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In key signature of C major the notes used are CDEFGABC and I came to know that in Chord signature become C Dmin Emin F G Amin Bdim and C. D is major second and A is major sixth interval but it is Amin chord . F is minor, but it become major chord why?

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You are mixing two things here. The interval to the note in the scale, and the triad that is formed from a note using the scale.

D is major second and A is major sixth interval

These are intervals from the root note C. This does not say anything about the chords formed from these notes. So A is a major sixth above C, but the A chord formed with notes in the C major scale is an A minor (A C E).

Similarly, D is a major second above C, but the D chord formed is D minor. F is a perfect forth, and forms a major chord.

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There are two facts here, that are separate, but you've mixed them. The intervals, between any two notes, are what they are. So - C>D= maj2, C>E=maj3, C>F=perfect4, C>G=p5, C>A=maj6, C>B=maj7. Those represent the space in between the lower and upper notes.

The chords built on the notes from any scale, C in your case here, have a root note, and a third and a fifth. C itself is CEG, Dm is DFA, Em is EGB, F is FAC, G is GBD, Am is ACE and Bo is BDF. All those chords are diatonic and as such, only contain notes from that scale.

Two separate bits of theory that are quite confusing to beginners - you're not alone! Try very hard not to 'fill the gaps' in the theory you know with your own 'theory'. It's easy to do, but see the problems it causes!!

Not sure what your last bit means.

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Major chords and major intervals are very different concepts and independent of each other.

A major interval is an interval that is the larger of two typical distances for given degree. For example, a major third is 4 semitones in distance from a given note while a minor 3rd is 3 semitones in distance from a given note.

A major chord has a specific definition based on the notes contained within it. The note contained within a major chord form the intervals of a major third and perfect fifth. For example, a C major chord had a major 3rd which is E and a perfect 5th which is G in it.

So are these two definitions in mind we can see that in C major, A is a major 6th while the chord built off of it is a minor chord due to the intervals contained within the chord.

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F is the 4th note of C major, its interval above C is known as a Perfect 4th. ANY note in a major scale is a major interval above the tonic, except that the "major" 4th and 5th have a special name - "perfect". We'll get to why later!

Yes, some of the triads are major, some minor, one is diminished.

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