Gaits in General for Dressage:
Math & Variations on a Theme of Walk, Trot, Canter
(or, Why the Old Classical Masters Were Right)
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Gaits of high quality are essential for dressage. Control of these gaits can be accomplished from aids given by the lower body (seat, legs). Gaits can be diminished in quality or damaged by overly active hands or a tense body. Basic understanding of what gaits are is of assistance in interpreting what you feel happening under your seat and legs.
Classical masters of equitation understood that there were important connections among theory, math and anatomy that could act as guides for the practice of riding and training.
Effective mathematicians, like effective (and affective!) riders or trainers, are patient with themselves as they explore the universe of patterns for those that form a foundation for a discipline. Then you can set to one side detours and distractions. But because mathematicians like to analyze gaits at the whole horse and spectrum of possible gaits, we should start with introductory material about terms and ideas.
Patterns are repeated motifs, designs, cycles (gaits are cycles of limb motions) or templates. It is this pattern match between the theoretical universe of mathematics and the actual physical world of gaits that allows us to have insights into their structure and function.
Patterns are a bridge shared between human and equine minds and bodies. So we turn to math applied to equine gaits and find a bridge to first principles about their cycling. The intimacy of connections between the nature of horses, of practice and of theory has kept ancient practice fresh while permitting new knowledge to expand current understandings of dressage.
That human minds can make the bridge between theory and practice permits practice to be informed activity in pursuit of a goal and not mindless repetition and drill with mediocre results. As La Guériniérè says in École de Cavalerie (1769):
All sciences and arts have principles and rules, by means of which one makes discoveries leading to their improvement [perfection]. Horsemanship is the one art for which it seems one needs only practice. However, practice without true principles [as a guide] is nothing but routine, the fruit of which is a strained and uncertain performance, a false diamond which dazzles demiconnoisseurs who are more often impressed by the accomplishments of the horse than by the skills of the horseman. From this comes the reason for the small number of well trained horses, and the lack of ability one sees at present in the majority of those who style themselves horsemen.
He continues:
This shortage of principles renders students unable to tell error from perfection. They have no other option but imitation; and sadly it is easier to descend into bad practices than it is [to devise, to learn] good ones.
Some, desiring to imitate those who would draw from a horse all its virtuosity, fall into the error of [maintaining] the hands and legs in continual movement, which offends the tact [grace, skill] of the rider, causes contact with the mouth to be false and results in uncertain gaits.
And from another 18th century master, Dupaty de Clam, in his Pratique de l'équitation ou l'art de l'équitation réduit en principes we have:
Geometry, anatomy and mechanics give us the first rules of horsemanship. Nobody in his right mind can doubt their validity. It is much wiser to take the known sciences as a guide, rather than merely following one's whims.
Understanding structure and function of equine gaits from their mathematical descriptions is a parsimonious <<réduit en principes>> way of understanding gaits from the first principles of logic (in this case, some math of gait timing) and functioning anatomy (biomechanics). The pen and paper math analysis of gaits allied with their physical expression allows the argument for invincibly soft, minimal aids to be made.
Question: Why do theory instead of practice on gaits first?
Answer: Because it simplifies understanding their biomechanical structure and function, so you have a clear guide on the path to your goal of the "compleat dressage horse." Practice requires humility, patience, unnatural calm.
The theory of gaits is founded on mathematics. As Neil Stephenson says in Cryptonomicon,
...He had figured out that everything was much simpler if, like Superman with his Xray vision, you just stared through the cosmic distractions and saw the underlying mathematical skeleton. Once you found the math in a thing, you knew everything about it, and you could manipulate it to your heart's content with nothing more than a pencil and a napkin. ...
Well, Stephenson is talking about the math of encryption of information and Türing machines, but the point he makes is accurate as a metaphor. Superficially, computation involving the concept of "number," the possible fates of a string of computational operations, like expressions of behaviors such as gaits, are full of "cosmic distractions" and detours. But look how mathematicians can describe gaits with a pencil and a napkin... using a boxandcircle diagram (my informal term is "roadkill" diagram because of its resemblance to flattened fauna).
The mathematical skeleton of gaits is amazingly simple, so simple that it might be embarassing when difficulties crop up on the way to a trained mount (the aforementioned need for humility). Finding the mathematical skeleton among the cosmic distractions in a training program is not easy on the ego or in the expenditure of the treasure of time (the need for patience welded to calmness of mind).
So what identifies a gait? And what makes a gait a dressage gait?
The answer to the first question is that horses are elite athletes like dogs, cats, or deer. These animals are capable of a continuum of limb combinations, some of which have arbitrarily been given names (gaits). We will not distract ourselves with all of those here, even though some of them have engaging names such as "bound" and "pronk" seen in horses at play, jumping or unloading a rider. As La Guériniérè points out in his treatise, not all things that horses can do are suited for classical equitation. However, bounding is appropriate for some of the "airs" of the classical school (but not competitive dressage).
The answer to the second question requires a description of leg behaviors that we can easily compare with each other, especially by their timing characteristics based on the order in which they participate in a cycle. By carefully widening the context of our discussion from
• 1) individual limbs, to the
• 2) whole horse, to the
• 3) spectrum of gaits that a horse commonly exhibits,
we can simplify and focus on understanding the special conditions for dressage gaits.
• 1) Individual Limbs in a Gait
BELOW LEFT is a diagram representing a horse's right hind leg moving through a cycle from mid stance to mid stance, based on digital video frame overlays. Out of context (relations) with the other three legs, it is unlikely that we could deduce the gait from which it comes. But in context with the other three legs, we can compare its timing position or phase and know if it comes from a walk, trot, canter, pace or other gait. In the table of gait images further on this page, the context is complete and the gait identity is especially clear from a "snapshot" frame of the horse at the instant of left hind toedown. These images can be turned into outline icons for comparison in a grid of gaits.
Cycle of right hind leg behavior. Legs alternate being grounded, then repositioning in swing phase. Gait forms that involve shifting the center of mass are very demanding of effort. Quick steps reflect the tendency of horses to demand less of their muscles by shortening ground contact time. We humans do this as well, we have all seen the "supermarket shuffle" where people take short strides while leaning on their shopping carts (2 legs, 4 wheels...).

