to get the Dorian Mode.
If we were to create a rule for every mode that could be applied to any major scale, it would look like the following:
Ionian Mode -----> 1 2 3 4 5 6 7 1
Dorian Mode -----> 1 2 b3 4 5 6 b7 1
Phrygian Mode ---> 1 b2 b3 4 5 b6 b7 1
Lydian Mode -----> 1 2 3 #4 5 6 7 1
Mixolydian Mode -> 1 2 3 4 5 6 b7 1
Aeolian Mode ----> 1 2 b3 4 5 b6 b7 1
Locrian Mode ----> 1 b2 b3 4 b5 b6 b7 1
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That's it!
You can now take any major scale and get the corresponding modes!