  Many different types of fractional calculus have been defined, which may be
categorised into broad classes according to their properties and behaviours.
Two types that have been much studied in the literature are the Hadamard-type
fractional calculus and tempered fractional calculus. This paper establishes a
connection between these two definitions, writing one in terms of the other by
making use of the theory of fractional calculus with respect to functions. By
extending this connection in a natural way, a generalisation is developed which
unifies several existing fractional operators: Riemann--Liouville, Caputo,
classical Hadamard, Hadamard-type, tempered, and all of these taken with
respect to functions. The fundamental calculus of these generalised operators
is established, including semigroup and reciprocal properties as well as
application to some example functions. Function spaces are constructed in which
the new operators are defined and bounded. Finally, some formulae are derived
for fractional integration by parts with these operators.
