Riemannian theory, in general, refers to the musical theories of German theorist

Hugo Riemann Karl Wilhelm Julius Hugo Riemann (18 July 1849 – 10 July 1919) was a German musicologist and composer who was among the founders of modern musicology. The leading European music scholar of his time, he was active and influential as both a mus ...

(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\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that ...

Dualism

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

harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...

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 an interval, chord, scale, or key. A composition, movement, section, or phrase may also be referred to by its key, including whether that key is major or minor. The words derive ...

, 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 A, called an A minor triad, has pit ...

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 conversely. 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 theory, music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. What binds these ideas is a central commitment to relating harm ...

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

References

Diatonic functions {{music-theory-stub