In music, function denotes the relationship of a chord or scale degree to a tonal centre. Two main theories of tonal functions exist today:The German theory created by Hugo Riemann in his Vereinfachte Harmonielehre of 1893, which soon became an international success, and which is the theory of functions properly speaking. Riemann identified three abstract tonal "functions"—tonic, dominant and subdominant—denoted by the letters T, D, and S, respectively, each of which could take on a more or less modified appearance in any chord of the scale. This theory, in several revised forms, remains much in use for the pedagogy of harmony and analysis in German-speaking countries and in Northern and Eastern European countries.
The Viennese theory, characterized by the use of Roman numerals to denote the chords of the tonal scale, as developed by Simon Sechter, Arnold Schoenberg, Heinrich Schenker, and others, practiced today in Western Europe and the United States. This theory in origin was not explicitly about tonal functions. It considers the relation of the chords to their tonic in the context of harmonic progressions, often following the cycle of fifths. That this actually describes what could be termed the chords' "function" is evident in Schoenberg's Structural Functions of Harmony (1954), a short treatise dealing mainly with harmonic progressions in the context of a general "monotonality".