Measuring complexities of classes of structures

How do we compare the complexities of various classes of structures? The Turing ordinal of a class of structures, introduced by Jockusch and Soare, is defined in terms of the number of jumps required for coding to be possible. The back-and-forth ordinal, introduced by Montalbán, is defined in terms of Σα-types. The back-and-forth ordinal is (roughly) bounded by the Turing ordinal. In this paper, we show that, if we do not restrict the allowable classes, the reverse inequality need not hold. Indeed, for any computable ordinals α ≤ β we present a class of structures with back-and-forth ordinal α and Turing ordinal β. We also present an example of a class of structures with back-and-forth ordinal 1 but no Turing ordinal.