Beginning to transcribe a passage from C. Thomas Cairney, Clans and Families of Ireland and Scotland: An ethnography of the Gael (1989). I will do this in installments. I anticipate starting and stopping as the day goes on, so this might take a bit of time.
Page 111: "The Clan MacDuff descends from Gillemichael mac Duff, Earl of Fife in about 1133. But the significance of the name Duff (Dubh) goes back to the line of Duff, King of Albany in 967, whose descendants' patrimony was in Fife (the "kingdom" of Fife). His line, the Clan Duff, was collateral with the line of King Duff's brother, King Kenneth II, and the two lines alternated the High-Kingship of Albany until 1034, as both lines had their ultimate origin in sons of King Malcolm I of the line of the Cineal Gabhran who had inherited the Picto-Gaelic crown (hence their traditional descent, in the female line, from Conall Cearnach, traditional ancestor of the Cruithne).
"Both of these lines ended in heiresses about the year 1034: The Line of Kenneth II ending in Bethoc, who married Crinan, hereditary Abbot of Dunkeld, of the Kindred of St. Columba, mentioned above; and the Line of Duff ending in Gruoch, who married Gillacomgan, Mormaer (King) of Moray, of the line of the Cineal Loarn. Their son, Lulach, was thus Chief of Clan Duff (in those presurname times of Picto-Gaelic succession) and King of Moray, and was as well a rival King of Albany. His daughter and heiress, the Princess of Moray and heiress of Clann Duff appears to have "married" Eth (Aedh, later Aodh, Gaelic form of Aethelred), Last Abbot of Dunkeld, who himself was the eldest of the four royal sons of Malcolm III (whose father was Duncan I, mentioned above, heir of the Royal line collateral to the Clan Duff) by his second wife, St. Margaret, a daughter of the Saxon King of England (Duncan II, son of Malcolm III by an earlier marriage, was the ancestor of the famous "MacWilliam" claimants."