This is a great example of how the Phrygian key is used in the usual place (the III of a major key). Jose uses some great chords here outlining this harmonic device. Let's take a look at the particulars:
Measure 1: You could just think of this as the IV chord in A but it is more about the function which determines what you are going to call it. You always have to look ahead to see where the chord is going before you name it. This is a Dmaj7/F#. It is dressed up in a very brazilian fashion which happens to be the same chord change(measures 1 & 2) found in Jobim's "Girl from Ipanema"(If you played it in the hip key of D not F).
Measure 2: This chord is seriously one of the coolest ever because it can be spelled in so many ways. It is always the function though that determines the spelling though as I said previously. I am going to call this chord E13/F which is how you would spell it in the Jazz. However, in Flamenco language many would think of this chord more of a extension/substitution for C#(b9) which is where it goes the second time in measure 4.
Measure 4: This progression from measures 1-3 finishes with a second ending on the C#(b9) Phrygian chord, but instead of staying in that mode, he uses it as the dominant chord to resolve to the Vi chord in the key of A. (F#min) (V of Vi).
Measure 9: Here you see an E diminished chord followed by an F#7 which leads to Bmin (V of ii)in the next measure. The interesting thing here is the relationship between the E dim and F#7. You could say that they are the same chord or rather an F#7(b9).
Measure 11-13: Here you see a typical cycle chord change that leads us back to the home key. The cycle is many about the II,ii, I progression but he borrows dominant chord to put in front of each so you end up with this: G#7(b9) - C#m - F#7(b9) - Bm - E7(b9) - Amaj
Measures 14-15: To end the falseta Jose outlines the meat and potatoes of all chord progressions. I-IV-V-I. (A,D,E,A)
Recorded in Santa Barbara, CA by Berto Boyd for more info visit: https://bertoboyd.com/