There are two standard ways to write chord progressions.
The first form looks like this: C Am F G7
* The capital letter is the root of the chord (C, A, F, or G)
* If it’s followed by a lowercase m, it’s a minor chord. If not, it’s a major chord.
* If there’s a 7, it’s a 7th chord, it means that you add the flat 7th note of the scale to the chord. (find the 7th note and go one half-step lower)
* You might also see a 6, meaning a 6th chord – add the 6th note
* You might also see other more obscure symbols like sus4, #9, M7 etc. – those are all just telling you to change a note in the chord or add an additional note. There are plenty of books and references telling you about all of those chords. You can play 95% of pop and rock music with only major, minor and 7th chords, but some music is more harmonically complex and has more complex chords.
The other form looks like this: I vi VI V7
* This form lets you play the song in any key, not just one key
* In this form, the roman numeral represents which note of the scale to start. So if you want to play in the key of C, I means C, II means D, III means E, and so on. If you play the same song in G, then I means G, II means A, and so on.
* Uppercase roman numerals means a major chord, lowercase means a minor chord
* The 7, 6, etc. means the same
If you’re playing chords on piano or guitar, you can just look up how to play any chord in many online references or cheat sheets that show you the exact notes and fingering. Of course if you’re studying theory it’s important to understand why, but if you’re just trying to learn to play you can just memorize what notes.
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