"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, notation, harmony, melody, phraseology, the history of music theory,''Geschichte der Musiktheorie im IX.-XIX. Jahrhundert'', Berlin, 1898. 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 In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \fr ...

Dualism

Riemann's "dualist" system for relating triads was adapted from earlier 19th-century

harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...

theorists. The term "dualism" refers to the emphasis on the inversional relationship between

major and minor In Western music, the adjectives major and minor may describe a chord, scale, or key. As such, composition, movement, section, or phrase may be referred to by its key, including whether that key is major or minor. Intervals Some intervals ...

, with

minor triad In music theory, a minor chord is a chord that has a root, a minor third, and a perfect fifth. When a chord comprises only these three notes, it is called a minor triad. For example, the minor triad built on C, called a C minor triad, has pitch ...

s being considered "upside down" versions of

major triad In music theory, a major chord is a chord that has a root, a major third, and a perfect fifth. When a chord comprises only these three notes, it is called a major triad. For example, the major triad built on C, called a C major triad, has pitch ...

s; this "harmonic dualism" (harmonic polarity) is what produces the change-in-direction described above. See also the related term

Utonality ''Otonality'' and ''utonality'' are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone (identity), respectively. For example: , , ,... or , , ,.... Definition ...

.Klumpenhouwer, Henry, ''Some Remarks on the Use of Riemann Transformations,'' Music Theory Online 0.9 (1994)

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'. 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 vice versa. Riemann's theory of transformations formed the basis for

Neo-Riemannian theory Neo-Riemannian theory is a loose collection of ideas present in the writings of music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. What binds these ideas is a central commitment to relating harmonies directly t ...

, which expanded the idea of transformations beyond the basic tonal triads that Riemann was mostly concerned with.

Sources

Diatonic functions {{music-theory-stub