Turlough Carolan is said to have derived the tune of 'Sí Beag sí mór' (Sheebeg sheemor) from 'The Bonny Cuckoo'

Edward Bunting collected tunes and songs were published between 1827 and 1840. The Bonny Cuckoo comes from the third and final collection published in 1840. The tune was in manuscript MS. 33 book 5, with one verse written over the words.

THE BONNY CUCKOO

My bonny cuckoo, I tell you true
That through the groves I'll rove with you;
I'll rove with you until the spring,
And then my cuckoo shall sweetly sing.
Cuckoo, sing girls, let no one tell
Until I settle my seasons well.

The ash and the hazel shall mourning say,
My bonny cuckoo, don't go away;
Don't go away, but tarry here,
And make the season last all the year.
Don't go away, but tarry here,
And make the season last all the year.

Notes from Bunting's Ancient Music of Ireland edited from the original manuscripts by Donal O'Sullivan with Micheál Ó Súilleabháin, Cork University Press (Ireland), 1983:

'' AIR: According to the Index the source is 'Ballinascreen and from the late H. Joy. Esq. Belfast, 1793'. Ballinascreen is in Co Derry and Henry Joy was one of the founders of the United Irishman. [In the Introduction …Bunting writes: 'From this ancient melody … another tune 'The little and great mountain' (Sí Beag, Sí mór) seems to have been arranged with some slight variations.' It appears that O'Carolan used this tune as the basis of one of his songs entitled 'Sí Beag, Sí mór' which Bunting printed as No. 63 in his 1796 collection.
   
''The Carolan song is dealt with extensively in DOSC No. 202. Other versions of 'the bonny cuckoo' are to be found in Neale's Celebrated Irish Tunes (c.1726), p.14 'cuckoo'; and Mulhollands's Ancient Irish Airs (1840), p.59…''

O'Sullivan & Ó Súilleabháin write that the words of The Bonny Cuckoo were ''Probably from Henry Joy''. Henry Joy McCracken himself is celebrated in at least a couple of songs (see the DT), and more recently Jane Cassidy composed a musical tribute to his sister Mary Anne, noted for her charitable work.

The 'DOSC' referred to is a book by Donal O'Sullivan, Carolan, the Life, Times and Music of an Irish Harper , two voulumes, 1958

DT refers to Mudcat's Digital Tradition

I sent MMario the sheet music, and he converted it to this:

X:1
T:The Bonny Cuckoo
N:O'Sullivan & Ó Súilleabháin - from Bunting
I:abc2nwc
M:6/8
L:1/8
K:D
z4zA|d3/2 e/2 f f e d|B2A F2A|A3/2 B/2 d e2d|f2e d2f|
B2e A2B|F2E/2 D/2 E2A|A B d e3/2 d/2 e|f2e d2|]