  A general form of the octupole collective Hamiltonian is introduced and
analyzed based on fundamental tensors in the seven-dimensional tensor space.
Possible definitions of intrinsic frames of reference possessing cubic symmetry
for the octupole tensor are considered. Cubic intrinsic octupole coordinates or
deformations are introduced. Shapes of the octupoloid are investigated. The
octupole collective Hamiltonian is expressed in intrinsic coordinates. An
intrinsic angular momentum carried by the octupole vibrations is discovered.
Small oscillations about an axially-symmetric pear shape are analyzed.
Formulation of a unified quadrupole-octupole collective model is discussed.
