  Two different tensions can be defined for a fluid membrane: the internal
tension, $\gamma$, conjugated to the real membrane area in the Hamiltonian, and
the frame tension, $\tau$, conjugated to the projected (or frame) area.
According to the standard statistical description of a membrane, the
fluctuation spectrum is governed by $\gamma$. However, using rotational
invariance arguments, several studies argued that fluctuation spectrum must be
governed by the frame tension $\tau$ instead. These studies disagree on the
origin of the result obtained with the standard description yet: either a
miscounting of configurations, quantified with the integration measure, or the
use of a quadratic approximation of the Helfrich Hamiltonian. Analyzing the
simplest case of a one-dimensional membrane, for which arc length offers a
natural parametrization, we give a new proof that the fluctuations are driven
by $\tau$, and show that the origin of the issue with the standard description
is a miscounting of membrane configurations. The origin itself of this
miscounting depends on the thermodynamic ensemble in which calculations are
made.
