Professor of Applied Mathematics (Univ. Autònoma de Barcelona), former dancer and dancesport judge
November 7th, 2013
Rythmic quality is universally regarded as an essential constituent of good dancing. However, it lacks a precise definition, which makes it liable to subjective opinion, even by the judges who officiate at dancesport competitions.
This article is part of a project that aims at changing this state of affairs. It is a sequel to a more technical paper that makes use of some mathematics. Here we will skip the mathematical technicalities, but we will still try to explain how do they result in dance-interesting conclusions.
Let us make clear from the beginning that we are aimed only at rhythmic quality in a strict sense, excluding any expressive deviations from the ideal pattern. Besides, this ideal pattern is not meant to be specific for a particular choreographic sequence, but it should have a generic character for a given dance genre.
Our project might eventually provide a system for effectively rating rhythmic quality in dance. Unfortunately, this requires overcoming certain technological difficulties that will be discussed in §9. Even so, we hope that the present preliminary analysis will already help towards a better understanding of rhythmic quality and towards showing that this concept is more objective than it seems.
1. Starting definitions
We intend to rate rhythmic quality as a degree of agreement between a performed accentuation and a specified rhythm pattern. By an accentuation we mean simply a distribution in time of certain accentuation variables, i.e. certain relevant measurable quantities. Accordingly, we view an accent as a strong concentration of these variables at a particular location in time. A rhythm pattern is an ideal accentuation, usually one that repeats periodically in time. Finally, the degree of agreement will be measured by a mathematical comparison between the performed accentuation and the ideal one.
The main aim of this article is finding out which are the accentuation variables relevant to dance. The issue of specifying a rhythm pattern for each dance and measuring the degree of agreement with it will be a matter of future work.
2. What is a movement accent?
In dance, there is a certain tendency to consider as rhythm-marking accents the steps of the feet. However, in dancesport and many other dance genres a good rhythmic quality lies more fundamentally in the movement of the centre of the body than in that of the feet. Generally speaking, however, the movement of the body centre is more flowing than the steps of the feet. So, the problem of identifying the rhythm-marking accents becomes a more difficult task.
The existing dance literature hardly goes into details about the precise meaning of a movement accent. Having said that, some authors make a try at it. For instance, Ann Hutchinson Guest (Labanotation, page 478 of 3rd edition 1977) defines a movement accent as “the result of a sudden momentary increase in the use of energy”, and, more interestingly, she distinguishes the following classes of accents: “A strong accent at the start of a movement: impulse; a strong accent in the middle of a movement: often a swing; a strong accent at the end of a movement: impact”. An impact merged with an immediately subsequent impulse makes a rebound. Besides these terms, there are other ones which have to do with sustained motion and the absence of an accent, or even the opposite of it: vibration, steadiness (motion without an accent), stillness (absence of motion), suspension. This language allows to describe movement accentuation in a qualitative way. For instance, Ruud Vermeij (Latin; Thinking, Sensing and Doing in Latin American Dancing, 1994, pages 119–126) uses it to describe the main rhythmic character of each of the five Latin dances of dancesport.
However, this is not enough for our purposes. In order to rate the rhythmic quality of dance movement, we have to rely on some measurable accentuation variable. And it is not obvious which variable is the right one. In the words of Ruud Vermeij (cited work, page 159): “Even if we understand rhythm only as timing, the question remains, what is it that changes?’”
3. Empirical observations
In order to answer the preceding question we will base ourselves on certain empirical observations about the human perception of rhythm in a motion. In this article we consider some very simple cases where the motion reduces to the translation of a single point (which one can think of as the centre of the body of a dancer). More specifically, we will look at bouncing balls and oscillating springs and pendulums. Such motions are universally recognized as typical examples of rhythmic ones. So, it is natural to ask where does one locate their rhythm-marking accents. We have posed this question to a variety of people, many of them dancesport experts, and we have got the following answers:
Empirical observation 1. For the oscillations of a mass suspended by an ordinary spring, the movement accents are perceived at the recoil points.
