An eleve is a simple exercise that involves nothing more than rising onto the balls of the feet. There is a very similar exercise that requires the same move with the knees bent in a demi-plie or a grand-plie position. This is called a releve. Many teachers will incorporate the eleve and releve into the plie exercise, and others will do an individual exercise specifically for eleves and releves. It depends on the age and advancement level of the class, or sometimes on the mood of the teacher.
Obviously, the purpose of the eleve is to develop the muscles of the lower and upper legs, along with the smaller muscles of the feet and the ankles. The eleve is a very common exercise in beginning ballet classes because it is so important for every other ballet position and movement. Without a strong and flexible lower body, the other ballet positions cannot be performed well.
During a ballet class, when the teacher specifies an isolated exercise each for eleves and releves, this will usually be for a beginning or intermediate class. In the eleve exercise, the pianist should provide music that is not too slow and which sounds like it is moving faster than it is. This gives the dancers the ability to hold the position a bit longer than they might otherwise be inclined to do. For beginners especially, the muscles can start burning quickly. A good pianist can help the dancers get through it without them being tempted to give up or lose form.
The pianist can achieve this effect by placing the downbeats of the music slightly ahead of the pocket and pushing toward the center of the phrase and then backing off into exact tempo for the second half of the phrase. This makes the music feel like it is driving forward even if the melody is lyrical rather than rhythmic. The melodic material can have an arcing effect to match the up and down movement of the ankles which makes the exercise more fun for the pianist and for the dancers. If everyone is having fun good things will happen.
To hear eleve music you can visit Ballet Class Music and hear free samples by composer/pianist/dancer Don Caron from his First Class Album series of ballet class music.
For more descriptions of how ballet class music is performed for the various ballet class exercises, read the posts about tendus, degages, and rond de jambes.