The German writing system is typically described as using two kinds of elementary units: graphemes, which are abstract entities, and graphs (or glyphs), which are their material expressions. Although this dichotomous model may be sufficient for orthographic tasks, it proves inadequate for mapping the writing system as a whole, for which it is neither differentiated enough nor consistent. This paper aims to enhance the customary model by demonstrating that the grapheme has to fulfill three incompatible systematic functions at once and, as a consequence, by establishing distinct levels within the model. In terms of graphemics, it is pointed out that although the grapheme can only be defined either as a phoneme correspondent or as a minimal semantically distinguishing unit, both contradicting definitions are often jumbled (for instance by using statistical criteria to determine minimal units). In terms of graphetics, it is argued that prototypical character shapes are systematically relevant entities as some graphemes show more than one prototypical shape. Furthermore, the choice between shapes can to some extent be influenced by morphological, lexical or other linguistic parameters, such as text length or region. As a result, the structural model of the writing system is extended to four hierarchic levels of elementary units, tentatively called "phoneme correspondents", "graphemes", "basic shapes", and "graphs". Even though the model is developed with respect to German, quintessential parts of the argumentation can be applied to other languages which use the Roman alphabet.