  We consider the Assouad spectrum, introduced by Fraser and Yu, along with a
natural variant that we call the `upper Assouad spectrum'. These spectra are
designed to interpolate between the upper box-counting and Assouad dimensions.
It is known that the Assouad spectrum approaches the upper box-counting
dimension at the left hand side of its domain, but does not necessarily
approach the Assouad dimension on the right. Here we show that it necessarily
approaches the \emph{quasi-Assouad dimension} at the right hand side of its
domain. We further show that the upper Assouad spectrum can be expressed in
terms of the Assouad spectrum, thus motivating the definition used by
Fraser-Yu.
  We also provide a large family of examples demonstrating new phenomena
relating to the form of the Assouad spectrum. For example, we prove that it can
be strictly concave, exhibit phase transitions of any order, and need not be
piecewise differentiable.