Functional positions of one leg (RH). Relative to other legs, these positions allow gait velocities to be adjusted. Cadenced dressage gaits involve lengthening the time of ground contact, which is demanding of muscle effort in terms of oxygen demand and flexible strength.


Suspension occurs when all four legs are off the ground (right canter shown here). Left hind toedown is the start of the gait cycle in the biomechanics and dressage literatures. 
• 2) Gaits in the Whole Horse
Each of the four legs involved in a dressage gait cycles its work of moving (braking, thrust) or supporting ((stance) the body and repositioning (swing) itself under control of patterns in the nervous systems (a very cool math of neural nets we will not discuss here). If we add the other three legs, with different times to land from the arbitrarily chosen start point of a cycle (T = 0) at different time intervals from the first limb, we can generate recognizable gaits, as in the table BELOW.
Leg positions at mid stance left fore.

Leg positions at mid stance left fore.

Leg positions at mid stance left fore.




Position of legs in walk at Left Hind Toe Down

Position of legs in trot at Left Hind Toe Down

Position of legs in canter at Left Hind Toe Down

And here we encounter four cosmic distractions plus a barrier to a simple understanding of these three dressage gaits. The barrier is the lack of a common language in several disciplines that have examined gaits. The distractions emerge from the lack of a common language across gait study disciplines, which produces different descriptions of the same thing. This can make one thing (walk, trot or canter, for instance) look dissimilar while simultaneously hiding similarities or obscuring defining differences.
1) The starting point of a gait cycle in the mathematics literature is the left foreleg touching the ground (toedown),
2) The starting point of a gait cycle in the biomechanics and veterinary literature is left hindleg toedown,
3) The starting point of a gait cycle in the dressage literature is designated in the asymmetric gait of the canter as the outside hind leg landing out of suspension, and
4) Dressage literature uses several sorts of diagrams to analyze gaits, each emphasizing a different feature, such as "ground contact time" graphs, or examining gaits in terms of "hoofbeats" (auditory analysis) while analyses from snapshots display leg positions at arbitrary moments.
The Problem: information describing basic questions of gait identity originating from different methods, chiefly the [arbitrarily chosen] place in a cycle that begins a description. Different methods yield results that are difficult to compare. Gait "purity" comes later in our analysis, when problems of consistent descriptive language across disciplines [for our purposes here] have been addressed.
The Solution: use the designation of the cycle start point for the biomechanics literature coupled with the little "roadkill diagram" in the mathematics literature. Then the dressage literature canter falls into line with left canter time one being the default gait and the outside hind landing out of suspension begins the cycle. "Left hind toedown" will be our starting point for gait cycles across these three literatures. Right canter is handled with time one being the landing of the outside hind, but for purposes of plotting gait identity outlines on a Hildebrand Grid, the right canter (left hind = time one), the left half of the walk or trot would be used. (Two formulae are possible for asymmetric gaits, but both fall in the range of dressage gaits.)
This scheme has the desirable feature of giving us a uniform descriptive framework for understanding a number of practical and observable features of gaits. Using a consistent descriptive language also indicates cycle features that might distract us from understanding the foundation features of gaits, and which might obscure special characters that distinguish dressage gaits from other gaits. For instance, pacing or running walks are not appropriate for the discipline of dressage, but why? Very athletic horses can perform these and other gaits in their own disciplines of flat racing, harness racing or gaited competition. But first things first...and sequence and timing are first, so we can decide which gait is defined by a particular order and timing of legs. Math is an ideal way to outline these gait features.
