  Atomic properties of $n=3$ states of the W$^{56+}$ $-$ W$^{61+}$ ions are
systematically investigated through two state-of-the-art methods, namely, the
second-order many-body perturbation theory, and the multi-configuration
Dirac-Hartree-Fock method combined with the relativistic configuration
interaction approach. The contributions of valence-valence and core-valence
electron correlations, the Breit interaction, the higher-order retardation
correction beyond the Breit interaction through the transverse photon
interaction, and the quantum electrodynamical corrections to the excitation
energies are studied in detail. The excitation energies and wavelengths
obtained with the two methods agree with each other within \approx 0.01 %. The
present results achieve spectroscopic accuracy and provide a benchmark test for
various applications and other theoretical calculations of W$^{56+}$ $-$
W$^{61+}$ ions. They will assist spectroscopists in their assignment and direct
identification of observed lines in complex spectra.
