  The properties of positive bases make them a useful tool in derivative-free
optimization (DFO) and an interesting concept in mathematics. The notion of the
\emph{cosine measure} helps to quantify the quality of a positive basis. It
provides information on how well the vectors in the positive basis uniformly
cover the space considered. The number of vectors in a positive basis is known
to be between $n+1$ and $2n$ inclusively. When the number of vectors is
strictly between $n+1$ and $2n$, we say that it is an intermediate positive
basis. In this paper, the structure of intermediate positive bases with maximal
cosine measure is investigated. The structure of an intermediate positive basis
with maximal cosine measure over a certain subset of positive bases is
provided. This type of positive bases has a simple structure that makes them
easy to generate with a computer software.
