In Evans v. Wilson, 776 S.W.2d 939 (Tenn. 1989), the Tennessee Supreme Court declared that "So long as [a Rule 59] motion is pending, there is no final judgment for purposes of T.R.A.P. 3(a)."
This statement appears to be a manifestly incorrect statement of law. More recently, the Supreme Court has said that a final judgment is one that "adjudicated all the claims raised in the pleadings and the rights and liabilities of all the parties." In re Estate of Ridley, 270 S.W.3d 37, 41 (Tenn. 2008). This definition (which the court appears to have cribbed from older cases) comports with the definitions in Rule 58.
Of course, no appeal will lie while a Rule 59 motion pends: but this is not because of the absence of a final judgment, but because Rule 59 and Appellate Rule 4 abate the timeliness of notices of appeal during the pendency of such motions. See Ball v. McDowell, 288 S.W.3d 833, 836 (Tenn. 2009). The time limits found in Rule 4 are, of course, jurisdictional. Thus, if an appeal is filed during the pendency of an unresolved Rule 59 motion, it is true that the Court of Appeals lacks jurisdiction, but it is not true that its lack of jurisdiction is the result of the absence of a final judgment: it derives instead from the untimeliness of the appeal.