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I am just starting to learn music theory so if I'm making an obvious mistake, forgive me.

Given the chord progression D major B♭ major F major C major, I have assumed that it is in the key of F major and is some form of the familiar chord progression vi IV I V. The only difference is that the D is major instead of minor.

I looked into secondary dominant chords to write the D major in the form "function/key" but I couldn't figure out how to write it since D is not the dominant of any chord in the progression.

Would it be correct to simply write it as VI IV I V with "VI" capitalized or am I missing something?

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Generally speaking, capitalized roman numerals (VI) represent major chords and lower-case (vi) represent minor.

However, you should never arbitrarily substitute chord qualities! Most of the time, a non-diatonic chord has some kind of secondary function and this must be notated in your analysis. Labeling a D major in the key of C as a II instead of a V/V will most likely cost you points on an academic assignment!

Now, there are a few ways to analyze D major in the key of F:

First, VI is a chromatic submediant of I and is perfectly viable...if it comes from or goes to a I. So I-VI and VI-I are just fine, but, unfortunately, VI-IV is not, so that's out.

Next, in a minor key, major V is often used instead of minor v as the dominant function, so V-i is perfectly acceptable. Because of this, we can use the V of any minor degree as a secondary dominant. In this case, we would use V/ii (D->Gm). But this would mean we have to resolve to ii in order to establish the dominant relationship and make it functional, so let's try again!

Clearly, context matters! So, finally, let's look at what comes next: Bb, or IV. Now that we know where we are going, let's assume we need to find some kind of secondary function to Bb. Since D major is a chromatic mediant of Bb, III-I is perfectly valid (just like VI-I above). But this is a secondary function, so in the key of F it becomes III/IV, a secondary chromatic mediant.

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You are correct: in F major, VI capitalized means D major, whereas vi in lower-case means d minor.

By the way, your A# major should be written as Bb major.

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