I think with a discussion of any sort of martingale you have to ask what you are looking for in terms of reaction – and what you are using it for, what you are DOING with the horse.
A standing martingale provides a limit to the amount of height a horse can raise his head above the wither. That is all. It is attached to the noseband. It limits the plane of of the head. As such, it keeps the mouth within contact of the reins especially if the horse is pulling, rooting, or pushing the nose forward. The horse feels the restriction and technically should react by releasing the pressure on the nose and dropping the face back, which puts his jaw and mouth, including the bars, back toward the rider’s hands, presumably soft and in contact. Some horses learn to feel that pressure and lean into it and develop a muscle system in the neck, wither and back to actually be able to jump restricted, but fairly inverted. The wonderful four foot show ring hunters with the martingales are not this way. The martingale really is for show, I believe, as these horses are properly muscled and jump in a round manner, which means the muscle development is correct and the martingale is not used to brace upon by the horse. A standing martingale is a reminder to a horse not to get above or beyond the aids, and can be used with a young horse that gets hopping, jumping forward, raising the head, or throwing the head to avoid the bit’s action. It’s useful to protect a rider that is riding forward, i.e., with the head and hands forward over the wither. It’s insurance – meant to be something that limits a horse from raising its head above the influence of the bit. It is used at all gaits and at many stages of a horse’s development.
The running martingale is used as an adjunct pressure piece that ADDS to the power of the bit, and reacts primarily on the BARS of the mouth when the reins are straightened and pressure is applied. It is important to note that a running martingale should never be adjusted where the reins are dipped downward in a V when light contact is made, because this makes the bit contact the bars in an unrelenting fashion, and actually can make a horse raise the head to avoid that pressure. Kind of counter productive. Instead, a properly adjusted running only comes into play when a horse runs his mouth out against the bit, fights any half halt, leans, or takes a hold; all a good rider has to do is raise the hands just slightly and the rings apply some leverage. A running is really meant to be used while galloping or jumping, when the horse HAS to raise its head as it strides forward, with the hips coming underneath and the back undulating, as the head and neck pump up and down to increase the power of the gallop. When you think about this, you see that it’s a specialty item of tack used in horses that “power gallop” – jumpers, perhaps steeplechasers at times, for sure foxhunters and eventers mostly – rather than in the showring hunter or a flatter strided sport horse.
In addition I use a running martingale to keep my reins on the correct side of the neck, and away from the horse’s front feet should I fall or get separated in some way. I had a horse get the reins over the head, step on them and break them and learned if the running martingale were on, even if loosely adjusted where it wouldn’t be a factor if the reins were in contact, it would keep them safely away from the front feet. I almost always hunt and go cross country in a running on everything for that reason – I like my reins and am a bit picky about them because of my arthritis, so the running martingale prevents breakage and saves the horse from a bad cut or bruise on the mouth should they step on a rein.
Can you tell that a horse’s conformation and way of going determines a lot of this stuff? There is a reason these items of tack were historically developed, the horsemen of yore had a darn good observation technique.
Sorry for the book! I love this stuff, can’t you tell. 
