In general, I use the regular right hand low putting grip which works great for longer putts (> 6 feet). But, when it comes to shorter putts with this style of grip I have tendency to go past the hole. The reason is that I am not able to keep the smooth tempo going for shorter backswing for these putts. The left hand low grip works wonders in this situation for me but I leave the putts short when I try longer putts with this grip. Does it make sense to use 2 different putting grips for different putting lengths? Can you comment?
Your question presents a very welcome opportunity to discuss the relationship between grip styles and the shoulder stroke. I think this may clarify your observations and suggest a fundamental fix for this problem.
In a shoulder stroke, the power of the stroke does not involve the hands or wrists or arms, or indeed the shoulder / upper back / pectoral muscles. If it did, the arm pits would open and close during the stroke. In a true pendulum-shoulder stroke, the arm pits remain closed throughout the stroke -- not tightly held shut, just inactive.
You can opt to move the shoulderframe back and thru with muscles in the gut and lower back (tugging in a spiral against the hips, thus rotating the whole upper torso as a unit), or to place the putterhead back to the top of the backstroke and then let the arms and shoulders rock down in a gravity-powered free-fall. Whichever pleases you -- I believe the gravity free-fall pattern is smoother, more consistent, and more accurate. But in EITHER case, the hands have nothing whatsoever to do, except possibly to mess up the stroke by twisting the face or guiding the stroke out of a straight path thru impact.
I personally have never had any faith or belief in the reverse overlap grip, which puts the right hand low and gives emphasis to the right hand for power and some nebulous form of "touch and feel," while the overlapping left index finger supposedly stabilizes the left wrist to guard against left-wrist breakdown or flipping thru. But I do subscribe to the idea that the hands should "oppose" one another on the grip. So, I devised a grip that has the palms opposed and that does not result in one hand especially lower than the other. When one hand is lower than the other, the shoulders start out tilted out of horizontal. I prefer to set up square, with shoulder sockets parallel left of the startline and shoulders about the same height. This grip, which I call, the "pistoleros" grip, is formed by making two "pistols" with each hand, opening the last three fingers on the right, then placing both hands onto the grip, with the right's last three fingers wrapping over those of the left in the crevices between fingers. It's a simple grip, since in my book the grip has little function.
For me, the function of the grip is to monitor the handle to make sure that my takeaway, transition, downstroke, impact, and thrustroke are performed with a smooth tempo without jerkiness, abruptness, or snatching. If there is any of this sort of action, the handle twists or flip-flops inside my grip and I know it immediately. In the shoulder stroke, the grip is there to keep the butt of the handle aimed at the pivot of my stroke in the area of the clavicle, monitoring the motion of the putter back and thru. Because I use a free-fall gravity stroke by which the "triangle" falls from the top of the backstroke by gravity, the handle has a natural and smooth acceleration downward that peaks right at the bottom of the stroke. There is nothing I add to the speed of the fall by muscle action in the hands or arms or shoulders, and my only thought is for my hands to "ride" the handle as it falls so that I "feel" absolutely nothing in my hands.
When I teach this, experienced golfers simply don't believe this way can possibly get a ball all the way to any hole, however close or far. It just doesn't feel powerful. But novices don't have this problem, for the simple reason that a shoulder stroke moving the triangle into and thru impact is far more powerful than a handsy stroke. Just ask anyone who hits their sand wedge 120 yards! But really, what matters for power is the putterhead's speed at impact -- not how that speed got generated. In the gravity free-fall shoulder stroke, the speed pattern is controlled by gravity, which is extremely consistent. The actual putterhead speed at impact always depends solely upon the length of the fall, or the length of the backstroke. So putterhead speed corresponds and varies very exactly with stroke length. That's quite a plus. On most greens, a twenty-foot putt is sent off with a backstroke that is no more than 1.5 to 2 feet long. So power is not really much of an issue.
Now, back to your question: the way you describe it, you use the right hand to power long putts and then back off with a left-hand low grip for short putts. What I would suggest is that you try a true shoulder stroke on long putts with the left-hand low grip. By this, I mean that thru impact and beyond you must power the stroke with an up-rocking of your shoulderframe, not with your arms or hands. When you power a stroke with the shoulders rocking, especially the thrustroke part with the lead shoulder moving vertically up thru impact, you should be frightened by how solidly and powerfully you can propel a ball across and off any green on earth!
So the problem that you describe -- a left-hand low grip on long putts making your putts too short -- is not really a grip problem but the manner by which the stroke is powered. Use the shoulder stroke instead of arms or hands and that particular problem disappears.
Now, for the second aspect: level shoulders or one shoulder high and one low. With a left-hand low grip, the left shoulder is lower than the right. This tends to deloft the putter. With the reverse overlap grip, right hand low, the right shoulder is lower and the left shoulder higher. This tends to add loft to the putter. With the "pistoleros" grip, the shoulders are pretty even. In all three cases, your shoulder setup will influence ball position forward or backward in your stance, and the precise biomechanics of a straight stroke. I simply find the level shoulders preferable and more conducive to consistency and accuracy.
