Modern Show Foxtrot vs. Old Style Foxtrot
Below are two video clips and several photos to help you to understand some of the different gaits which are all called a "foxtrot".
VideoShowFoxtrotters.WMV Trail (Natural/Old Style) Foxtrot.WMV
Many show foxtrotters have the potential to be very smooth, but usually are trained and shod in a manner that decreases the quality of the ride. This is done to increase the animation of the legs, head nodding, ears flopping, and other show-ring trends. The stride is increased so much that one foot is still in the air when the opposite foot is lifted up, thus creating a falling motion. Ironically, this is the same thing that makes non gaited horses rough to ride. Some show horses will be doing a correct foxtrot motion, but in the air rather than on the ground. Others actually have exaggerated their trot to the point that they are not doing a true foxtrot any more, but are long trotting in the rear and paddling air with their front feet. The result is a bouncy ride that is only good for the show ring and nowhere else!
The next 3 pictures (which match the Official Logo) are of the good old style foxtrot gait w/weight placed on the left front hoof prior to the right hoof being lifted up & right rear hock just passing the hock of left rear. An important fact to understand is that when a horse gaits correctly it can stumble with the foot being put down, but will be able to catch itself because it still has weight on the opposite foot, and that foot is still underneath its body. In a modern show gait the stride has been lengthened so much that the foot being set down is way out in front and the other foot is so far behind the horse's center of gravity that they will often fall on their face if the front foot slips or stumbles.
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If you take a slow motion video of the show gaits you will often see that at no point do these horses match the Foxtrot Symbol. See the photos below and observe the silhouettes.

This show horse's rear legs are so far out of a true gait sequence that it is actually very close to a long trot like a Thoroughbred or Lanky Quarter Horse. It has its left rear leg forward way too much, as the left hock should just be passing the right one at this time when the front right leg is fully extended. On the front end its left leg is being lifted up before the right leg bears weight, causing free fall and a bounce, or rough gaiting.
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This show horse is almost hard trotting, with a foxtrot motion in front. In the front end its left foot has left the ground and its right is falling towards the ground, so even though it is in gait it is a rough ride due to pushing the horse past its natural ability. In the rear end it has almost the same problem with the right hoof being picked up before weight is placed on the left. Also notice how the horse has been pushed way past the "Old Time Shuffle" to a hard trot motion. (You can tell this because at the very moment when the front right leg is fully extended and (should be) touching the ground, the left rear hoof should be at only 3/4ths of its stride with the hock just past the right hock & the right rear leg should be straight (vertical) from the hock to the ground. In recent years quite a few horses with this type of gait have placed well in the world show and celebration in Ava, Mo.
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This show horse is foxtrotting with the correct motion, but because of the animated head nodding and exaggerated stride, the right front hoof is still in the air when it should be setting down. This "rough gaiting" causes the horse to free fall until the right front hits the ground, thus a bouncy ride with lots of animation. This is a good example of a wonderfully bred animal that's been shod and trained to "show gait". This is the type of horse that has been winning the world championships in recent years.
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There are several silhouettes being used to represent the foxtrot by various organizations and clubs. Let's compare some of them to the official breed logo silhouette (on the right):

If we look at this one we'll see that the horse's hocks are in the correct position for a smooth foxtrot, but the front legs are spread too far apart. This is very common in the modern show gait and is simply a case of trying to get a longer stride. The problem with this gait is that by the time the forward front hoof bears weight the other front hoof is so far behind the horse's center of gravity that if it stumbles on the first hoof, the second can not catch it from falling. In the Western states I often hear cowboys that ride the range make fun of foxtrotters saying they fall down in rough country. This unfortunate reputation has come about because of this type of show gait.
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Looking at this one we'll see that the horse's hocks are past the point of a smooth foxtrot. The left hock is too far forward when the right front hoof hits the ground. This is a good example of being trotty or too diagonal. Also, the front legs are taking too long a stride for a smooth sure footed ride, even though the horse may be doing a "correct" foxtrot by modern standards.
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"I would like to see the show trend turn back to the original idea that a foxtrotter is supposed to be a smooth ride, and judge them accordingly. If a horse is a world champion foxtrotter it should be among the smoothest horses in the world. After all, isn't that the reason we ride gaited horses?" Alma DeMille