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I was looking at Itzhak Perlman playing Shalom Aleichem It starts (0:14) in E Phrygian Dominant with the i chord, but ends (4:16) in A minor, the iv chord.

I've seen the same thing happen with Hava Nagila where they play it in D phrygian, starts (0:28) with i chord, but ends on iv chord, G minor (2:31). did it modulate?

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Actually, the Perlman piece begins with the tonic chord of A minor; the entire piece begins and ends in A.

This piece uses A harmonic minor, it's just that that collection is equivalent to the E Phrygian dominant collection; the only difference is which pitch is tonic.

With that said, it is possible for a piece to begin in a key and ultimately modulate to its subdominant—this does happen in "Hava Nagila." In such cases, we often understand the opening of the piece as conceptually being a large dominant to the final tonic. In other words, the opening D, with a close in G, is just a big V–i (D–G) motion in G minor.

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    @foreyez I hear D as the opening tonic of "Hava Nagila," but it ends in G. The Perlman, however, definitely begins in A. The melody might begin on E, but that's definitely within an A-minor context. – Richard Jan 19 '19 at 19:08

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