Figure 1
An arbitrary rhythm
tum tiddy tum tiddy tum tum
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DANCE APPRECIATION
A dance is an aesthetic entity existing in the four dimensions of space-time. Different styles of dance have different degrees of concern for the spatio-temporal symmetry of the movements. For example, the communication of a variety of emotional feelings to the viewer has been suggested as a major aim of American Modern Dance (Mazo, 1977,166), and Classical Ballet evolved through a dramatic form with the purpose of narrating a story. However, other styles of dance appear to be concerned solely with the manipulation of abstract patterns in space-time, for example Ballroom Dancing, Scottish Country Dancing, and ballets such as those by George Balanchine.
PERCEPTION
The aesthetic appreciation of a dance is complicated by some curious differences in the human perception of symmetry in space and time.
Time by itself is of only one dimension, so the symmetries possible of a pattern in time are only translation and reflection. If one considers an arbitrary pattern in time, e.g. the rhythm in Figure 1,
and then considers the results of appending a translated copy to itself (i.e. repeating it), as in Figure 2, the human ear perceives the repetition easily.
If however, the pattern is reflected and appended to itself (i.e. repeated backwards), as in Figure 3, the human ear has difficulty in appreciating the symmetry.
The human eye has the opposite facility. This may be seen in Figures 4 and 5, which show a random pattern of dots in a square replicated four times in the four quadrants of the Figures. (Julesz, 1971, 57).
In Figure 4, the replication is by translation. The human eye only with difficulty perceives the translation symmetry.
In Figure 5, the replication is by reflection. The human eye easily perceives the reflection symmetry.
This difference in the ease of human perception of different types of symmetry in space and time is a great complicating factor for any intending choreographer.
WIDDERSHINS
Although the human form has approximate reflection symmetry across the sagittal plane, human culture associates different meanings to movements done on the right and on the left.
Also, most people are more adept at performing complex movements using their right arm or leg than using their left. This carries over into dance. For instance, the tour de force in Swan Lake of 32 fouettes en tournant is normally done on the right leg.
In Scottish Country Dancing, circling to the left (clockwise viewed from above) is normal ('deasil). Circling to the right is called 'widdershins', meaning 'the witches way'. Being the direction of apparent motion of the sun as viewed from below in the Northern Hemisphere, this is perhaps a hangover from sun worship (Milligan, 1976, 11).
In Ballroom Dancing, circling to the right around the floor is considered normal, and to the left is antisocial. Oddly, the circling is typically performed by the couples taking a 3/4 turn to the right at each corner of the room, rather than a rather simpler 1/4 turn to the left. Sequences of steps turning to the right this way are called 'natural' in Ballroom Dancing, whereas turns to the left are called 'reverse' (Moore, 1951, 48).
The origins of the convention of progressing anti-clockwise around the ballroom dancefloor are obscure. It was clearly already evident in the the early 19th Century, when the Viennese Waltz became popular. Paradoxically this was achieved by only dancing 'natural turns', in which the couple turns clockwise. Not until the 20th Century did the 'reverse turn' become equally popular. Progressing anticlockwise while dancing the anticlockwise 'reverse turns' requires considerable skill and practice, and until the 20th Century was considered antisocial.
Perhaps the convention arose much earlier when dancing was disapproved of by the Church, and dancing was the associated with witches, and hence had to progress 'widdershins'. Alternatively, perhaps doing 'natural turns' was considered more natural, as the turning direction was the same as the movement of the shadow of a sundial gnomon (in the Northern Hemisphere), and the anticlockwise progression around the floor was found by inexperienced dancers to be easier doing these turns.
Another possibility is that it has something to do with the men wearing swords on their left hip. This was the normal position for a scabbard, as it enabled the sword to be drawn more easily with the right hand than if the sword were worn on the right hip. Thus in an Allemande, the lady would normally be on the man's right to avoid tripping over the scabbard, and it would be reasonable to progress anticlockwise around a room, putting the man inside the circle, to avoid hitting the legs of the audience with the scabbard also.
REPETITION
The ability to recognize an acoustic sequence when repeated in time but not when reflected carries over into dance. Many Classical Ballets, particularly those from the 19th Century, have sequences of steps repeated three or four times. Usually the fourth repetition is modified as an introduction to the next sequence.
