I'm pretty sure that Richard plays this in a drop D tuning (ie 6th tuned down to a D). The last time I saw him do a solo gig, he had a special machine head for the sixth string that was fitted with a lever to move from E to D (and back again) quickly. Listening to the chord voicings when he accompanied Linda, I reckon he played with a capo on the first fret and used the chords D, G and A for the verse and A E D and G for the "other bit". When playing the A, the sixth string would have to be avoided, for the E it would be played as 222100 going from 6th to 1st and for the G (as alluded to already) as 550030.

  This old house is falling down around my ears
           D                               A
  
  I'm drowning in the river of my tears
      G               D           A
  
  When all my will is gone you hold me sway
       D                               A
  
  I need you at the dimming of the day
    G               D       A      D   G D
  
  
  You pulled me like the moon pulls on the tide
      A                                    E
  
  You know just where I keep my better side
      A                                EADG
  

and for the last two lines

  I need you at the dimming of the day
    D               A              D  G
  
  I need you at the dimming of the day
    D               A              D
  

Something else I have noted from the lyrics on his web site, he uses the words "I'm drowning in the river... whereas every other web site where I have found the words (as recorded by other people) they say I'm drowning in a river.....

It was only when I tried learning this song did I realise how skillful a singer Linda is. As has already been said, you need a good range to get all of the notes.