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Motivated by model-theoretic properties of the BSR class, we present a family of semantic classes of FO formulae with finite or co-finite spectra over a relational vocabulary Σ. A class in this family is denoted EBS_Σ(σ), where σ is a subset of Σ. Formulae in EBS_Σ(σ) are preserved under substructures modulo a bounded core and modulo re-interpretation of predicates outside σ. We study properties of the family EBS_Σ = EBS_Σ(σ) | σ⊆Σ, e.g. classes in EBS_Σ are spectrally indistinguishable,arXiv:1002.4334v2 fatcat:erzc5b3h45cnhnoxwbctygb3ou