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Whereas most of the classical commentators regard the infinitive noun tasnim as the proper name of one of the allegorical "fountains of paradise", or, alternatively, refrain from any definition of it, it seems to me that the derivation of tasnim from the verb sannama - "he raised [something]" or "made [it] lofty" - points, rather, to the effect which the "wine" of divine knowledge will have on those who "drink" of it in paradise. Hence, the tabi'i 'Ikrimah (as quoted by Razi) equates tasnim with tashrif, "that which is ennobling" or "exalting".
Tasnim literally indicates height, fulness, opulence. Here it is the name of a heavenly Fountain, whose drink is superior to that of the Purest Wine. It is the nectar drunk by Those Nearest to Allah (n. 5227 to lvi. 11), but a flavour of it will be given to all. See n. 5835 to lxxvi. 5 (Kafur fountain), and n. 5849 to lxxvi. 17-18 (Salsabil).