What is A flat to C minor?

I know that Cm is the relative minor of Eb. I noticed the Eb chord is made up of two of the same notes of Cm (beginning on the third degree). Ab is also made up of two of the same notes, but on the opposite side of the Cm chord.

I wondered if Ab/Cm have a particular relationship (like 'relative' for Eb/Cm)?

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About the chords sharing notes, that is Ab and Cm sharing C and Eb, in a way it's related because together you've got an Ab7. If you build a minor triad from the third, you end up with a 7 chord. That is, if Ab is the one, the third would be C, so you build a Cm and you end up with Ab C Eb G or Ab7. – Mike Hildner Dec 19 '12 at 22:00
I should have said a major 7th chord - AbM7 – Mike Hildner Dec 19 '12 at 22:26

Key Signatures

"Relative major" and "relative minor" are terms typically used to describe keys with identical key signatures and root notes a minor third apart. So, when we compare C minor and Eb major, the key signature is 3 flats and C and Eb are a minor third apart, so we say C minor is the relative minor of Eb major, or vice versa, that Eb major is the relative major of C minor. Another common example is the `0b/0#` natural key signature and the relationship between C major and A minor.

Now, if we compare Ab major to C minor, we can't get too far off the ground since Ab is a major third away from C, not a minor third. However, if we get a little creative we can come up with a logical answer:

Let's consider that both chords come out of the same heptachord (set of 7 notes, a.k.a. key signature). We need the notes Ab, C, Eb, and G, which means the fewest number of flats we can use is going to be 3 (see also: Order of flats). This is convenient, because the relative major of C minor is Eb major, or 3 flats.

Now, the "creative" part is that I'm going to extend our terminology to make use of the diatonic "church modes" (lydian, ionian, mixolydian, dorian, aeolian, etc.), so that we can talk about any keys relative to one another that share the same key signature. Aeolian and minor as terms here are interchangeable, since their key signatures are identical.

In this system, I would say that Ab major triad outlines the relative lydian to C minor, because in our key signature of 3 flats (we've already established that the relative major of C minor is Eb major, or 3 flats), the mode that starts on Ab is the lydian mode.

Sidebar: As another example of how I'd use this terminology, consider the chords G minor and A minor. It is possible to draw a relationship between them in the key signature of F (one flat) by saying the G minor triad outlines the relative dorian of A phrygian, and that the A minor triad outlines the relative phrygian of D dorian. This should work for any two chords that are possible within a single valid key signature.

Chords

This, I imagine, is going to be closer to the answer you were really looking for.

When we talk about chords, we don't use the terms "relative minor" or "relative major", instead, we talk about functional relationships. For example, G major is the dominant of C major. C minor is called the submediant of Eb major. In the other direction, Eb major is called the mediant of C minor.

These names, and a much more in-depth explanation can be found on the above linked Wikipedia article. Mediant relationships involve distances of a 3rd.

So then, if we look at the chords in your question (Ab major, C minor, Eb major) and we are in the key of C minor, we would say Ab is the submediant of C minor and Eb is the mediant of C minor.

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Actually, relative minor/major (and parallel major/minor) can also be applied to chords, at least in Neo-Riemannian theory. I've added answer to explain this. – Caleb Hines Sep 28 '14 at 17:19

Cm is the relative minor of Eb because they have the same key signature. I don't think the chords overlapping is anything other than coincidence. For example, the "relative minor" of G Mixolydian could be said to be Am, and the G Major chord (which fits G Mixolydian) does not share any notes with the Am chord. Conversely, D Dorian also has the same key signature as Am, and both Dm (in the key) and D Major (not in the key) share only the A with Am.

You could certainly give a name to this relationship you're seeing and describe other keys according to it. You can look at the differences/similarities between any two keys and transpose up or down to find another pair of keys with exactly the same characteristics, in fact. But that doesn't make the relationship useful :). In this case I don't think there is a name or obvious use for this relationship, and Cm / Ab are not closely related keys.

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In Neo-Riemannian theory, this relationship is called a Leading Tone Exchange (often abbreviated "L"), and is one of the three fundamental transformations that can be performed to a chord. The other two are Parallel (P) and Relative (R) which are much better known.

If you look at the geometric triangle that a chord makes on a Tonnetz map, these three operations (L, P, and R) can each be visualized as creating a new chord by flipping the existing one over one of its edges. This is because each of the transformations only involves changing a single note.

• A Parallel exchange (P) involves switching chords by raising or lowering the third of the chord.
• A Relative exchange (R) involves switching chords by raising the 5th of a major chord (or dropping the root of a minor chord) by a whole step.
• A Leading-Tone exchange (L) involves switching chords by raising the 5th of a minor chord (or dropping the root of a major chord) by a half step.
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