The Neapolitan Sixth chord (usually N6) is a major triad built on the lowered 2nd degree of a major or minor scale/key. The sixth, as you'll recall from this post, meaning that the triad is in first inversion. It still functions as a II chord: that is it is usually followed by the dominant (or I64), and like any II chord it can occur in either major or minor settings.
Here's an example from Coste's Quadrille No. 2, op. 3 -- this one's in E minor, meaning that the Neapolitan chord is an F major, and since it's in first inversion it has an A as its lowest note. (click on the image to make it bigger):
And here's a nice extended usage from the end of Tárrega's Maria (Gavotte).
The key here is A minor, so the Neapolitan sixth is a B-flat major triad (with D in the bass).
Here's another example which uses the Neapolitan as a pivot chord in modulating. Because the Neapolitan is a major triad it is found in other keys diatonically. The following is from Carcassi's Etude no. 9 op. 60. It's in A minor and goes to the N6 chord, but instead of going on to cadence in A minor it pivots here to D minor (B-flat is the sixth degree of D minor).
(N.B. In the above Roman numeral analysis both of the VII chords are diminished and probably should have little circles next to them, too, to help call attention to that fact.)
So these are some uses of the Neapolitan: they are, in fact, widespread, and even though the above examples don't illustrate it they are used also in major keys. Also they aren't always used in the first inversion: Segovia in his Remembranza (Etude II) uses the Neapolitan in 2nd inversion (i.e. N64). Molino in his first Guitar Sonata (2nd movement) uses the Neapolitan in root position (the N/V...an F major chord going to E major, overall in the key of A major).
That appellation of Neapolitan has no significance, btw...just like the augmented sixth chords which are called Italian, German and French. More on those in a later post.
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