have trouble understanding the concept of "modal triads". Some people use terms like Lydian or Phrygian triads, but if you build tertian triads on a Lydian scale you end up with "standard" (major/minor/dim/aug) ones. If you use other triads, like quartal triads for example, you end up with different triads, but not 1-#4-5 (Lydian triad). Thank you.
The standard meaning of 'triad' is a set of three notes that can be stacked vertically in 3rds. 1,3,5 of some (7 note) scale.
So it's a bit of a misnomer to speak of the 'Lydian triad' (1-♯4-5) and the 'Phrygian triad' (1-♭2-5). But the terms are generally understood as describing the chords, built on the root of a Lydian or Phrygian scale, that may be seen as stating the essence of those modes. The 'tonic triads' of those modes if you like.
So we start of with understanding what I triad is. It's just a 3 note chord and the way we label these triads are with numbers and their relations to the scale degree (the major scale to be precise) for example a C major triad is 1 3 5 (CEG). I'm assuming you understand that so if not type a comment and I'll explain it in more detail.
So if we look a the "modal triads" what we are doing, is extracting the main flavours from the modes themselves and putting them into chord form.
Start with the Ionian Triad. well that's just 1 3 5 (a major triad)
Aeolian or minor triad is 1 b3 5 (pretty standard)
then lets look at a dorain triad. The dorian Mode is 1 2 b3 4 5 6 b7 8 so in the first 5 notes theres nothing different about it from the Minor scale. Meaning you still get a minor triad but you can add a 9th (2nd note) or the 11th (4th note) which you still get from an aeolian triad. Therefore we don't get dorian traids.
Phrygian is different, the scale is 1 b2 b3 4 5 b6 b7 8. So in this, we know we can get a minor triad but looking at it theres a b2. Meaning we can have a triad which is 1 b2 5. Which varys from the sus 2 triad from a minor or major scale being 125.
Lydian is the same but just as a major scale. Having the formula 1 2 3 #4 5 6 7 8. The #4 can be replaced from the 3 giving the lydian triad as 1 #4 5.
Mixolydian doesnt have one.
Locrian is is 1 b2 b3 4 b5 b6 b7 (you could have a feild day with this) therefore we have the diminished chord. Then you can insert a b2 giving 1 b2 b5 or a 4 giving 1 4 b5.