I got sucked into two books last week, followed by a trip to the in-laws for our third Christmas celebration of the season ("On the twenty-eighth day of Christmas, my true love gave to me, a garbage bag full of used bows"). Thus I already am behind on my new series. So I'll try to catch up today. Related to last week's question, I had two visitors wondering about mediant relationships, one specifically about chromatic mediant chord progressions.
The mediant relationship describes two chords or two keys that have roots or tonics a third apart. The interval of a third divides the perfect fifth in half, thus it is the middle or mediant. Plus, way back in the Renaissance when theorists like Zarlino were first conceiving of the idea of chords, one way of generating a triad was by finding the harmonic or arithmetic mean of the root and fifth. One gives the major triad, the other the minor triad (I can't remember which is which right now, any pointers would be helpful). So the interval of the third was considered the middle of the fifth, thus the mediant.
Mediant relationships are special, because the two chords or tonalities share many notes in common. As an example, the two chords that have the diatonic mediant relationship with C major are E minor and A minor. Each of these chords have two notes in common with C (E and G, C and E respectively). While sharing these notes, the two chords or tonalities do not form the typical tonic-dominant relationship, so there is something slightly exotic about them.
Chromatic mediant relationships are even more exotic. E minor and A minor both fit in the key of C major, thus they are diatonic relationships. Chromatic mediant relationships do not share as many notes in common, and in fact could not be within the same key. The full definition of the chromatic mediant relationship is two chords or keys with the following characteristics: 1) the roots or tonics are a third apart, and 2) the chords or tonalities share the same mode (major or minor). There are four tonalities that are chromatical mediants of C major: Ab major, A major, Eb major, and E major. Each of these tonic triads only share one note with C major, though the other notes are very close. Because these tonic chords do share one note in common, smooth transitions can be made between these very foreign keys. Beethoven often enjoyed using common-tone modulations to move from one key to the next, such as in the third movement of his seventh symphony. Here, the two major themes are in F major and D major respectively, with A as the common tone. Within the first theme the key modulates from F to A and back, another chromatic mediant relationship.