Riemannian theory

Riemannian theory, in general, refers to the musical theories of German theorist Hugo Riemann (1849–1919). His theoretical writings cover many topics, including musical logic,^[1] notation,^[2] harmony,^[3] melody,^[4] phraseology,^[5] the history of music theory,^[6] etc. More particularly, the term Riemannian theory often refers to his theory of harmony, characterized mainly by its dualism and by a concept of harmonic functions.

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" (harmonic polarity) is what produces the change-in-direction described above. See also the related term utonality.^[7]

In the 1880s, Riemann proposed a system of transformations that related triads directly to each other. Riemann's system had two classes of transformations: "Schritt" and "Wechsel".^[7] A Schritt transposed one triad into another, moving it a certain number of scale steps. For example, the "Quintschritt" (literally "five-step" in a mixture of Latin and 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 exchange sides") mapped a triad on to its parallel minor or major, transforming C major to C minor and conversely.^[7] 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.

• Schenkerian analysis
• Undertone series

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