  In the 1960's Igusa determined the graded ring of Siegel modular forms of
genus two. He used theta series to construct $\chi_{5}$, the cusp form of
lowest weight for the group $\operatorname{Sp}(2;Z)$. In 2010 Gritsenko found
three towers of orthogonal type modular forms which are connected with certain
series of root lattices. In this setting Siegel modular forms can be identifed
with the orthogonal group of signature $(2,3)$ for the lattice $A_{1}$ and
Igusa's form $\chi_{5}$ appears as the roof of this tower. We use this
interpretation to construct a framework for this tower which uses three
different types of constructions for modular forms. It turns out that our
method produces simple coordinates for the graded rings of modular forms.
