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I'm writing a piano app for a high school class I teach and I'm trying to figure out the best algorithm to name chords related to a specific key. I'll use C major as the example for this question.
I've figured out how to name some of the chords that exist in neighbouring/related keys. For example, the chord D major is borrowed from the key of G acting as a V of V. What's the connection for a chord like Db major (the bII chord)? Or a chord that begins on F# (or Gb)?
Hmm, interesting question. I assume you're using uppercase and lowercase Roman numerals to distinguish chord quality? If so, I see two options.
D major, as you've correctly said, is best shown as V/V and not II. But F♯ wouldn't be V/vii°, because that doesn't exist. But it would be silly to write it as V/V/iii, which sometimes does happen. Instead, this suggestion would be to write F♯ as ♯IV. If this F♯ chord were diminished, however, you could show it as vii°/V (not ♯iv°). D♭ would be ♭II, D♭ minor would be ♭ii, and so on.
This approach follows in the tradition of Heinrich Schenker; you can find his use of these Roman numerals in scholarly discussions of "secondary" and "double" mixture.
But that leads me to wonder how helpful that would be for high school students. (Maybe this is unfair of me to say, since I don't know the class or the students.) I wonder if throwing in all possible chords is that helpful? Are you really intending to use chords like ♯vi and ♭iii? Which leads me to my second recommendation:
By doing this, you could get rid of the nonsensical chords like ♭iii that either don't really happen, or aren't encountered until much later in one's music theory training. By doing this, it seems you could stick to the following chords in a major key:
This latter option, in my opinion, seems to mimic real musical practice, and therefore the actual music theory that they'll be using.
Modal mixture chords can be determined through the tonic minor. The issue with writing an app is that you have to determine rules that work in every situation hopefully in an elegant way.
In C, Db is the Neapolitan chord. Normally it occurs in first inversion and gets the symbol N6. F# is often called vii0/V analogous to II being V/V.
However, the sound and effects of these chords is far more importan than their names.
Sounds like a fun project, but it would be a mistake to give your students the impression that you can compute the proper, key-relative notation of a chord based only on the notes that comprise it.
For example, in the key of C major, consider a chord containing C, D, F#, and A. This could be a V of V in 42 position, or it could be a common tone chord written as I4+2. The chord has to be interpreted for its function in context. There are many other examples like this, especially if you're going to be looking at certain periods, e.g. the dissonant tonalism of the impressionists like Debussy often add minor sevenths and major sixths to chords, and telling which chord is which is sometimes pretty hard-- intervallically, they are identical. For example, a computer would have difficulty distinguishing an Am7 from a C with an added 6th.
Perhaps the program could work as an introduction to the concept, and then you could use the program to show how it isn't always correct. That would be a good lesson!
I suggest you name chords as what they ARE - a non-judgemental 'bVII' or 'II7' - let the student (and the context) decide what they DO. Sometimes it's a 'cycle of 5ths', dominants of dominants thing. Very often it isn't. For instance, in a C major song it's very common to find a Bb chord. It MIGHT be acting as 'subdominant of the subdominant' (The Beatles 'Get Back'). It may just be a pleasant-sounding 'chord next to C' with no functional reason whatsoever. Don't leap on the 'IV of IV' possibility with a 'Good! That's given THAT one a reason to exist!' attitude.
Chords don't have to be justified as 'borrowed'. Where did that idea come from? A chromatic chord MIGHT be a gateway to another key. But it equally likely might not. Call II 'V of V' if that's what happens. But not if it doesn't.
It seems like you really have two questions here. The first, what do I name these "out of key chords" for the app. The other, what relation do these chords have to the key, or what purpose do they serve. As for the the first question it seems that you have a lot of good advice. I think you want to stick to classical music theory and harmony theory,especially if this is for educational purposes.
As to the second point we often use "out of key chords" in progressions. This could be due to a modulation in key (see Max Reger). It could be a classic chromatic plagal cadence (like IV to iv (minor IV) to I). Here the minor IV is out of key but serves a purpose. Chord substitutions, especially tri-tone. Hence you may see ii --> bii^o --> I6 (or I Maj7) or something similar. For your app stick to the advice of your top answer but it would be cool if, in a future version, the app could identify these patterns in a chord chart or music. These oddities have a specific function in music. Just an idea.
