Maybe this well help those like me who sometimes need visual aids to understand this stuff. It's a graphic I found somewhere on the web that includes modes in a Circle of 5ths chart. It helped me understand why the music I play -- mostly Irish and Scottish trad -- often has sheet music where the number of sharps doesn't mean what you might initially think it means for the key (mode).
For example, Irish trad music (and related music like many OldTime tunes) is generally based around the keys of D and G Major (Ionian) with their relative minors, but what that really means in practice is more often the related Dorian mode. This is probably due to the influence of the pipes, whistle, and flute with a bell (lowest) note in D, and the ease of playing tunes in G and D Major in first position on the fiddle.
So on this chart you can see the relation of the outer keys G and D Major (Ionian) with their Dorian relatives just below and to either side in the circle -- A Dorian, E Dorian, and B Dorian, which you'll hear in many Irish traditional tunes. D Dorian is also there to the left of G, although not as common as the others.
Now, step way down the inner line from G and you'll find D Mixolydian, another common mode for Irish tunes. Step the same way down from D and you'll see A Mixolydian, common in Scottish pipe tunes. These aren't the only modes used in Irish trad, there are a few tunes in things like F# Dorian and G Dorian, but you won't find Dorian or Mixolydian modes far from the "home" keys of D and G.
This explains why you can't trust a key signature on Irish trad (or OldTime) sheet music like one sharp and think it's "in the key of G Major," because it could also be in the minor-sounding A Dorian or the "bright major" sound of D Mixolydian. Similarly, if you see two sharps in sheet music it might be the key of D Major, but it might also be E Dorian instead, and so on.
One might ask why are these the only modes used in Irish trad? Why not Phrygian, Lydian, Aeolian or Locrian? Well, each folk musical culture has "a sound," and for whatever reason, the sound of say the Phrygian mode just wasn't as attractive to these players. I'm no ethnomusicologist, so someone else will have to explain this one.
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