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We extend the study of the multifractal analysis of the class ofequicontractive self-similar measures of finite type to the non-equicontractivesetting. Although stronger than the weak separation condition, the finite typeproperty includes examples of IFS that fail the open set condition. Theimportant combinatorial properties of equicontractive self-similar measures offinite type are extended to the non-equicontractive setting and we prove thatmany of the results from the equicontractive case carry over to this new, moregeneral, setting. In particular, previously it was shown that if anequicontractive self-similar measure of finite type was {\em regular}, then thecalculations of local dimensions were relatively easy. We modify thisdefinition of regular to define measures to be {\em generalized regular}. Thisnew definition will include the non-equicontractive case and obtain similarresults. Examples are studied of non-equicontractive self-similar generalizedregular measures, as well as equicontractive self-similar measures whichgeneralized regular in this new sense, but which are not regular.

Local dimensions of measures of finite type III -- Measures that are not equicontractive