  In representation theory of algebras, there exist two types of mutation
pairs: rigid type (cluster-tilting mutations by Iyama-Yoshino) and
simple-minded type (mutations of simple-minded systems by
Sim\~oes-Pauksztello). It is known that such mutation pairs induce triangulated
categories, however, these facts have been proved in different ways. In this
paper, we introduce the concept of ''mutation triples'', which is a
simultaneous generalization of two different types of mutation pairs as well as
concentric twin cotorsion pairs. We present two main theorems concerning
mutation triples. The first theorem is that mutation triples induce
pretriangulated categories. The second one is that pretriangulated categories
induced by mutation triples become triangulated categories if they satisfy an
additional condition (MT4).
