Proposal: Transformation Theory in Dance Analysis and Composition
Transformation theory explains tonal relations in terms of transposition and inversion, which correspond to symmetries: rotations and reflections in pitch-class space. We represent these transformations in the group T/I, which has a group action on a set of pitch-classes or pitch-class sets.
Neo-Riemannian theory also casts music in terms of reflection and rotation. Each operation (P, R, LT) corresponds to a reflection or rotation/translation on the Tonnetz, creating patterns of symmetry across the plane. These in turn correspond to patterns of plane symmetry that Escher exploits in his work.
Art, then, is another manifestation of musical chemistry. I have undertaken to translate these concepts into the medium of dance. Symmetries and transformations have abundant precedent in both folk and classical dance (see Fig. 1), but they have never been presented and studied in any form so developed as transformation theory. I feel this undertaking offers benefits to both music theory and dance composition. Dance is a visually engaging and flexible medium with which to illustrate music theoretical concepts. Music theory offers choreographers organized models with which to structure their work.
Michael Buchler and Nancy Rogers have shown that movement patterns in square dance correspond to transformations in music theory (“Square Dance Moves and Twelve-Tone Operators: Isomorphisms and New Transformational Models,” 2003). I will extend this line of enquiry to contradance, and use group theory to model movement patterns on the level of the couple, square, and line formation (see Fig. 2).
In Part II, I will apply two general mathematical concepts to dance. The first is plane symmetries, which have already been thoroughly studied in music theory and in geometric art. I will show that reflective and rotational symmetry in dance are a natural result of the symmetry of the human body and the nature of social dance. (See Fig 3-4.) The second is group theory, which underlies Lewinian transformation theory. Instead of a set of pitch-classes, I will define a set of ten movement motifs, notated A through J. I then define a group of operators analogous to those found in transformation theory and tone rows. The generators of my group are as follows: Rf (reflection, or the mirror image of a gesture), Rg (retrograde, or the gesture performed in reverse), Fa (faster), S (slower), Ro90 (rotate 90 degrees clockwise), Ro180, and Ro270. By combining these in all possible combinations and adding the identity element, I define a group under the conditions of associativity, closure, identity, and inverses, of cardinality 39. By applying this group to my ten set generators A through J, I produce a set of 390 elements that map to each other. The 390 resulting motifs form the vocabulary of my piece Dance in Modular Space, which I will show and analyze in the following video:
| Fig. 1. Choreography by Busby Berkeley in the film Dames (1934) |
| Fig. 2. Expressions of contradance movement patterns |
| Fig. 3. Rotational symmetry in ballroom dance – this arises when the couple travels forward/backward. |
| Fig. 4. Reflective symmetry in ballroom dance – this arises when the couple travels at right angles to their original orientation. |
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