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The major II chord (whose root is the supertonic) is actually a temporary modulation to the key of the Dominant, and is called a secondary dominant.

Given your chords I'm assuming we are in C Major, which already has the major I, IV and V triads (the chords C, F and G). The two, three and six chords are normally all minor (IIm, IIIm and VIm or in C Major: Dm, Em and Am). In C Major, the minor two triad (IIm in jazz / ii in classical) consists of the notes D, F and A, but here the D major triad has an F# instead of the F natural as it's third.

One of the things that happened to music when harmony became part of it, about five centuries ago, was the ability to change keys. The Baroque and Classical mentality was to present a theme in the home key and then at some point modulate to the dominant and subdominant or to the relative minor or tonic minor and then eventually modulate back to the home key. Over time, the idea of modulating to a different key has been modified to just "borrowing" a chord or two from another key. That's basically what's happening when you use a Major II chord.

The raised F to F# in C Major is actually the proper Leading Tone of the key of G Major (it's "ti" note). If we were in the key of G major, that D triad (the V triad) would be the notes D, F# and A - the dominant chord. Notice also that G is the dominant of C major. So the D major triad in the key of C Major is the dominant of the dominant. We have temporarily modulated to the key of G major and used its dominant chord (the D major triad), which is usually followed by the G chord, making a proper V-I cadence in the key of G Major. The G chord is also the V chord in C Major and becomes the "pivot chord" to get back into the key of C Major. Because the modulation is so temporary, we don't usually perceive that we have gone to another key (even though we sort of did). The II Maj. Triad is frequently used in this manner as a secondary dominant and is notated as the V/V chord (read as "the five of five" chord - meaning the dominant chord of the dominant chord).

That same II Major chord can also be "borrowed" out of a key's relative minor. If we go to the relative minor of C, we end up in A minor, which has a minor V chord in the natural minor - in this case the D minor triad. However, if we use the harmonic version of the scale this gives us a Major V triad (the notes D, F# and A). When a II major triad goes to the V chord it is being used as a secondary dominant. When it goes to a different chord it has been borrowed from another key, but in both cases it is notated as if it was a secondary dominant as in: I - VIm - V/V - V - I.

As far as that goes, any chord in the scale can be approached by it's own secondary dominant: In C Major the A major triad would be the V/VI chord making a progression like: C - F - C - Em - E - Am - G - C possible, where E major is the secondary dominant of the Am (VIm) chord.

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