"**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.

## Dualism

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]}

Some people today (such as Jacob Collier) refer to the theory of dualism as the theory of "negative harmony".^{[citation needed]}^{[clarification needed]}

## 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: 'Schritt' and 'Wechsel'.^{[7]} 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.^{[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.

## See also

## Sources

**^***Musikalische Logik. Hauptzüge der physiologischen und psychologischen Begründung unseres Musik-systems*, Leipzig, 1873.**^***Studien zur Geschichte der Notenschrift*, Leipzig, 1878;*Die Entwickelung unserer Notenschrift*, Leipzig, 1881;*Notenschrift und Notendruck*, Leipzig, 1896.**^***Skizze einer neuen Methode der Harmonielehre*, Leipzig, 1880;*Handbuch der Harmonielehre*, Leipzig, 1887;*Vereinfachte Harmonielehre*, London/New York, 1893.**^***Neue Schule der Melodik*, Hamburg, 1883.**^***Katechismus der Phrasierung*, Leipzig, 1890, 2d edition*Vademecum der Phrasierung*, 1900, 8th edition as*Handbuch der Phrasierung*.**^***Geschichte der Musiktheorie im IX.-XIX. Jahrhundert*, Berlin, 1898.- ^
^{a}^{b}^{c}Klumpenhouwer, Henry,*Some Remarks on the Use of Riemann Transformations,*Music Theory Online 0.9 (1994)

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