IV. Translation and Rotation
V. Analytical Methods
VI. Analysis of Unspecialized Works
Bach's Invention No. 1
VII. Analysis of Works with Intensive Applications:
Bartok's Music for Strings, Percussion & Celesta, and Webern's Variations for Piano, Op. 27
VIII. Quadrate Transformations
IX. Some Psychological Considerations
Symmetry is shown to be the major determining factor in composition. Some of the compositional parameters that are demonstrated to be symmetry operations are: all aspects of serialized composition, all contrapuntal operations (including imitation, canon, rounds, cancrizans, melodic inversion, invertible counterpoint, augmentation, and diminution, and cantus firmus composition), all musical forms (including all sectional, contrapuntal forms, and arch forms), isorhythm and isomelos, ostinati and passacaglia, mirror chords, planing and fauxbourdon, vibrato, scale formation, invertible counterpoint, meter and pulse, timbre, trills and other ornaments, Alberti bass and other accompaniment figurations, antiphony, the circle of fifths, and pitch itself.
Definitions, descriptions, and mathematical formulations of the different types of symmetry are provided, and each of the major types is explored with examples. Parallels are shown in nature and other art forms. An analytical methodology is developed, and specific works are examined to demonstrate intensive applications. These include Bartok's Music for Strings, Percussion, and Celesta and Webern's Variations for Piano, Op. 27.
Some new transformations are developed, called the Quadrate transformations, which are 90 degree rotations of a basic set, exchanging the time and pitch dimensions. An essay on these also appeared in Perspectives of New Music, 1973, under the title "New Symmetric Transformations". Chapter II (on Definitions of musical symmetry) was also published in the Journal of Transfigural Mathematics (Berlin) Vol. 3/3 (1997-98). A chapter is also devoted to the possible psychological effects of musical symmetry.
The first edition was originally published as a dissertation for completion of a PhD in music at West Virginia University in 1973. The original title was Symmetry as a Determinant of Musical Composition. This new edition preserves the original ideas and adds some new materials and discoveries. The New Transformations names have been transposed so that the Quadrate Prime (QP) is the form that reflects around an ascending diagonal, thereby switching time and pitch axes. This makes the Quadrates more consistent with their own transformations and with the Prime form. Mathematical descriptions of these forms have been added.