Qualitative "roadkill" math diagrams of gait phases using the biomehanics convention. Numbers indicate RELATIVE phase or fraction of a cycle of a reference foot (*) toedown and the next foot of the gait (or of interest) toedown.  The fourth phase in biomechanics literature terms for Left Canter: in dressage terms this would be Time One or "outside hind." For those who have noticed the sums of fractions in the symmetric gaits = 1, this is not true of asymmetric gaits.  The third phase in biomechanics literature terms. Time 4, "suspension" or "silent time" in dressage terms. OK one more frame to go. Now you see why the Right Canter is the asymmetric gait to compare across literatures: its phase description matches!  The second phase in biomechanics literature terms: Time Three or "inside fore" in dressage terms.  The start of the Left Canter cycle with LH toe down in biomechanics literature terms: Time 2 or "diagonal pair" in dressage terms. Formulae are from digital video at 29.95 fps. The trot is about 76 strides per minute, the canter is about 95 strides/minute. 
Cross Canter or disunited canter shows order of legs different from true canter, and with 4 hoofbeats PLUS suspension. This is not a dressage gait.  Fourth Phase (suspension) occurs with a different relative phase placement of limbs. Compare with left canter frame 3 above.  In "Third" Phase four feet are on the ground. This does not occur in true canter. Left fore is the final leg to land (not labeled). RH 5/17 = 0.3; LF 7/17 = 0.4  Second Phase: disunited or REVERSED diagonal pair formed with the initial leg plus diagonal foreleg than the normal UNITED diagonal of true canter. 4/17 = 0.2  Begin cross canter: LH lands out of suspension in contrast to left canter where RH is the first leg to land, as in frame 4 of left canter above: about 101 strides per minute. T = 0 
This table ABOVE removes cosmic distractions encountered when trying to interpret different models describing DRESSAGE gaits and reveals them to be an internally consistent set of limb actions. Further, DRESSAGE gaits are a very limited set (mathematical term again) of the universe of gaits horses are capable of performing. Cross canter or disunited canter has its limb step sequence diagram "rotated 90 degrees." This gait, with its reversed hind leg sequence has no moment when a transition to trot or walk can be level (note also that relative elevation during cross canter is downhill). The gait itself, with minimal overlap of limbs, is more staccato (bumpy to ride) than legato, and so is inherently not fluent. The next analysis based on timing of limbs helps us understand why.
There is a method for developing gait timing formulas, pioneered for many species (including horses) by Milton Hildebrand. Applied to canter, cross canter or gallop the formula involves using the right half of the gait (left half with cross canter) with LH toedown as its initial condition so the analysis fits the biomechanics and dressage literatures. One or the other of the two hind legs may theoretically be used to generate the gait formula, which can then be plotted on a graph grid. In canter and gallop, contact times close enough that they do not substantially modify the position of the formula coordinates. For instance, if the left canter were used in a gait formula, its initial conditions are RH toe down, so the biomechanics literatures and dressage literatures would use different methods to initialize a stride (RH and LHRF, respectively. Intuitively, this can be confusing although it is technically correct, given an arbirtrary starting point for a cycle.
A plotted grid of timing formulas for symmetric gaits (trot, pace) and asymmetric gaits canter, gallop, cross canter). Symmetic gaits have legs on the left side of the body doing the same things as the legs on the right side in the second half of the gait. Asymmetric gaits have limbs on the two body halves performing differently, as in the diagonal pair of canter, cross canter or gallop.
After M. Hildebrand 1965, Symmetrical Gaits of Horses, Science 150:701708. 
