Riemannian theory

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Illustration of Riemann's 'dualist' system: minor as upside down major. Half-steps are indicated by slur marks, other notes are separated by whole steps. The descending minor scale shown has the same order of half steps and whole steps as the ascending major scale.

Riemannian theory refers to the musical theories of German theorist Hugo Riemann (1849-1919).

Riemann's "dualist" system for relating triads was adapted from earlier 19th-century harmonic theorists. (The term "dualism" refers to the emphasis on the inversional relationship between major and minor, with minor triads being considered "upside down" versions of major triads; this "harmonic dualism" is what produces the change-in-direction described above. See also: Utonality) ^[1]

Transformations

In the 1880s, Riemann proposed a system of transformations that related triads directly to each other. Riemann's system had two classes of transformations: 'Schritts' and 'Wechsels.' ^[1] A Schritt transposed one triad into another, moving it a certain number of scale steps. For example, the 'Quintschritt' (literally "Fifth-step" in German) transposed a triad by a perfect fifth, transforming C Major into G major (up) or F major (down). A Wechsel inverted a triad according to the Riemann's theory of dualism, mapping a major triad to a minor triad. For example, Seitenwechsel ("die Seiten wechseln" translates as "to change ends") mapped a triad on to its parallel minor or major, transforming C major to C minor and vice versa. ^[1] Riemann's theory of transformations formed the basis for Neo-Riemannian theory, which expanded the idea of transformations beyond the basic tonal triads that Riemann was mostly concerned with.

Sources

↑ 1.0 1.1 1.2 Klumpenhouwer, Henry, Some Remarks on the Use of Riemann Transformations, Music Theory Online 0.9 (1994)

Degrees and functions of the diatonic scale

${\hat 1}$	${\hat 2}$	${\hat 3}$	${\hat 4}$	${\hat 5}$	${\hat 6}$	(♭) ${\hat 7}$	${\hat 7}$

Tonic	Supertonic Sp	Mediant Dp, Tkp, tP, [D](Sp)	Subdominant	Dominant	Submediant Tp, sP, tCp	Subtonic dP	Leading D̸⁷

C	D	E♭/E♮	F	G	A♭/A♮	B♭	B♮

v t e Riemannian theory

Functional harmony Counter parallel Dominant parallel Subdominant parallel Tonic parallel Harmonic dualism Hugo Riemann Klang Neo-Riemannian theory Terzschritt Tonnetz

Handbuch der Harmonielehre Lehrbuch des Contrapunkts

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