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For exponentially closed ordinals $\alpha$, we consider recognizability ofconstructible subsets of $\alpha$ for $\alpha$-(w)ITRMs and their distributionin the constructible hierarchy. In particular, for $\alpha$-ITRMs, we showthat, there are lost melodies that are recognizable without parameters for all$\alpha$, that the iterated recognizability is absolute between $L$ and $V$ formost values of $\alpha$ and generalize "all or nothing"-phenomenon known fromITRMs occurs for a proper class of $\alpha$. For $\alpha$-wITRMs, we offer acomplete characterization of those $\alpha$ for which lost melodies exist andthat the relation between the sets of computable and recognizable subsets of$\alpha$ varies wildly, depending on $\alpha$: The computable sets may beincluded among the recognizable sets (which is usually the case in ordinalcomputability), but there are also class many values of $\alpha$ for which theset of recognizable sets is empty and such for which the set of recognizablesets is non-empty, but disjoint from the set of computable sets.  %for class many values of $\alpha$, the sets of $\alpha$-wITRM-computable and$\alpha$-wITRM-recognizable subsets of $\alpha$ are both non-empty, butdisjoint, and, also for class many values of $\alpha$, the set of$\alpha$-wITRM-recognizable subsets of $\alpha$ is empty.  This paper is an extension of our paper in the CiE 2023 proceedings.

The strange world of transfinite Melodies -- Recognizability for weak and strong infinite time $\alpha$-register machines