Let’s talk a little about the min6 chord. The minor 6 chord is technically spelled 1-b3-5-6 (notice the major 6). There is a complication here though. If we take the three diatonic minor chords, the ii, iii and vi chords and harmonize them to min6 chords, we find that only the ii chord harmonizes to a true min6 chord. The other two have minor sixes, which makes them into minb6 chords. These are nice chords, just a little less common. But the thing that you really have to keep in mind here is that, unlike the three major diatonic chords, you can’t replace any minor chord with a min6 chords. You can only really do this with the ii chord. You can, of course with a little common sense replace the iii and vi chord with minb6 chords (just be weary of melody notes and context).
A lot of time they pop up in progressions like this (the root of the minor chord moving down in half steps):
When we look at the whole diatonic system harmonized as 6 chords we get this:
Difference between 6 chords and 13 chords
The difference between the two chords are simple, the 13 chords contains 7ths. A lot of people worry about where the 6th is placed (above or below the 7th in the chord) but I don't think it makes any difference.
By the way, the 9th and 11th have no bearing on 13th chord, they are total options.
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