Remark. Having said that, there is a certain bias in favour of the bottom recoil instead of the top one (see §7).
Empirical observation 2. For a ball that bounces on the floor, the movement accents are perceived at the bottom rebound points.
Empirical observation 3. For a pendulum that oscillates with a small amplitude, the movement accents are perceived at the recoil points of both sides.
Empirical observation 4. For a pendulum that oscillates with a large amplitude, the movement accents are perceived at the bottom.
Empirical observation 5. For a pendulum that oscillates with an intermediate amplitude, there is a general hesitation or division of opinions between locating the movement accents at the recoil points of both sides or locating them at the bottom.
On the basis of these facts, we claim that the movement accents are determined by the magnitude of the acceleration vector. Other candidates for accentuation variables —such as speed, kinetic energy, their respective rates of change, or the magnitude of the sensed acceleration vector that will be considered in §9— may happen to fit some particular perceived movement accents, but not all of them. In contrast, the magnitude of the acceleration vector does explain all those empirical facts.
As it is shown in the already mentioned more technical paper, the preceding claim is supported by a mathematical analysis of the above considered motions using their so-called differential equations. Here we will try to explain as much as we can in plain words for a dance audience.
4. Velocity, acceleration and the two components of the latter
Let us recall that acceleration is the rate of change of velocity. And that velocity is the rate of change of position. Both of them are vectors, which means that they have not only a magnitude but also a direction. The magnitude of velocity is usually called speed, and its direction is usually called simply the direction of motion. Having said that, one can admit of negative speeds: a negative speed in a given direction should obviously be understood as a positive speed of the same magnitude in the opposite direction. One can have a motion in a constant direction with a varying speed, as well as a motion with a constant speed in a varying direction. In general, both the speed and the direction of motion can change in time.
For a general motion, the acceleration vector has two components, one in the direction of motion and another in a perpendicular direction. They are called respectively the tangential and normal components of the acceleration.
The tangential component gives the rate of change in speed and the normal one gives the rate of change in the direction of motion itself.
For a rectilinear motion, the normal component vanishes. Therefore, the acceleration can then be viewed simply as the rate of change in speed, and a motion accent is then nothing else than a marked change in speed.
However, for a curvilinear motion it is not so. In fact, curvilinear motion allows for accents with a vanishing rate of change in speed. Such accents are the main point of the swing motion that is so characteristic of such dances as Waltz, Viennese Waltz, Foxtrot and Quickstep.
5. Swing and rebound accents
Regardless of the technique for achieving it, in a swing motion the centre of the body is supposed to move like the bob of a pendulum. The mathematical analysis shows that such a motion contains an accent at the lowest point. Since at this point the speed is maximal, it follows that in this moment the tangential acceleration vanishes and therefore the acceleration vector is purely normal. Now, in order for such a situation to really exhibit an accent, i.e. a peak in the magnitude of the acceleration vector, a specific mathematical condition must be satisfied that in the case of a pendulum occurs only for large enough amplitudes. In general terms, this necessary condition requires a combination of both a large enough speed and a large enough curvature (i.e. a small enough radius of curvature). In dance terms, the latter translates into a strong enough rise and fall.
For small enough amplitudes, the oscillations of a pendulum don’t satisfy that condition. In this case no accent at all occurs at the lowest point and the only accents of motion that are present, which are rather weak, occur at the side recoil points, where the acceleration vector is purely tangential. For intermediate amplitudes, the pendulum motion contains both types of accents, which explains the hesitation or division of opinions mentioned in Empirical observation 5.
The main feature of a swing accent is that the acceleration vector is perpendicular to the direction of motion. As we have seen, this requires a curved trajectory. In a pendulum-like motion the trajectory is contained in a vertical plane. In other cases, however, it could be contained in an horizontal plane. Such accents can be called rotational swing accents. Having said that, here we are dealing with a motion of rotation about the centre rather than the motion of the centre itself.
