The Sandwich-Lemma: The Recursive Structure of Super-Syntonic and Super-Diatonic Automorphisms

Abstract

In the context of the transformational modelling of Pairwise Well-Formed (PWWF) modes, we study transformations $\sigma _f$ and $\tau _g$ on four letters a, b, c, d, which we call super-syntonic and super-diatonic morphisms, respectively. They are constructed from underlying syntonic and diatonic morphisms f and g on two letters a, b. From f and g, which are instances of Sturmian morphisms, one also constructs the Well-Formed (WF) syntonic and diatonic projections of the PWWF mode in question. From a super-syntonic morphism $\sigma _f$ one obtains the PWWF authentic triadic mode $\pi _{d \rightarrow a}(\sigma _f(c)|\sigma _f(b)||\sigma _f(a))$ and the PWWF plagal triadic mode $\pi _{d \rightarrow a}(\sigma _f(d)||\sigma _f(c)|\sigma _f(b))$. In Sect. 2 we complete an earlier investigation by proving that the kaleidoscope maps $\sigma$ and $\tau$ are monoid homomorphisms, i.e. $\sigma _{f_1 f_2} = \sigma _{f_1} \sigma _{f_2}$ and $\tau _{g_1g_2} = \tau _{g_1} \tau _{g_2}$. In Sect. 3 we show that the recursive “sandwich” construction of syntonic and diatonic morphisms can be lifted to the level of super-syntonic and super-diatonic morphisms. This result enables us to show, that super-syntonic and super-diatonic morphisms, which are constructed in this way, are automorphisms of the free group $F_4$ with four generators.