The chord progression that accompanies the first section of Greensleeves (What Child Is This) is a common chord progression known as the Andalusian Cadence. You might have heard or played it in the song “Hit the Road Jack.” Can you think of other songs that use it?

The Andalusian Cadence comprises of four chords starting with a minor chord, descending until it reaches the dominant of the starting minor chord. That final chord can be a major or dominant seventh chord. Sometimes it’s easier to play the seventh, such as E7 instead of E major on the ukulele.
It’s also possible to have a modal sequence, using the E minor rather than E major for a progression starting on the A minor chord.
In my book on common chord progressions, I listed 40 songs that use this chord progression, alphabetically ordered by title from Paul Simon’s 50 Ways to Leave Your Lover to Cat Steven’s Wild World. The strange thing is that I did not list Greensleeves (What Child Is This) !
Here’s the arpeggio accompaniment using the chord progressions in Greensleeves as an exercise in the Andalucian Cadence.

Try substituting the four chords listed above for other keys of F minor, E minor, and G minor.

More about the Andalusian Cadence
More information
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