One can define a rebound as a sudden change in the direction of motion. This can be seen as the limit of a swing accent where the radius of curvature has become infinitely small.
6. Impulse, impact and recoil accents
Swing accents, i.e. motion accents where the acceleration vector is purely normal, lie somehow at one end of a spectrum. At the other end we have the accents where the acceleration vector is purely tangential. In this case a motion accent can be identified with a marked change in speed.
The case of an accent of this kind where the speed becomes larger corresponds to the notion of an impulse. In the case where the speed becomes smaller, one can call it an impact accent (even in the case of a relatively weak slowdown or a non-vanishing final speed). Finally, it can also happen that the speed becomes negative, i.e. positive in the opposite direction, as in the side recoil points of a pendulum, which case can be called a recoil accent. A recoil can be viewed as a particular case of a rebound where the direction of motion changes in 180 degrees. In dancesport it is present in check and rock actions.
Technically speaking, a walk means in dancesport a special kind of step. What makes it special is mainly the presence of an impulse accent and/or an impact one. Consider, for instance, the Rumba walks, or the Tango ones. In a sequence of several of them in the same direction the speed of the body centre is not uniform, but it oscillates between higher and lower values, with one oscillation for each walk. This entails that the acceleration oscillates between positive and negative values. So its magnitude will exhibit a couple of maxima for each walk, one of them being an impulse accent and the other an impact one. The impulse accent is caused by pushing from the standing foot and the impact accent is caused by putting weight on the moving foot (which becomes the next standing foot).
Having said that, usually one of these two accents predominates over the other. In Rumba, for instance, the impulse acceleration is usually more marked than the impact deceleration. In fact, the established gliding of the moving foot in this dance indicates that this foot collects the weight in a gradual way. In contrast, Tango often emphasizes the impact accent. In fact, the staccato interpretation of this dance puts a sharp accent at the end of the movement, and not at the beginning of it.
Anyway, the rhythm-marking accent of a walk need not coincide with the landing of the moving foot on the ground, but it lies in the initial acceleration of the body and/or its final deceleration.
Let us look now at bounces and hops, as in Samba, Jive and Quickstep. We assume that the dancer is travelling at constant (possibly zero) horizontal velocity, in which case the horizontal motion does not contribute to the acceleration and therefore the accents are the same as if the bounces or hops are done in place. So it amounts to consider a purely vertical rectilinear motion, where we know that the acceleration is simply the rate of change in speed.
The two basic models for bouncing are: (1) a ball, and (2) a mass attached to the floor by means of a spring that operates vertically. The latter is mathematically equivalent to a mass suspended from a certain height by means of another spring. The main difference between models (1) and (2) is that the ball can leave the floor, whereas the mass remains connected to the floor (through the spring).
While the ball is flying, its acceleration vector remains constantly equal to the acceleration of gravity (we assume that the friction with air is negligible). In particular, the magnitude of the acceleration vector remains constant. So there is no accent of motion during the flight. The only such accent occurs at the bottom rebound.
In contrast, the spring oscillations contain an accent not only at the bottom but also at the top. If the spring is soft enough, the motion is symmetric about the mid point. From now on we will assume that this is the case. So the two accents differ from each other only in the direction of the acceleration vector, but not in its magnitude.
In spite of that, when such a motion is produced and the observers are asked to locate the rhythm-marking accents, some of them indicate only the bottom rebound. One can convince them of the contrary by videoing the motion and then turning it 180 degrees. However, the fact remains that they initially preferred the bottom rebound. Such a preference for the bottom rebounds rather than their top counterparts is also present in certain dance practices. In fact, when the Samba bounce is taught and demonstrated by alternately flexing and extending the knees, without any foot steps, the timing counts are usually uttered at the bottom rebounds. And when this stepless repeated bounce is done to music, the bottom rebound is usually in synchrony with the strongest beat. This does not agree with the established technique of such Samba figures as the Whisk, where the top rebound is the one that is scheduled to happen at the time of the strongest beat (see Walter Laird’s Technique of Latin Dancing, 6th ed. 2003, revised 2006, page 67).
