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I have read that the name originates from that of the owner of a "Tenement", that is to say a house with multiple occupancy. Today we would use the term "Landlord". Derived from old French and used in the times of the Romans.
The surname seems to have started in Yorkshire England and the central belt of Scotland.
The earliest record is William Tenant de Crofton in the county of Linlithgowshire.
I suspect the Crofton could be todays Corston, a ruined farm of that name is linked by a half mile path to Linnhouse Tower home of Francis Tennant in the 16th centuary
My notes can be found at http://freepages.genealogy.rootsweb.com/~tennant/index.htm sorry not had time to copy to Geni yet.
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Balázs Déri adds :
Tennant family might have got the name of Annet-Tenna , one of the sons of king Ed-Daya-Wa-Da , inspirer of some of Edda texts mentioning him as Odín , called perhaps as prince of Mauritania and of anttessar or tengeradiv tribe also A-War-a-Ma-yat.
He might have issued tennet and nándi tribes in Africa as well as some of náni and udi people in Asia all speaking somewhat similar languages respectively.
Jews and christians call them Natan and Dawid.
See Magyar,David,Wodianer,Vodnansky,Törzsök,Denkstein,Waldner,Vajda,Czigány,Rácz,László,Drust,Durst,Gáldy,Reuther,Mohoray,Antic,Kotromanic !