In the context of horses' locomotion, "natural cadence" is an expression that can be replaced with "natural frequency".

What does Natural Frequency mean?

Natural frequency, a term of physics, is the rate at which a system wants to oscillate without external forces acting on it. In steady state movement, the horse can be viewed as a system with a natural frequency. In the video, the system is the bucket with a bag of grass seeds suspended by a spring from a screen door. It’s an easy to make simple mass-spring system used to illustrate natural frequency. The frequency remains the same at any amplitude of motion. This is not necessarily an intuitive concept. Many people, when asked to predict the behavior over time, will predict that it oscillates faster as the amplitude (distance traveled from the lowest point to the highest point) gets smaller. The video demonstrates that the frequency remains almost exactly the same as the amplitude diminishes.

How does "natural cadence" apply to horses?

The natural cadence is the tempo that is most efficient for horses. It uses the least amount of muscular effort and the maximum elastic energy. It is the tempo that can most benefit the musculoskeletal system by providing the healthy repetitive loading needed to strengthen bone and tissue, with the caveat that the limb and spinal column loading is optimal.

What is optimal loading during natural cadence?

Loading of bones and joints fluctuates between near zero (if there is a flight phase) and some maximum.  We don't need to know the value because it is established that maximum ground reaction force occurs when the fetlock is at the lowest point in dorsiflexion.  Long structues tend to be strongest in compression - when they are at maximum load while in the vertical position. 

Can riders influence this?

Yes. What sets the horse up for optimum limb loading is a high trunk posture, sufficient downward rotation of the pelvis, and balaced muscle coordination of the forelimb and back muscles that control lift, rotation, and bend. Too much tension in the reins, as determined by the horse, impairs this. When all is properly synchronized, the fore limbs can propel the body upward with elastic energy that is created until maximum loading. The hind limbs can store and recycle elastic energy and the back can manage the forces such that the croup and withers bounce together in the flight phase of trot. The rider's seat and posture can be out of the way or they can hinder. The best rider body organization is a neutrally balanced seat or a very low and quiet posting.