The term "functional harmony" derives from Riemann and particularly from his
Harmony Simplified. Riemann's direct inspiration was Moritz Hauptmann's dialectic description of tonality. Riemann identified three abstract functions: the tonic, the dominant (its upper fifth), and the subdominant (its lower fifth). He also considered the minor scale the inversion of the major scale, so that the dominant was the fifth above the tonic in major, but below the tonic in minor; the subdominant, similarly, was the fifth below the tonic (or the fourth above) in major, and the reverse in minor. Despite their complexity, Riemann's ideas had huge impact, especially where German influence was strong. A good example are
Hermann Grabner's textbooks. More recent German theorists have abandoned the most complex aspect of Riemann's theory, the dualist conception of major and minor, and consider the dominant the fifth degree above the tonic and the subdominant the fourth degree in both minor and major. In
Diether de la Motte's version of the theory, the three tonal functions are denoted by the letters T, D and S, for Tonic, Dominant and Subdominant respectively; the letters are uppercase for functions in major (T, D, S) and lowercase for functions in minor (t, d, s). Each function can in principle be fulfilled by three chords: the main chord corresponding to the function and the chords a third lower and a third higher, as indicated by additional letters. An additional letter P or p indicates that the function is fulfilled by the relative (German
Parallel) of its main triad: for instance Tp for the minor relative of the major tonic (e.g., A minor for C major), tP for the major relative of the minor tonic (e.g. E major for c minor), etc. The other triad a third apart from the main one may be denoted by an additional G or g for
Gegenparallelklang or
Gegenklang ("counterrelative"), for instance tG for the major counterrelative of the minor tonic (e.g. A major for C minor). Triads a third apart differ from each other by one note only, the other two being shared. In addition, within the diatonic scale, triads a third apart necessarily are of opposite mode. In the simplified theory where the functions in major and minor are on the same scale degrees, the possible functions of triads on degrees I to VII of the scale could be summarized as in the table below (degrees II in minor and VII in major, diminished fifths in the diatonic scale, are considered chords without fundamentals). Chords on III and VI may have the same function as those a third above or a third below, but one of these two is less frequent than the other, as indicated by parentheses. In each case, the chord's mode is denoted by the final letter: for instance, Sp for II in major indicates that II is the minor relative (p) of the major subdominant (S). The major VIth degree in minor is the only one where both functions, sP (major relative of the minor subdominant) and tG (major counterparallel of the minor tonic), are equally plausible. Other signs (not discussed here) are used to denote altered chords, chords without fundamentals, applied dominants, etc. Degree VII in harmonic sequence (e.g. I–IV–VII–III–VI–II–V–I) may be denoted by its roman numeral; in major, the sequence would then be denoted by T–S–VII–Dp–Tp–Sp–D–T. As summarized by
Vincent d'Indy (1903), who shared Riemann's conception: • There is only
one chord, a
perfect chord; it alone is consonant because it alone generates a feeling of repose and balance; • this chord has two
different forms,
major and minor, depending on whether it is composed of a minor third over a major third or a major third over a minor; • this chord is able to take on
three different tonal functions—tonic, dominant, or subdominant. ==Viennese theory of degrees==