The same plot as ABOVE LEFT, but with the area of dressage gaits shaded grey. Note that "normal" symmetric horse gaits fall inside the dotted line established by Hildebrand. I have used Hildebrand's method* of computing gait formulas for asymmetric gaits and added my calculations to his data. Mature gallop and gaits like cross canter fall outside the arbitrary Hildebrand boundary. A colt's gallop plots near running walk. Dressage gaits can be seen to have timing formulae that create a limited set within the larger set of "normal" gaits. *This method applied to canter, cross canter or gallop involves using the right half of the gait (left half with cross canter) with LH toedown as its initial condition so the analysis is consistent for biomechanics and dressage literatures. 
The axis label "HIND LEG STANCE PHASE, % stride duration" means how long a hind leg is grounded. Hildebrand did not use asymmetric gaits, possibly because the hind legs have different duration. I chose the same hind leg he used, left hind in order to plot the asymmetric gaits.  The axis label "LATERAL ADVANCED PLACEMENT" means that the plot formula indicates the time interval between left hind toedown and left fore toedown (lateral interval or the plot uses the legs on the same side of the animal). 
In the 1950s and 1960s, Milton Hildebrand analyzed symmetrical gaits of horses (walk, trot, pace) and published a diagram in the journal Science that was a distillation of more than 120 samples of "gaits." From this continuum he selected nonoverlapping icons to fill his grid. Each gait was given a "coordinate formula" that identified the timing of contact times of specific legs or leg pairs that placed it on the grid. I have added the asymmetric gaits of canter (right), cross canter and gallop to his diagram (darkest outlines ABOVE) from the same type of motion analysis he used (he used high speed film, I used digital video technology). Tracings on the chart ABOVE show the phase instance of legs at the same moment of left hind toedown, making the display uniform for comparison. Dressage gaits are confined to the lower left sector of this display.
[At last!] Why dressage gaits are different from the gaits of other equestrian disciplines and why they are linked to fluent (legato) transitions.
Dressage gaits turn out to have two consistent features that make them useful for the discipline:
a) They have relatively long contact times with legs (long stance times as their plotted formulas above indicate), and
b) There is a diagonal pair grounded that forms the basis in its timing and spacing for the limb swaps that mark fluent, prompt, level TRANSITIONS between gaits.
c) Transitions between gaits correctly performed are evidence of throughness.
Pacing, galloping or running walks offer no opportunity for dressage transitions in their phasing and stride frequencies. This does emphatically not mean that horses (or other quadrupeds, for that matter) cannot perform transitions at low ground contact times and high stride freqencies. Those sorts of transitions are part of the whole spectrum of transitions and are amenable to statistical analysis. Dressage transitions are a fraction, a limited set, of all the possible transitions and are the ones we wish to understand.
The difficulty of riding correct DRESSAGE transitions emerges from the need for maintaining deliberate, PREDICTABLE tempo (this means regularity! And good, consistent footing!) so the horse can both make the phase shift of limbs from one gait to the next as well as manage the velocity differences between walk, trot and canter.
Transitions are the most frequently ridden dressage movement. For example, in the table below, several 1995 tests are compared in terms of the kind of transitions (whether or not they are scored separately) and their proportion in the whole tests. Note that some transitions are abundantly incorporated into some movements, such as the "enter in trot, halt through the walk, proceed [through walk] in trot" requirement (4 transitions!). When individually scored transitions between movements is added to those between gaits within movements, it could be argued that dressage tests are about transitions first and quality of gaits second. The point here is that quality of gaits and quality of transitions is, among several factors, inextricably linked via tempo.

