Copied from Gen-Medieval/soc.genealogy.medieval by Stewart Baldwin at www.rootsweb.com/~medieval/man.htm:
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The Ancestry of Godred Crovan
The ancestry of Godred Crovan (d. 1095), king of Dublin and Man, is not well documented, and there are differing opinions regarding his parentage and immediate ancestors. Rather than try to give a definitive solution to the problem, the basic evidence will be outlined, and several possible alternatives will be given, along with their strengths and weaknesses. Since the basic raw data is itself somewhat contradictory, none of the suggested alternatives will fit all of the primary evidence.
The basic raw data is as follows. First, the Annals of Tigernach [AT] for the year 1091 refer to him as the son of the son of Harald ["Goffraidh mac Maic Arailt, rĂ Atha Cliath."]. Then, there is the Chronicle of the Kings of Man [CRM], which states that Godred was the son of Harald the Black of "Ysland" (Iceland), without further identifying this Harald. Finally, there is the Welsh collection of Norse pedigrees in "Achau Brenhinoedd a Thywysogion Cymru" [ABT, in EWGT, pp. 95-110], which includes a genealogy of the kings of Man, as follows:
ABT.6c: Rhanallt m. Gwythryg m. Afloyd m. Gwrthryt mearch m. Harallt ddu m. Ifor gamle m. Afloyd m. Swtrig.
Changing the names from these Welsh forms to the more familiar English forms gives:
Reginald [king of Man, d. 14 Feb 1229], son of
Godred [king of Man, d. 10 Nov 1187], son of
Olaf [king of Man, d. 29 June 1153], son of
Godred [Crovan, king of Dublin and Man, d. 1095], son of
Harald ddu [i.e., the Black], son of
Ivar gamle [i.e., the Old], son of
Olaf [presumably Olaf Cuaran, king of Dublin and York], son of
Sitric [d. 927]
It may be that the above genealogy was composed during the reign of Reginald (d. 1229), since he is the latest person mentioned in the genealogy. There is no way of knowing whether copying mistakes were made between that time of composition and the surviving manuscripts.
We now list several possibilities regarding the ancestry of Godred Crovan, along with the strengths and weaknesses of each possibility (some of which are valid for more than one case, and are therefore repeated). While there are other scenarios which could be listed, they would seem less likely than the ones given below.
Possibility 1: The genealogy of ABT is to be accepted as it is.
Strengths: It requires no emendation of the genealogy in ABT. It agrees with the Chronicle of the kings of Man in making Godred the son of Harald "the Black".
Weaknesses: No son of Olaf Cuaran named Ivar is known from the Irish records. The generations are a bit long (but not drastically so). It disagrees with the Annals of Tigernach, which make Godred the grandson of a certain Harald.
Possibility 2: In the process of copying the pedigree, a "Harald" was accidently omitted between Ivar and Olaf Cuaran, so that the pedigree should read Godred son of Harald the Black son of Ivar son of Harald son of Olaf [Cuaran].
Strengths: Olaf Cuaran had a son named Harald, who in turn had a son named Ivar, both known from the Irish annals, so the agreement with the Irish annals would be excellent. It agrees with the Chronicle of the kings of Man in making Godred the son of Harald "the Black". The chronology fits better than Possibilities 1 and 3.
Weaknesses: It requires an emendation of the pedigree in ABT. It disagrees with the Annals of Tigernach, which make Godred the grandson of a certain Harald.
Possibility 3: In the process of copying the pedigree, Harald and Ivar were accidently switched, so that the pedigree should read Godred son of Ivar son of Harald son of Olaf [Cuaran].
Strengths: Olaf Cuaran had a son named Harald, who in turn had a son named Ivar, both known from the Irish annals, so the agreement with the Irish annals would be excellent. It agrees with the Annals of Tigernach, which call Godred the grandson of a certain Harald.
Weaknesses: It requires an emendation of the pedigree in ABT. The generations are a bit long (but not drastically so). It disagrees the the Chronicle of the kings of Man, which make Godred the son of Harald the Black.
Possibility 4: The pedigree in ABT is wrong, and Godred was not a descendant of Olaf Cuaran, but was instead descended somehow from the kings of the Isles who ruled in the late tenth and early eleventh centuries.
Strengths: It would explain Godred's claim to the kingship of Man. The known names used by the early dynasty of the kings of the Isles were Guthfrith (i.e., Godred), Harald, Lagman, Olaf, and Rognvald, which were exactly the names which were common in the family of Godred Crovan (including the rare name Lagman), so this possiblity has some onomastic support.
Weakness: It requires abandoning the manuscript genealogy of ABT, so there is no direct supporting evidence. The onomastic argument is mitigated somewhat by the fact that the names Guthfrith, Harald, Olaf, and Rognvald were all common among the Hiberno-Norse in general, so that only the rare name Lagman carries significant weight in the onomastic argument.
Before I was aware of the genealogy in ABT, I favored possibility 4. Now that I know about the ABT genealogy, I think Possibility 2 is the most likely one. However, I think that none of the four possibilities can be ruled out, given the currently known evidence.