The reason for this bias between top and bottom is gravity. Even if a perfectly symmetric vertical oscillation is performed, with the top and bottom accents having exactly the same magnitude of the acceleration, the dancer feels them in a different way. In fact, the motion of the body centre is the result of two acting forces. One of them is gravity. The other is the force exerted by the dancer with the help of the floor. Gravity is certainly directed downwards. In contrast, the force exerted by the dancer-floor system is directed upwards (in order to direct it downwards one would need either some kind of anchor at the floor or a ceiling within reach to push from). So the downward force comes for free, and the dancer feels mainly his upward effort. This effort is minimal at the top and maximum at the bottom. So the dancer really feels the two accents in a different way in spite of the fact that the magnitude of the acceleration is the same.
Our daily experience with gravity is somehow the reason that both accents are perceived in a different way not only by the dancer but also, to a certain extent, by the spectator.
Jive bounce is very similar to the Samba one. Both of them follow the spring model, where the dynamics of the body remains connected to the floor,
By the way, Samba and Jive bounces are a clear example of the fact that the movement accents of the body centre can diverge in time from the foot steps. In fact, the bounce accents (alternately bottom and top) occur at the beginning of the quarter beats 1 3 5 7, whereas the foot steps occur at the beginning of 1 4 5 8.
In contrast to Samba and Jive, Quickstep bounces —as in the Scatter Chassés and the Pendulum Points— follow the bouncing ball model, which allows for a momentary disconnection from the floor and contains accents at the bottom rebounds only.
8. Music conducting
In dance, the movement must follow the music. In music conducting, it is the other way around: here it is the music that must follow the movement. In both cases A following B means that A’s rhythm-marking accents must conform to those of B.
So in music conducting the movement accents of the baton must be very clear and accurate, so that they are clearly read by the musicians. Accordingly, conducting methods make a frequent use of well-defined recoil and swing accents. This is specially the case of the Saito conducting method, that makes a careful use of different kinds of movement accents.
9. Towards measurement
Having identified a physical variable that matches the human perception of the rhythm-marking accents of motion —namely, the magnitude of the acceleration vector— it is natural to go for measuring it. Today’s smartphones contain certain micromachined accelerometers that are supposed to measure the three components of the acceleration vector. One should be able to use these sensors to produce an application that keeps track of the accentuation of the motion of the body centre and even compares it with a dance-specific ideal pattern. As a result, the dancer would get a number that would rate the rhythmic quality of his/her motion.
However, this is not as simple as it seems. The main source of problem is that, in actual fact, these accelerometers do not measure the acceleration vector of the actual motion, but only its difference with respect to the acceleration vector of gravity. In the following we will refer to this difference as the sensed acceleration vector.
By the way, by multiplying the sensed acceleration vector by the mass of the moving body, we get the total of the acting forces other than gravity. In the case of a solo dancer, this is nothing else than the force exerted by the dancer-floor system that we were considering in §7. The opposite of this vector is the force that acts on the floor, which is sometimes called dynamic weight.
In order to get the acceleration vector of the actual motion, it is as simple as taking the sensed acceleration vector and adding to it the acceleration vector of gravity. Since the latter is perfectly well known, there should not be any problem.
However, it happens that the accelerometer measures the sensed acceleration vector in a coordinate frame that is bound to the device and can rotate with it, whereas the gravity acceleration vector is known to us by its coordinates in another frame that remains fixed (more properly, it remains bound to the Earth). In order to help with this problem, accelerometer sensors are usually accompanied by gyroscope sensors. In principle, the latter allow to keep track of the changes in the orientation of the device. In practice, however, orientation errors accumulate over time, which makes it difficult to efficiently make use of such a procedure for our purposes. As it is shown in the already mentioned more technical paper, except for very simple motions one needs either more accurate and better calibrated sensors or more elaborate methods. ❀