This high percentage of transitions in dressage performance means that they play a potent role in the definitions of the dressage discipline itself. Further, level, prompt, fluent transitions are only possible at certain deliberate tempos (see the annotated Hildebrand grid) found in the realm of the dressage gaits. This is not coincidence. Phase and velocity shifts enabling level, prompt, fluent transitions lie in the realm of the slower tempos. For those who imagine the slow tempos (about 52 strides per minute) of passage and piaffe represent long suspension times (all 4 legs off ground), data from video records show long ground contact for passage/piaffe, with longest suspension times going to extended trots (still longer suspension times in racing trotting and pacing).
So in a way, a brief look at the mathematics of gaits has led us to identify unique characters of dressage gaits, which has led us to understand the role of transitions or gait changes in defining the discipline of dressage. Classical equitation came to this conclusion pragmatically: we can justify the intuitive assertions of classical equitation by logical argument and inspection of data about the physical manifestations of the mathematics of gaits. And we did this with digital "pencil" and "napkin!"
Rider coordination of seat and leg are strongly influential in creating and maintaining dressage gaits. This is because a rider is subject to the same math of gait and control as are horses and sits directly on muscles that are major movers of legs. A horse is aware of rider relaxation or tension, of motion patterns or position of a rider's body.
Walk with some major muscles controlling the legs indicated.  A rider sits in a position to strongly influence the coordination of muscles and thus the purity of gaits.  Canter as an asymmetric gait is usually an unfamiliar coordination pattern for humans. 
The difference between human and horse coordination being that horses know more patterns because they are tetrapod (4limbed) quadrupeds (go on all fours) and riders are tetrapod bipeds (toddle around on two legs). Humans generally abandon true walking (crawling developmental stage for babies) for the adult upright stepping trot. Check this by "walking" and observing that you swing an arm with its diagonal paired leg. Riding an equine walk as though it were a trot is disturbing to the timing purity of the gait. Cantering, as an asymmetric gait, is generally foreign to humans and needs to be learned kinesthetically from an experienced horse with a clear gait. As collection is approached, mismatches of coordination between horse and rider become increasingly problematic, especially in walk and walk or canter pirouettes.
The timing of pacing, involving lateral pairs of legs, makes fluent dressage transitions impossible. Again, horses can make transitions between pacing and trotting or cantering, but those limb swaps are not level, fluent transitions as required for dressage. Further, a lateral walk or syncopated is a gait fault of timing in dressage. A lateral walk or lateral canter ("Pogo canter") is considered "impure," a deviation from the dressage set of gait patterns. Lateral tendency is often found when the collected walk is misunderstood, ridden with inappropriate rider coordination or the horse is stiff for some reason.
Collected Trot. Simultaneous
stance support of a diagonal pair. A Collected
Walk partakes of the swing and loft of the trot but has the
hind leg braking the gait while ther foreleg is at
mid stance. 
Passage. This gait shares walk and collected trot character. The elevation of the forehand reflects the additional time (cadence) spent on diagonal pairs. 
Collected Walk. This is a gait transitional between walk, passage and piaffe because of nearly simultaneous midstance support from the diagonal pair. 
Free Walk. There is symmetry of phase (thrust/braking) of the diagonal pair. Compare to Collected Walk to see adjusted phasing of swing legs. 
Some trainers frown on or advocate teaching "Spanish Walk." I prefer a Suspended Walk because of its timing relation to collected walk at diagonal support.  

Suspended walk.

Collected walk.

Piaffe. 
As another example of the importance of timing for dressage gaits, the phasing of legs is different for collected walk compared to other walks. A collected walk has fore and hind limbs reaching mid stance nearly simultaneously and is not merely a shortened walk. It is allied to collected trot, to piaffe and passage in the position and balance on the diagonal pair. It also shows some of the elastic character of trot in the freedom of swing limbs. Impression of a cadenced march is indicated by an expressively raised forelimb and a grounded hind limb at mid stance under its hip (engagement). If collection work makes the walk lateral, a horse's training is moving away from the classical path.
Now we are in a position to assert that, theoretically and practically, transitions are crucial for dressage and are bound to gait tempo because phase and velocity shifts can be accomplished in a narrow range of timing modes. Further, we might ask what transitions actually mean for the physical world of dressage training. Theory and practice complement one another as guide and test of the accuracy of the guide. Understanding transitions are a specific case of how theory informs practice based on observed equine behavior. How are transitions capable of advancing gymnastic ability? What might be the distinguishing characteristics of correct transitions that are fluent and in stride? Are there some kinds of transition riding that is not beneficial for horses?
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