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Riemannian theory : ウィキペディア英語版
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,〔''Musikalische Logik. Hauptzüge der physiologischen und psychologischen Begründung unseres Musik-systems'', Leipzig, 1873.〕 notation,〔''Studien zur Geschichte der Notenschrift'', Leipzig, 1878; ''Die Entwickelung unserer Notenschrift'', Leipzig, 1881; ''Notenschrift und Notendruck'', Leipzig, 1896.〕 harmony,〔''Skizze einer neuen Methode der Harmonielehre'', Leipzig, 1880; ''Handbuch der Harmonielehre'', Leipzig, 1887; ''Vereinfachte Harmonielehre'', London/New York, 1893.〕 melody,〔''Neue Schule der Melodik'', Hamburg, 1883.〕 phraseology,〔''Katechismus der Phrasierung'', Leipzig, 1890, 2d edition ''Vademecum der Phrasierung'', 1900, 8th edition as ''Handbuch der Phrasierung''.〕 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.
==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.〔Klumpenhouwer, Henry, ''Some Remarks on the Use of Riemann Transformations,'' Music Theory Online 0.9 (1994)〕

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