I learned ”The First House in Connaught” from a one of Doolin’s legends, concertina player Packie Russell. It is an old favourite amongst concertina players in County Clare and has been recorded by Paddy Murphy, Tom Carey and Dympna O’Sullivan among others.
See the video below of Packie Russell playing this tune (after “The Milliner’s Daughter”) at Gus O’Connor’s Pub back in the early 1970s.
T:First House In Connaught (The)
|:D~G3 A~G3|BGdG B~G3|D~F3 A~F3|ABcA BGAF|
D~G3 A~G3|BGdG B~G3|d2 (3efg agfd|1 cAFA G2GE:|2 cAFA G2Bd||
|:g2ga gfdf|g2bg afde|~f2de f2de|f~a3 bgaf|
~g3a gfdf|g2bg agfe|d2 (3efg agfd|1 cAFA G2Bd:|2 cAFA G2GE||