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A minMaj7 chord doesn't have both a minor and major seventh. It is a minor triad with an added major seventh. For example CmMaj7 (which, as you show, can also be written Cminmaj7) has C Eb G B. I think it is simpler and clearer to write the shorter name for this chord. I'll make a few short observations about the use and theoretical aspects of this chord: ...


1

One handwaving characteristic of the "root note" is that its frequency is the greatest common divisor of the chord notes. It takes a lot of handwaving since for one thing, with an equal-tempered tuning there is at best an approximate gcd (you have to use some scale with just intontation, and of course the choice of scale influences the result), for another, ...



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