relative measure of the differential thermal conductances of devices, as distinct from the thermal fluxes, assuming of course that the heat capacities of the devices are the same.

Experiment {#sec:Experiment}
==========

Transition Edge Sensors having a variety of patterned phononic legs were fabricated on 200nm thick, low-stress, amorphous SiN$_{\rm x}$ membranes. Every TES had an identical $80 \times 80$$\mu$m MoAu bilayer with 3 gold bars deposited on the upper surface, giving transition temperatures $T_C = 135 \pm 4$mK. Phononic structures were classified according to the number of filters in series per leg, m, and the filter style. Distinct filter styles were termed either ‘interferometers’ (mI), with pointed elliptical loops connected by collinear microbridges, or ‘ring resonators’ (mR), with typically angled connections intersecting circular rings: Fig. \[fig:TES\_Images\]. The primary difference between the two styles lies in the way in which power is divided and recombined upon entering and leaving a filter section: see later. A number of straight-leg control devices (mIC) were also fabricated, with lengths equal to the direct end-to-end lengths of the interferometers mI.

In previous work we have always used optical lithography (OL) and reactive ion etching (RIE) to pattern the the SiN$_{\rm x}$, followed by deep reactive 