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What is it called when a chord progression in a specific key uses root/tonic notes that are in-key, but other members of the chord are not?

For example, say I am in the key of C major and I pay C Maj, D minor, E Minor, and then F Minor instead of F Major. In this instance, F is in the key of C major, but the chord should be F major, but I am playing an F Minor...

A) Is this concept used often in music?

B) Does it have a name?

C) Are all of the notes that make up every chord in a given scale always in the scale themselves?

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A) Is this concept used often in music?

Very often. Enough to say it's the norm.

B) Does it have a name?

The most general term would be "chromatic alteration", but there are other terms to describe specific uses. For example, the "F minor in C major" example would likely be attributed to "modal mixture" — the interchange of chords from the parallel major and minor modes. F minor is a part of C minor.

There is also the concept of "secondary chords", which are chords that function as though the music has moved to a different key. D major in the key of C major, for example, might very likely operate as a chord leading to G major.

Searching this site for terms like "chromatic chords" or "secondary dominant" will give additional and more detailed information.

C) Are all of the notes that make up every chord in a given scale always in the scale themselves?

When we speak of a chord being "in a scale" (or more likely, "in a key") it means that all of the notes in the chord come from that scale/key. However, just because a piece of music uses a certain scale or key does not mean that every chord will use notes exclusively from that foundation.

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