Until early 20th century study on various works had been focused on melody, rhythm, or harmony. Recently, however, metric has actively been studied among many music theorists. A typical type of symmetry commonly found in pre-18th century works is simple in a way that words are paired every so often. This pattern had been diversified with the adverse of Wiener Klassik era when the theoretical work on symmetry began systematically. Koch and Naegeli laid foundation on the study of symmetry and the concept was further developes to include even a spacial symmetry suggested by Riemann at the late 19th century. Later on the early 20th century a diverse application of symmetry and asymmetry was treated importantly in 12-tone music. There are three types of symmetry. The first type is a rotational symmetry along the horizontal axis. The second type of symmetry is a horizontal-moving motif along the vertical axis. And the third one is a mirroring type along any line or plane. These are presented in various forms such as motif iteration, translation, sequence, inversion, retrograde, retrograde inversion, augmentation or diminution. This article presents a detailed analysis of "Ludus tonalis", a Hindemith's work for piano, from the viewpoint of symmetry techniques. In his book "Unterweisung im Tonsatz" which publisched in 1942, Hindemith was trying to prove his theory with the case of "Ludus tonalis". The work begins with praeludium and ends with postludium. Here the postludium is simply a half rotation of praeludium and this is, therefore, a typical case of retrograde inversion. There are twelve fuges and eleven interludiums between praeludium and postludium. A diverse form of symmetry are found in both inter-fuge and intra-fuge relation.

Ⅰ. 들어가는 말 
Ⅱ. 음악에서의 시메트리(Symmetrie)의 일반적 의미 
1. 시메트리에 관한 연구의 발전 
2. 개념 및 형태 
Ⅲ. 시메트리의 관점에서 본 힌데미트의 음악 
1. 힌데미트의 음악적 발전 약사 
2. "음의 유희"의 탄생배경과 구성 
3. "음의 유희"에서의 시메트리 
3.1. 프렐루디움(Pra¨ludium)과 포스트루디움(Postludium) 
3.2. 인터루디움(Interludium) 
3.3. 푸가(Fuge) 
Ⅳ. 나오는 말 
참고문헌 
영문초록

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