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In studying music theory I have hear the term Neapolitan 6th a quite few times, but I don't know what it is. What is a Neapolitan 6th and how is it used in a composition?

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It is a Major triad built on the lowered 2nd scale degree. It's usually in first inversion, hence the "6th" part of the name. So if I'm in C-minor, the Neapolitan 6th (sometimes analyzed as N6 or bII6) would be a Db-major triad, probably with the F in the bass. They are chromatic harmonies, and their primary function is to go to V.

EDITED TO ADD: There is a very particular voice-leading involved also, usually the voice that has the b2 note moves directly to the leading tone in the upcoming dominant harmony. This is a rare example of a diminished melodic third being standard practice.

A good example is in beats 3 and 4 of the 3rd measure of the first movement of Beethoven's "Moonlight" Sonata. The piece is in c#-minor, and the 3rd measure starts on A (VI) and moves to a D Major chord in first inversion (the Neapolitan). The D-natural root of that chord is chromatic to the key, and is what causes the harmony's somewhat exotic quality. It's an alteration of the diatonic d#-dim (iio) chord that could just have easily have been used here, but would have been far less dramatic. Either way, the next harmony is V (with some cadential 6/4 action thrown in for seasoning).

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Good answer - just want to expand on the first paragraph a little bit for clarification that since an N6 usually goes to V it is considered a predominant chord. –  jjmusicnotes Feb 4 at 1:03
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Mini extra question: Can a Neapolitan chord be applied to mode that has a minor second i.e. Phrygian and Locrian? –  Dom Feb 4 at 2:00
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That's an interesting question. Technically, I suppose the lowered-2nd scale degree that primarily gives the chord it's unique quality is just natural to Phrygian and Locrian, and the weird thing would be to build a chord on the raised second scale degree. But a lot of music in Phrygian and Locrian (I'm thinking primarily of metal and industrial music) uses bII as a dominant function--a chord leading straight to i. Ultimately, once you're talking about modal stuff like that, the general functionality of common-practice chords is at best shifted substantially. –  Pat Muchmore Feb 4 at 2:35
    
Maybe it should be asked as a separate question eventually. –  Dom Feb 4 at 14:19
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@BraddSzonye, well no, 6/4 would mean second inversion. A 6 by itself would mean first inversion. In every case, it's a reference to intervals above the bass. Technically, the full basso continuo or Roman numeral exponent for a first inversion should be 6/3, but the third is generally implied. –  Pat Muchmore Jul 19 at 23:37

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