Riemannian theory

Musical theories of Hugo Riemann

For the mathematical conjecture, see Riemann hypothesis.

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 C melodic minor scale shown has the same order of half steps and whole steps as the ascending C major scale.

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

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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]

## Transformations

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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.

## See also

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* Schenkerian analysis
* Undertone series

## References

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

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Degrees and functions of the diatonic scale

| I / i
(Major/Minor)| ii / iio| iii / III| IV / iv| V / v| vi / VI| viio / VII

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|
|
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Tonic

Supertonic

Mediant
Dp, Tkp, tP, [D](Sp)

Subdominant

Dominant

Submediant
Tp, sP, tCp

Leading tone
D̸7

Subtonic

C| D| E♮ / E♭
(Major/Minor)| F| G| A♮ / A♭
(Major/Minor)| B♮ / B♭
(Major/Minor)

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Riemannian theory

* Functional harmony
* Parallel and contrast chords
* Harmonic dualism
* Hugo Riemann
* Klang
* Neo-Riemannian theory
* Terzschritt
* Tonnetz

* Handbuch der Harmonielehre
* Lehrbuch des Contrapunkts

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