The Interstellar Dust Emission Spectrum, Going beyond the single
  temperature greybody
  Context. Most of the modelling of interstellar dust infrared emission
spectrum is done by assuming some variations around a single temperature
greybody approximation. For example, the foreground modelling of Planck mission
maps involves a unique dust temperature T along a given line-of-sight with a
unique emissivity index \b{eta}. The two parameters are then fitted and
therefore variable from one line-of-sight to the other. Aims. Our aim is to go
beyond that modelling in an economical way. Methods. We model the dust spectrum
with a temperature distribution around the mean value and show that only the
second temperature moment matters. We advocate the use of the temperature
logarithm as the proper variable. Results. If the interstellar medium is not
too heterogeneous, there is a universal analytical spectrum, which is derived
here, that goes beyond the greybody assumption. We show how the Cosmic
Microwave Background radiatively interacts with the dust spectrum (a
non-negligible corrective term at millimetre wavelengths). Finally, we
construct a universal ladder of discrete temperatures which gives a minimal and
fast description of dust emission spectra as measured by photometric mapping
instruments that lends itself to an almost linear fitting. This data modelling
can include contributions from the Cosmic Infrared Background fluctuations
