The main result of this paper is the representation of the free algebras of a certain class of discriminator varieties, called \({\mathcal M}\)-spectral varieties by the author, as certain Boolean products. \({\mathcal M}\)-spectral varieties are defined as follows: If \({\mathcal L}\) is a language of algebras, \({\mathcal M}\) a class of \({\mathcal L}\)-algebras and \(A \in {\mathcal M}\), define the normal transform \(\eta^A\) of \(A\) by \(\eta^A (x,y,z,w) = [z\) if \(x = y\), and \(w\) if \(x \neq y]\). Let \(A^\eta\) be the corresponding \({\mathcal L} (\eta)\)-expansion of \(A\), and \({\mathcal M}^\eta = \{A^\eta |A \in {\mathcal M}\}\). If \({\mathcal M}\) is axiomatized by a set \(\Sigma\) of sentences of the form \(\forall x \exists!y (\varphi (x,y))\) with \(\varphi\) open, this defines new function symbols, and a corresponding class \({\mathcal M}_\Sigma\) of the canonical \({\mathcal L}_\Sigma\)-expansions of the algebras in \({\mathcal M}\). Then the \({\mathcal M}\)-spectral variety relative to \(\Sigma\) is defined to be the \(({\mathcal L}_\Sigma (\eta)\)-) variety generated by \(({\mathcal M}_\Sigma)^\eta\). The author gives several natural examples of \({\mathcal M}\)-spectral varieties, defined from simple \({\mathcal M}\)'s, as well as some applications of his result. The author also shows that every quasivariety in a discriminator variety can be axiomatized by sentences of the form \((\forall y(u = v) \vee \forall x (p \neq q))\). There are a few minor misprints, most of them being ``\(n\)'' written for ``\(\eta\)'', and vice-versa. Also p. 397, line --2, the expression ``\(p_1q \in T (x)\)'' appears by mistake.