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I am learning the construction of 7th chords and this Cminmaj7 chord made me really confused when I came across in my studies. Why do both minor and major 7 exist together? Also, is this a consonant or dissonant sound? Can you please explain theoretically the reason for its use and where to apply it in practice?

For this question I used C as the tonic and illusrated some of the formulas that I know and are commonly used.

Tonic + major 3rd + perfect 5th + major 7th = Cmaj7

Tonic + minor 3rd + perfect 5th + minor 7th = Cm7

Tonic + major 3rd + perfect 5th + minor 7th = C7

Tonic + minor 3rd + diminished 5th + minor 7th = Cm7b5

But what is Cminmaj7?

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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:

  • this chord can be thought of as being non-diatonic - it cannot be created from the notes of any of the related diatonic modes. It is, however, tonal; it is chord I7 of the Harmonic Minor scale (or Melodic Minor Ascending).
  • it is this non-diatonic basis which gives this chord its "astringent" character, making this chord an evocative sonority in its own right, irrespective of context within a key or chord sequence. Specifically, it is the non-diatonic relationship between the seventh, root and minor third (a semitone and minor third in closed position), which give the chord this character (and contribute strongly to the character of both the Harmonic and Melodic Minor scales, too).
  • this chord is often created as the result of chromatic movement of a single voice (note within a chord) with a minor triad. (This is particularly common in popular music, but certainly not exclusively.) See the example below:

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This chord is the tonic seventh chord of the harmonic and melodic minor scales, because both have a minor third and a major seventh. As you've noted it has a very special sound to it, often perceived as quite dissonant. If the corresponding chord scale is melodic minor, then it can be replaced by a m6 chord, which has a major sixth (in C: C-Eb-G-A).

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