  We study the multifractal analysis of a class of equicontractive,
self-similar measures of finite type, whose support is an interval. Finite type
is a property weaker than the open set condition, but stronger than the weak
open set condition. Examples include Bernoulli convolutions with contraction
factor the inverse of a Pisot number and self-similar measures associated with
$m$-fold sums of Cantor sets with ratio of dissection $1/R$ for integer $R\leq
m$.
  We introduce a combinatorial notion called a loop class and prove that the
set of attainable local dimensions of the measure at points in a positive loop
class is a closed interval. We prove that the local dimensions at the periodic
points in the loop class are dense and give a simple formula for those local
dimensions. These self-similar measures have a distinguished positive loop
class called the essential class. The set of points in the essential class has
full Lebesgue measure in the support of the measure and is often all but the
two endpoints of the support. Thus many, but not all, measures of finite type
have at most one isolated point in their set of local dimensions.
  We give examples of Bernoulli convolutions whose sets of attainable local
dimensions consist of an interval together with an isolated point. As well, we
give an example of a measure of finite type that has exactly two distinct local
dimensions.
