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If in the key of A major the chord C#minor forms part of the chord progression, then how does the chord progression A C D E work also in the key of A

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It works because it can sound good! The theory behind it is that chords 'borrowed' from the parallel key will fit within both major and minor. Thus the C major chord comes from the key of A minor, and brings with it other chords, such as G.

So, the list of chords available to be played in a piece in A major includes - A,D,E,F#m,Bm,C#m and G#o AND Am, Dm,Em,C,F,G and Bo.

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Remember the theory in circle for fifth which helps you to understand the theory of Chord progression.

For any Major scale – the progression can be achieved by the chords - I (major), ii (minor), iii (minor), IV (major), V (major), vi (minor) and viio (diminished).

A major is a major scale based on A, with the pitches A, B, C♯, D, E, F♯, and G♯. Its key signature has three sharps.

Relate to circle of fifth. –

it's an example of subdominant (IV) and dominant (V) chords leading back to the tonic (I). If we have a tonic note we can use the circle of fifths to give us the subdominant and dominant chord of that key. Just locate the tonic in the circle, let's use A as an example, and then locate the keys immediately adjacent on each side. So, if we're using A as the tonic the subdominant would be D, immediately counter-clockwise, and the dominant would be E, immediately clockwise.

Also just to remember the Chord progression – you move in a sequence of notes in A Major scale **A, B, C♯, D, E, F♯, and G♯. ** Such a way by viewing the circle of fifth **A (I - Major) in outer circle - B (ii - Minor) in the inner circle - C♯ (iii - minor) in the inner circle – D (IV - major) in outer circle - E (V -Major) in outer circle - F♯ (vi -minor) in the inner circle.

http://musictheorysite.com/the-circle-of-fifths

https://en.wikipedia.org/wiki/Circle_of_fifths

The most common chord pattern is 1-4-5-1 pattern found in all types of music. So it forms I (major), IV (major), V (major), I (major), So the chord progression will be A Major chord - D Major Chord - E Major Chord - A Major chord. The other chord pattern are I - vi - IV – V and it forms I (major), vi (minor), IV (major), V (major) and the chord progression will be A Major chord – F# Minor Chord - D Major Chord - E Major chord.

This is also can be a pattern 1 - 3 - 4 - 2 - 1 So it forms the Chord progression I - iii - IV - ii - I Chord Patterns), i.e. A Major chord - C# Minor Chord - D Major chord - B Minor chord - A Major chord.

This is another chord progression pattern and it forms I - vi - ii - V - I chord progression.

As per the theory - There is no C in the chord progression and it is C# Minor. Sometimes accidental chords can be used in some songs (Which is out of the basic rules & theory).

It all depend upon how it sound to the ear and every rule can override the if it sounds pleasant to the ears.

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    How does the circle of fifths relate to M,m,m,M,M,m,o? I'd also question your statement in para 5. That's not a particularly common seqence. – Tim Nov 22 '16 at 6:50
  • Dear Tim- i have explained the arrival of tonic (I),subdominant (IV) and dominant (V) chords using circle of fifth.Earlier - in the 5th para - i was intend to write "one of other example chord pattern" and not the common pattern. Apologies for confusion. – robert winsly Nov 22 '16 at 9:04
  • Thanks, that makes more sense, although it doesn't address the question about how 'C' fits into the key of A. – Tim Nov 22 '16 at 9:12

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