The motivation for the paper comes from the following question: can one obtain a tight wavelet frame with three generators of compact support arising from any given B-spline \(B_m\) of order \(m\), such that the generators are symmetric and have vanishing moments of (the highest possible) order \(m\)? The main result proves the following. There exist three finitely supported sequences \(b^1,b^2,b^3\) on \(Z\) such that the functions \(\psi^\ell (x)= \sum b^\ell(k) B_m(2x-k), \;\ell=1,2,3\), have the following properties: (i) \(\{\psi^1,\psi^2,\psi^3\}\) generates a tight wavelet frame and has vanishing moments of order \(m\), (ii) \(\psi^1,\psi^2,\psi^3\) are real-valued, symmetric, and compactly supported, and \[ \psi^1(1-t)=(-1)^m\psi^1(t),\quad \psi^2(m-t)= \psi^2(t),\quad \psi^3(m-t)=-\psi^3(t). \]