of incommensurate potentials on 1D p-wave superconductors have been studied.[@DeGottardi1208.0015; @Cai1208.2532; @Satija1210.5159] *Commensurate* diagonal or off-diagonal Harper model has also attracted theoretical attention.[@Lang1207.6192; @Ganeshan1301.5639]

Here we are interested in characterizing the new TS regions further, focusing on when they emerge, and what happens when the quasiperiodically modulated quantum wire is connected with other wires with different modulation phases or an unmodulated one. Particularly, we would like to (i) clarify the correspondence between the energy spectrum of the single particle states and emergence of the TS states, in the presence of either a quasiperiodic modulation with a general wavenumber or the external magnetic field to a general direction, (ii) understand the effect of phase jumps of the quasiperiodic modulation and that of a quasiperiodic modulation applied to only a limited part of the one-dimensional system, and (iii) study the effect of a quasiperiodic modulation of the *hopping* parameter.

In section II we define our model and introduce our Bogoliubov-de Gennes (BdG) equation approach to study the system. We find that with a lattice-site energy modulation with a generic wavenumber, quasiperiodic or periodic, new topological superconducting regions with end Majorana fermions emerge. We also study the effect of the direction of the 