Halliday & Matthiessen (1999: 320-1):
Just as the semantic system is functionally diversified (into the ideational, interpersonal and textual metafunctions), so the context in which language is 'embedded' is also diversified. The context encompasses both the field of activity and subject matter with which the text is concerned ('what's going on, and what is it about?') and the tenor of the relationship between the interactants, between speaker and listener, in terms of social roles in general and those created through language in particular ('who are taking pan?'). The field is thus the culturally recognised repertoires of social practices and concerns, and the tenor the culturally recognised repertoires of role relationships and interactive patterns.Now, both these contextual variables are, in some sense, independent of language, even though they are constituted in language and the other semiotic systems of a culture. That is, they concern realities that exist alongside the reality created by language itself, semiotic reality.
However, there is a third contextual variable that is specifically concerned with the part language is playing in any given context — the symbolic mode, how the linguistic resources are deployed. This covers both the medium (spoken, written, and various subtypes such as written in order to be spoken) and the rhetorical function — persuasive, didactic, informative, etc..
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Note that Martin misunderstands field, tenor and mode as dimensions of register, and misunderstands one dimension of mode, rhetorical function, as a dimension of genre. In SFL Theory, varieties of language, whether viewed as register or text type (genre), are modelled as language, not context.