I'm learning about secondary dominants and noticed references to secondary mediants and secondary submediants in a resource (Wikipedia). However, I haven't found any other references to them in internet searches. I think I could guess at the function of these types of chords but the lack of information leaves me wondering if they really exist in the lexicon of music theory. I also wonder if there isn't a different and more practical explanation for whatever function secondary mediants and submediants might play. Can anyone offer any clarity here?
These chords theoretically exist, but in practice they're pretty rare. Logically, they make sense: secondary submediants function as vi/x, while secondary mediants function as iii/x, x being whatever chord is being tonicized.
Consider one possible example here:
We begin with a clear i–V–i in the key of C minor, and then we move to a V–I in the key of E♭ major. But notice that beat 3 of the first measure also fits into the context of E♭ major; more specifically, that C-minor triad can be understood as vi in the key of E♭. This is one possible use of a "secondary submediant," but most would tend to view that C-minor chord as i of tonic as opposed to vi of E♭. In other words, there's typically an explanation more simple than "secondary submediant."
The other instance where secondary submediants might be a real thing are in what we call extended tonicizations. In short, an extended tonicization is halfway between a tonicization and a full-fledged modulation. Instead of just one or two chords that tonicize a new key, we have several chords, but we still never reach the cadence that's required for a full modulation. During such extended tonicizations, we'll often have chords like ii or vi or iii of the temporary key. In that sense they are "secondary submediants," but typically we just address them as submediants within an extended tonicization.
In short: they exist in theory, but they're pretty rare in practice.
Lastly, note that "secondary submediant" is distinct from "a secondary chord built on the submediant." "Secondary submediant" suggests vi/x, whereas "a secondary chord built on the submediant" ultimately means V/ii. I didn't consider this latter interpretation because it's clear from the Wikipedia link that "vi/x" was intended.
The chords have other spellings depending on their function... these type of secondary relationships are more common in the later/impressionistic/neoclassical era.
A few examples: * Secondary mediant ie, Major: III/IV or III/V (as opposed to VI or VII, or V/V/V or V/iii) * Secondary submediant Major: VI/IV or VI/V (as opposed to II or III, or, V/V or V/vi)
For examples of styles incorporating this expanded notion of mediant motion, see https://www.jstor.org/stable/745872
Think 'six degrees of separation' ... what chords can't be related to one another?
A fifth is the dominant ... how about a fifth of a fifth? ... just like a family tree they're related, they have to be, but just another level removed.
I'm stating it like this because you're asking 'why' ... well it's because music theory wants to name EVERYTHING and no matter the chord it has some relationship to every other chord.