A La Modal and other questions

Now in the article the Green Day song uses the chord progression A - G - D - G, and it's in A mixolydian. A mixolydian is the 5th mode of the D major scale, so the song is essentially using a V - IV - I progression but it's using a mode, how is this possible?

No, if the song is in A anything (major, minor, Mixolydian, whatever) then A major (or minor) is the "one" chord. The key centre is always the I (or i) chord. So the progression is I - bVII - IV - bVII.

Sorry, that's right. What I meant to say was that the progression is using the I- IV - V chords of D major, even though it's a I - bVII - IV - I progression in A. I thought that playing a mode other than the major scale over the dominant chord (5th chord in the key, which for D major would be A major) in a progression would not work, since the dominant chord would want to resolve back to the tonic (D major), hence making any use of a mode simply sound like it's relative major scale.

Yes, if the chord was A7, instead of A, (and if it resolved to D) there would be a tendency to hear D as the tonic instead of A, making A Mixolydian sound like the D major scale.

Hello. Just a quick reply here. Harmony is not seperate from modes. Listen to Miles Davis "So What" From Kind of Blue. This is one of one million examples of a modal composition. You will notice in these Modal compositions many times that the chords are not built in a conventional manner either. In other words the chords are not built in thirds (Tertial) however built quartaly (in fourths).

Also, it is not true when moving from V-I that you can only use one mode. You could, however that does not offer the coolest sound or the best voice leading. Think of Modes strictly as sounds that work over certain applications. For example: V-I in D, you could use an A Altered scale over A7 which happens to be the 7th mode of Bb melodic minor, and then resolve to D Major (Ionian)

Just some thoughts

I'm going to sharply disagree with you on all four points you've made.
Harmony is not seperate from modes

Modes are melodic devices, and they always have been - early modal music, like Gregorian chant, was monophonic. It didn't have harmony at all. Modern modal compositions, like "So What" use a restricted set of chords (sometimes referred to as 'modal harmony') to avoid having conflicting tonal centers.

If you think of modes as Glareanus did - the major scale 'starting from' a different point*, you can see why this is true. The tonal center of a melody will be its tonal center, or 'starting point'. Change the 'starting point', you change the mode.

*this is not a practical way to think of modes while playing, because you don't need to start a melody from the tonal center, but it helps in understanding the point I'm making here.

When you deal with harmony, you're adding a vertical dimension to the music. There are two ways to think about this extra dimension...

If you think only about the vertical dimension, you're dealing with a chord. The equivalent of the tonal center (or 'starting point') will be the root note. But with an individual chord, if you change the starting point you only change the inversion; the chord remains the same. If you 'start' a C major scale from G, you get a G mixolydian mode; if you 'start' a C major chord from G, you still get a C major chord.

The other way to view harmony is in a functional sense, considering both the vertical chords and their evolution through time: the 'chord progression'. Looking at a chord progression, you can identify a tonal center - it's defined by the V-I, the "authentic cadence". This chord motion is often enhanced by using V7, adding a tension to the dominant chord (which is why these chords are called "dominant sevenths").

If you harmonize a mode, you get the identical tension as you do in a major scale - but that V7 chord, which leads strongly to I, won't fall on the V. In D Dorian (the key of "So What") it'll fall on the IV.

Because using a dominant 7th chord would create a functional dominant chord, which in turn will define a functional tonic (or tonal center), "modal harmony" avoids the use of functionally dominant chords. You'll notice that "So What" does not use any chords with a G root... and that's why: using one would define a harmonic tonal center of C, regardless of the mode, and that would conflict with the melodic tonal center of D.
n these Modal compositions many times that the chords are not built in a conventional manner

Again that's a separate issue. Quartal harmony uses chords constructed in fourths - the most basic quartal chord would be two perfect fourths, creating what's often noted as a sus7 chord (C-F-Bb rather than C-E-G-Bb). Although this building block has '7' in the name, there's no third... so there's no internal tritone, and the chord doesn't have the tension of a dominant 7th.