The interesting point is that a left-hand low grip works better with the sort of shoulder stroke I have been describing than the reverse overlap. In the left-hand low grip, the left arm and wrist remain pretty stable throughout the stroke and there is no powering with the left hand, whereas in the reverse overlap grip the right hand is sort of given its "head" to control the thrustroke with powering thru with so-called "touch and feel." The powering thru of the right hand expresses itself sometimes as left-wrist breakdown, but not always. Sometimes the left wrist stays stable and the right hand sends the extra "umph" up thru the left elbow. None of this is taking place with the left-hand low grip. The point of all this is that a left-hand low grip is a lot closer to a "dead-hands" shoulder stroke than a reverse overlap grip with right-hand power.
Consequently, if you go for a true shoulder stroke with dead hands, it will be good on short putts (like the left-hand low but without the shoulders being unlevel) and also good on long putts (plenty of power but smooth and straight just like left-hand low).
If all this bewilders you or doesn't seem to answer your question, then the precise answer is no, there's nothing wrong with using two different grips -- left-hand low for short, and reverse overlap for long. But it's better to use dead hands and a level grip.
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I take up the putter first with the left hand (I'm right handed), laying the side of the handle down the lifeline of my palm so that it runs naturally down alongside my extended left index finger. Then I close the left hand by wrapping the last three fingers beneath the handle in a certain way. In particular, I make sure the meaty part of my thumb pad closes down to shut the handle against the meaty part at the base of my three fingers. This action flattens my left thumb at the base of the thumb and it lays pretty straight down the top of the grip handle, which is also a flat surface. I then "lock" this thumb pad down onto the flat handle surface by cocking my wrist downward (like a fly fisherman would in casting). This levers the higher part of the handle harder against my thumb pad, and lossens the last finger or so. I then retighten the last fingers to press the meaty pad at the base of the fingers back against the handle at this new angle with the wrist set.
This is the key hand and grip that needs to stay intact with constant pressure during the stroke. It has to be set onto the grip down low enough so that your arms and hands can hang freely without sticking the putterhead into the grass. The sole of the putter ought to rest lightly on top of the grass blades without pressing then down very much.
Then I place my right hand onto the handle. I do this by pointing my index finger like a pistol, opening the last three fingers, and moving the whole sideways against the left hand's last three fingers. I have the right hand's last three fingers now closed over top of the left three, knuckles on top of knuckles. Then I slip the right hand down a bit to move all three right-hand fingers into the valleys between the left-hand fingers. The right thumb pad rides partially on top of the big knuckle on the left thumb and the right's next thumb knuckle before the nail folds on top of the left's thumb knuckle. The right thumb does not completely cover the left thumb, but just abuts its mostly from the side, and only partly is on top. The right thumb is then extended past the left thumb only about one fingernail's length, and the same is true of the right index finger compared to the left index finger.
The index fingers do not exactly have the fingerprints at the tips pressed flush against the side of the grip handle (or metal in my case). I have found that doing this encourages finger muscles and hand activity during impact, usually causing a pull action thru impact. Instead, I make sure the index fingers are "resting" down the shaft and the tips sort of hang below the line of the shaft a touch. The feeling emphasizes the insides of the two palms, especially the left, and not the "pointing" of the index fingers. They are really just out of the way. You could do the same grip with the index fingers also wrapping beneath the handle - I just don't like that feeling, and feel the inside of the hands better with the index fingers out of the way, "parked" down the shaft. I suppose the extension of the index fingers is paralleled psychically by the extension of the left thumb down the top of the handle, and this parallelism probably reinforces control and straightness in the stroke.
You will notice that I had something of a job to press the thumb pad and base of fingers to enclose the handle in the left hand. Now that my grip form is set, I have to back off that tension without losing the form. So in my routine I have incorporated this step, as follows: once the putterhead is placed and the grip assumed, I relax and "soften" my grip without allowing the form to alter (other than the "puffiness" that comes with relaxing the hand muscles). I then use a Don Pooley trick of "seeking" the correct grip pressure for the putt I am facing. Ordinarily, the grip pressure will be pretty light (maybe a 2 on a scale of 1-10), but sometimes you need a little more (maybe 3 or higher). The way Pooley thinks about it is to relate the grip pressure to the green speed, but I don't really think that's the best idea. I relate the grip pressure to the stroke I am planning on making, depending on my setup posture and the length of the putt and the speed of the green. If the ball is above your feet, you will have a slightly different forearm muscle tone than you would if the surface at the ball is level or the ball is below your feet, and this forearm tone directly influences grip tone or pressure. At any rate, try relaxing and then finding your ideal pressure for the specific putt. This is not the same as "milking" the grip, but sort of similar if you "milk" the grip by progressively settling down to a final grip pressure (which is an ok trick, also).