An air by Mozart is playing and the chandeliers are glittering above the huge oval arena as the riders, immaculately clad in the traditional bi-corn hats, white gloves, coffee-coloured double-breasted tail coats and high boots, begin to put their mounts through their paces.
It is the utterly harmonious co-ordination between man and horse which is so entrancing to behold. The riders are as if glued to the saddle and seem to convey the messages to their mounts almost by telepathy. This is haute ecole riding, which developed in the 16th century from both the agility needed by the cavalry in battle and the fact that equitation became a fashionable pursuit for the nobility.
Some of the horses are grey, which means they are young, as all Lipizzaners are born black. One in a hundred remains dark for life, and these are considered the lucky horses. As we watch these graceful, disciplined beasts executing the shoulder-in or sideways walk, the crossed legs or travesale and the cadenced trot of the piaffe, I remember from my childhood fixation that these movements are known as airs on the ground. Then, and this is lucky, we see two riders on foot who, using long wands and words of encouragement, persuade a stallion to rise on its haunches then jump into a courbette without lowering its forelegs. Lucky because these balletic movements are airs above the ground and not regularly practised at morning exercise. Everyone applauds and one of the riders reaches into the pocket in his tailcoat for a piece of sugar to reward the stallion.
Next, we embark on a guided tour of the stables, where the stallions which have not been at the morning exercise are being taken by their grooms to the purpose-built horse-walker. We smile as one whinnies and the call is taken up by all those looking out from their stables; soon the whole yard is echoing to the sound.
The fact each groom leads two stallions speaks to their docility, an essential quality as so many males living together (there are currently 72) could cause havoc. This, besides tradition, accounts for the absence of mares at Vienna.
The elegant stable building dates from the 1560s and was first used by the Hapsburg Emperor Maximillian II for his menagerie, which included elephants and the Andalusian horses he brought back from Spain, which form one of the bloodlines of today's Lipizzaners. Something of this is explained to us when we see that each stable bears two names for each occupant. The first is the name of one of the six sire's lines and the second, written larger, the dam's.
Beyond, in the inner stables, are rows of dark wooden boxes, each with a marble drinking trough and a decorative horse's head above the door. The grooms are working hard around us - the stables are mucked out every hour, as this is easier than cleaning stains off the white coats if the horses lie down and get dirty. Each beast is fed three meals a day consisting of specially energised hay, carrots, linseed, horse nuts and a sort of muesli.
Next we visit the tack room. Each horse has two made-to-measure saddles hand-crafted in Switzerland: a white deerskin one for performances and a dark one for exercise. These are adjusted as the horse develops his muscles. The dressage snaffle bits on each bridle are quite soft; gone are the days of bitting to produce results.
It is in the tack room that we see another rare creature - Hannah Zeitlhofer, the first girl to have been accepted as a student and the first to qualify as an assistant rider. We had observed her slim and elegant riding impeccably at the morning exercise. In 2008 Elisabeth Gurtler, who had taken over as director, broke the all-male tradition and now four girls are undergoing the arduous training. It takes four to six years to reach the level of assistant rider and 25 to 30 years to reach that of chief rider. It is not just the riders who have to prove themselves - the stallions too must pass through a series of strict tests before reaching Vienna.
To learn more about this we make the journey down to the Federal Stud Farm at Piber in Styria, where the mare families are accommodated and the foals are born. The herd came to Piber in the 1920s prior to which some of them were at Lipica in Slovenia, hence the name Lipizzaner. The Piber stud, amid forests and rolling meadows, is open to the public and offers tours plus a museum, cinema, shop and cafe. All buildings are impeccably maintained and in one we delight in seeing the gentle white mares suckling their dusky long-legged foals.
We then enjoy a visit to the upper meadows in the company of Erwin Klissenbauer, the chief of operations. There we see a group of colts and fillies, running free. Having been used to people since birth, they are surprisingly tame and friendly. At one year old they will be divided into mare and stallion herds and at 18 months, Klissenbauer tells us, the first selection will take place.
At this the stud manager, the stable master and the two first riders from Vienna assess movement, intelligence, size, behaviour and temperament. Similar controls will take place until the horse is four, at which time 10 stallions will be selected and taken to the training centre at Heldenberg for breaking-in by the riders - something which takes up to nine months. The education of a stallion is not rushed and in fact continues in Vienna where he will perform until he is well into his 20s. When his performing days are over he will be brought back to Piber to end his life peacefully in the meadows - although some have been observed leaping into an impromptu capriole or courbette as if to relive their glory days.
What, I ask, happens to young mares and the colts which do not make the grade? Well, just as much importance is given to the selection of some of the mares for the "mother herd" - they will bear their first foals at five or six years of age. Some stallions are sold and others gelded and, together with some mares, trained for another of Piber's traditions - carriage driving.
This we witness for ourselves later that afternoon at Piber's annual autumn parade, where not only elegant carriages but also mares and foals are shown off by their grooms and visiting teams demonstrate skilled horsemanship to an enthusiastic public.
Returning to Vienna, one last treat awaits us - enjoyable though the morning exercise had been, I still longed to see a choreographed gala performance - and by chance we learned this had become a possibility. We see horses and riders, wearing their smartest and most opulent outfits, undertake moves, each more breathtaking than the last until they reach the unforgettable culmination of the gala, in which eight perfectly synchronised white stallions dance the complex steps of the finale quadrille. An unforgettable end to the trip of a lifetime.
BA (ba.com) flies to Vienna from Edinburgh and Glasgow via London Heathrow. From Glasgow this costs from £270 return and £271 from Edinburgh. Piber is a 155-mile drive from Vienna or can be reached by train from Vienna to Korflach via Graz.
What to do
All ticket prices and opening times for the Spanish Riding School can be found at www.srs.at/en. For The Federal Stud, Piber, see www.piber.com/en. Patricia Cleveland-Peck travelled with assistance from austria.info.