Repetition is very important in learning dance and other movements. Studies have shown that coordination and efficiency improve with repetition, even after millions of previous repetitions, (Schmidt, 1975, 45).
Because of the approximate left-right symmetry of the human body, left-right symmetry in dance has a special significance.
A symmetric shape of a human figure has been equated with an emotional feeling of security and repose. As such, it is an anathema to dance: it has been suggested that too much left-right symmetry in a dance is inclined to put an audience to sleep. In order to stimulate and excite an audience, the shapes and movements in a dance should be unsymmetrical (Humphrey, 1959, 51).
Symmetry plays a large part in dance teaching because pupils find it easiest to copy and learn movements while standing behind the teacher. This teaching situation has the problem that the teacher cannot then watch the progress of the pupils. This problem can be rectified by having the teacher face a large wall mirror. Thus many dance studios have mirrors not only for the pupils to see and assess their own movements, but also for a teacher to see the pupils while demonstrating facing away from the class.
Dance teachers working in studios without mirrors frequently demonstrate facing their class, but performing the actions with left-right reversed. This way, the pupils see a mirror image of what they are to copy. The success of this teaching method presumably depends on the familiarity that the pupils have with coordinating their own movements while using a mirror. It also leads to dance teachers who are very adept at performing left-right reversals of movement sequences. This task is much harder than one might suppose; for example few people can do mirror writing with their non-dominant hand.
Another facility developed particularly by teachers of Ballroom Dancing, is that of doing a simultaneous left-right and front-back reversal of a movement sequence. This enables them to derive the lady's part from the man's when a couple is in closed ballroom hold.
DANCE NOTATION
Although only a small proportion of dancers know how to write down dances on paper using notation, most major dance companies in the world now depend on notated scores for the maintenance of their repertoires. Many dance notations have been devised over the centuries (Guest, 1986, vi). All have the problem of reducing a four dimensional manifold to two dimensions. The basic method of compressing the dimensionality is by quantization. This process also limits the detail representable, control of which is then given by the number of symbols used. Any process for reducing the dimensionality must also lose some direct representation of the symmetries inherent in the higher dimension domain. Many notations have extra symbols for representing commonly used symmetry operations.
The two notations in most common use are Benesh notation (Brown, 1986, 79) and Labanotation (Brown, 1984, 9). Examples of these are shown in Figures 6 and 7. These notations are very different in the ways they treat space and time. Benesh quantizes time and Laban quantizes space. Both retain the left-right symmetry of the human form, mapping left-right movements of the body on the left-right dimension of the paper.
The figure is initially standing facing downstage left on a bent left leg, the right leg extended in front. The right leg is swung to the side and then brought in rapidly as the figure rise on point, and then pirouettes.
Benesh uses notional orthogonal projection of the figure onto a frame on a music-like stave. Each frame is embellished with signs representing detail about positions and movements of the various parts of the body. The frames may be viewed as a series of snapshots of the figure in time. Thus each frame has an analogue representation of left-right and up-down movement. It also has an analogue representation of forward-back movement using the foreshortening of fixed length limbs, and a simple binary symbol to show if the foreshortening is due to the limb being in front of or behind the coronal plane.
Laban notation has time running continuously up the page. The vertical length and positioning of a symbol represent when and for how long the action occurs. Different columns of the vertical stave represent different parts of the body. Additional signs can be added to represent additional detail. The primary division of space is into 27 directions: high, centre, low; forward, centre, back; and left, centre, right, thus referencing a quadricosahedron:
Both Benesh and Laban notations have special symbols for copies and repeats of a movement, and for copies and repeats with left-right reversed, and front-back reversed.
DIMENSIONALITY
The mindset of people in various disciplines might be characterised by the dominant dimensionality of thought needed to perform their work. Thus for example, computer programmers typically need to think in one dimensional character strings. Engineers and architects are inclined to reduce everything to two dimensional drawings. Sculptors and surgeons need to think in three dimensions. Four dimensions are the domain of, amongst others, physicists and dancers. One can scarcely conceive of two disciplines in which the participants are further apart in terms of ability to communicate with one another in terms of their working paradigms.
It is tempting to imagine that the study of symmetry might be an element where they can find common ground.