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The piece is Chopin's waltz in A minor op. 34 no. 2, measure 70:

image of score snippet

We seem to be in the key of A minor due to the cadences in measures 71 and 73. But B-flat major is not in the key of A minor (right?). Is there a concept in music theory that explains this or is it just an irregularity? I must be missing something.

Link to full score on IMSLP.

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    There's nothing in 'the rules' - there are no 'rules'! that say anything shouldn't be wherever it is. Just because it's not 'in key' (whatever that might be) does not mean it can't be played. Does it sound o.k? Fine, then! – Tim Jan 11 at 20:03
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This is a very common concept known as the Neapolitan chord.

In short, the Neapolitan chord is typically a major chord built on the lowered second scale degree; you'll occasionally see/hear it called the ♭II. It's also commonly in first inversion, so you'll also occasionally hear it called the "Neapolitan six(th)," the "six" indicating the figured bass for a first-inversion chord. The Neapolitan chord functions as a predominant, often moving to V immediately afterwards (as it does in this example).

In the A minor of this example, scale-degree 2 is normally B♮, but lowering it one half step gives us B♭, the Neapolitan scale degree.

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    Suggested edit: add a link to this youtube video which explains exactly what you're talking about with examples: youtube.com/watch?v=_0fWoa5jUBM – Kevin Jan 12 at 5:18
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    @Kevin Why? Since when is YouTube a normative reference? A reference to a well-know composition, or a citation from a widely accepted textbook or dictionary, might be nice, but YouTube has no more status in this discussion than this answer, or indeed this question. – Marquis of Lorne Jan 13 at 8:50
  • @user207421 - It was just a suggestion, since it illustrates exactly what this answer is talking about with examples. It's a lot easier to hear an example alongside the score than it is to simply see a score and try to picture it. – Kevin Jan 13 at 13:40
  • @user207421 I just watch the video and to be honest, the extent of my music experience is a few years of concert band in a small high school but I have to say it was broken down very well. I think it's a great addition to Richard's answer for someone who wants more background. – corsiKa Jan 13 at 16:45
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The "Neapolitan chord" answer deserves to be the accepted answer, but I'd like to add another use for Bb in Amin.

The bII can be used as a tritone substitution for the V, and particularly the V7. In Amin, for example, E7 resolves to Amin (or Amaj, but here Amin) basically because E7 has G# and D, but Bb7 also has Ab and D (assuming G# and Ab are the same note on your instrument), so you can resolve Bb7 to Amin. This is much more common in jazz than in music of Chopin's era, so I'm sure that's not what Chopin is doing in your example, but it's sometimes the right answer to the title of the question.

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The chord sequence C - Bb - F (or variations on it) is common in modal music where the scale more often uses the minor 7th. It's also a frequent element of rock music, for example in the song Can't get enough of your love.

The relative major/minor relationship between C major and A minor should be clear, of course.

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In addition to the other answers, there is a more general concept of full tone and semitone chord movement in chord changes in general. This is frequently employed as a major triad or one of its inversions. Often, the bass note sometimes is and sometimes is not a part of the harmonic triad. You also often see the inverse of this, where only the base note moves in full tone or semitone steps while the harmony sticks to the primary chords of the key with a more pedestrian chord progression (e.g., ii-V-I or vi-ii-V-I). This happens a lot in jazz but you also see it in rock, pop and classical.

I'm not sure of any theoretical basis for this other than that it's really just progressive movement along the chromatic scale in the case of semitone movement and the diatonic scale in the case of full tone movement.

From a compositional perspective, it's a relatively simple matter to just try a semitonal or full tone chord change within the context of the song you are writing and just "see" if it "works" in the context of whatever song you happen to be composing.

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