Since there's no tension, quartal chords aren't functionally dominant, and therefore they don't define a tonal center. That makes them suitable for use over modal melodies (because there's no harmonic tonal center to clash with the melodic tonal center), but the presence of quartal harmony doesn't mean you're in a mode - many artists like Chick Corea, Horace Silver, etc. have used quartal harmonies over major scale melodies.

Although there are some folks (like Mark Levine) who look at "So What" and see quartal harmony at work, most theorists aren't in that camp. Miles' group used m11 chords voiced without ninths, a very common chord simplification. If you play the chords Em7 - Dm7 over the melody, you hear the chord progression; if you play the implied quartal harmony (Esus7 - Dsus7) you get a result that's a bit more ambiguous. That's because you're taking the chord simplification a step further, and dropping TWO tones from the chords - the 9ths they didn't play, and the 3rds that they did.
it is not true when moving from V-I that you can only use one mode. You could, however that does not offer the coolest sound or the best voice leading

"Voice leading" as applied to modes opens up a whole 'nother can of worms. Yes, there is such a thing, and no, it doesn't apply in the way that you're throwing it into the mix.

Voice leading can refer to counterpoint - how one voice moves against other independent voices. It can also refer to how a piece is arranged - how an individual voice fits into a chord progression. The first use results in "modal counterpoint", which is the style of music of the late medieval, early baroque, and all of the renaissance periods of music. If you want to write a fugue like Palestrina's, you're using modal counterpoint - but you don't have "chords" to worry about ("chord progressions" are a feature of homophonic, or chord-based music, not polyphonic music composed of multiple independent voices)

In the sense of arranging, there are lots of approaches, and the "best" voice leading depends on what the chords are doing. I'll dumb this down to some simple rules: rule #1: the melody should end on its tonic. Rule #2: the melody must contain a chord tone for each chord used.

Now let's look at a V-I progression, G -> C. There are lots of modes that fit those two rules: any mode rooted on C, E, or G can end on the tonic and agree with the last chord. Since "modal" music typically uses one of the four church modes (Dorian, Phrygian, Lydian, and Mixolydian), that gives us 12 possibilities - each of the four church modes times each of three possible root candidates.

Now let's look at simple voice motions: a melody line can rise, fall, or stay put. Staying put (called "oblique" motion) means the note must be in both chords - for any G-rooted mode, that works, and you can get the "best" voice leading by simply staying on G. If the melody rises at the end, "best" voice leading generally means it rises by a half step; since the only chord tones that are a half step apart in a G -> C are B and C, you could use either C Lydian or E Phrygian and still have the "best" voice leading. And if a melody line falls at the end, best practice is for it to fall by a whole step or a third. That means we can move from chord tone D to chord tone C (C Dorian, C Lydian, or C Mixolydian), or from chord tone B to chord tone G (G Dorian or G Phrygian). That's still nine possible modes, each of which can result in the "best" voice leading.

Now if you want the mode to sound like a mode against the harmony, it's a good idea to have the tonality of the melody agree with the tonality of a chord progression. Since we're in a major key with a G->C cadence, we can toss out all the minor modes. But that still leaves four choices: C Lydian, C Mixolydian, G Lydian, and G Mixolydian. Any of these can still have the "best" voice leading.

In practice, G Mixolydian is going to end up sounding like C major. They have the same tones, so it'll sound like you're simply ending the melody on the fifth. But the other three modes CAN and DO work over a G->C cadence. They just sound a bit off, like you're not sure where the tonal center is... which is exactly why modal music usually avoids harmonic cadences, and modal tunes tend to use simple chord vamps (like "So What" does) instead of functional harmonic progressions.
you could use an A Altered scale over A7 which happens to be the 7th mode of Bb melodic minor, and then resolve to D Major

Although you can think that way when you play - it's a perfectly legitimate approach to improvisation - the result isn't modal music. Using a scale over a single chord doesn't establish a tonic - which is why you have to resolve somewhere else. Seen as a whole, the result will show chromaticism, rather than the use of a mode.