Sorry for creating the apparent confusion, what I meant to say was that the bits you hear in the clip are not an Irish tune, they're an intro, counter melody or whatever you want to call it the Chieftain played on that particular track, The Morning Dew or the Hare in the Heather or whatever name you want to use for the tune. The tune itself is only represented there by the opening bar you can hear the fiddle start towards the end of the clip.