Although Paul, 73, is one of the most revered musicians of all time, Heather - his former wife - says that she was responsible for showing Beatrice, who was born in 2003, how to play the instrument.
She told The Guardian: ''I think she's got the best of both of us, we're both very musical. I taught her the saxophone, because her father can't read music so I do all the music teaching, and I'm good with languages. She's a brilliant poet so obviously gets that from him, but I think she's got the best of both of us.''
Paul and Heather married in 2002 and divorced acrimoniously in 2008 and Heather - who was married to Alfie Karmal in 1989 for two years - admitted she has nothing nice to say about her second husband.
She said: ''I don't want to go into my second marriage too much because if you haven't got something nice to say about someone you shouldn't say anything at all, and it had a lot to do with the person I married.'
Music legend Paul - who gained worldwide fame as the bassist of British rock band the Beatles - went on to marry Nancy Shevell in 2011.
'Smalls Change (Meditations Upon Ageing)' arrives in April.
The two awards have made for a great 72nd birthday present for the country music icon.