The primary theme here is of weight (and the limitations of what it can accurately account for in objective reality), although the relationships between pianism and martial arts/tennis may well be of interest to some. In short, I can agree with very little asserted within the article and thus wish to debunk numerous claims within, via simple practical illustrations. I certainly hope that this post won't end up as a tirade of negative vitriol. However, I'm publishing it specifically because of how interesting I consider some of the off-shoots of my disagreement to be, with regard to my own alternative conclusions about how the analogies can best be related to pianism (in a manner that is both simple to understand and consistent with credible science). Also, I have no personal beef with the author and would like to start by making it clear that my intense disagreement with the validity of what he draws from these analogies should not be interpreted as any direct slight upon his capabilities as either a teacher or as a pianist. I've often witnessed teachers (whose verbal or written explanations of technique I would dispute) using practical approaches that are so effective that the direct plausibility (or lack of) in the explanation itself doesn't in any way interfere with what they draw out of students. My disagreement is in no way intended to read as any kind of professional condemnation. Indeed, if he wishes to provide any follow up comments to what I have to say here, in the spirit of two-way dialogue, it would be most warmly welcomed.
However, the following arguments were triggered for two primary reasons. Firstly, there are numerous issues being portrayed as if having some kind of real life grounding in science- most notably the outrageous pseudoscience (nb I don't use such a word lightly, and will substantiate that assertion fully) that is quoted from boxer Jack Dempsey. I have a big problem with ultra-subjective imagery portrayed directly as a reality for two reasons. Firstly, a subjective method does not become any more effective on a practical level, due to having flawed analytical logic tacked on. It if does offer something useful on a practical level (which it may very well, to those who come at it from the right starting point), there will be ways to trigger the same practical effects, without also creating confusion. But above all, if it is not a real truth, even on a practical level it's not only very unlikely to work for everyone, but may well be least effective of all for those who pay the closest attention. Advice should not punish those who follow it the most correctly. A great many of the ideas dealt with here are extremely prevalent beliefs in modern pianism, yet represent just about everything which I consider to have severely held my technique back over a period of many years. Indeed, while I'll go on to give theories as to why scientifically implausible ideas about (supposed) use of weight might well help a boxer or tennis player, I'll also show quite how easily they could hinder some of the most important fundamentals of pianism (unless the pianist has already been lucky enough to have developed them).
Anyway, the first assertion which I take overwhelming issue with is as such:
"Third, body weight rather than finger strength gives a much greater sense of control at the piano – weight-based technique returns more feedback to the body than finger technique alone."
Firstly, we have a false dichotomy. It's not a simplistic choice between one option or the other. As I illustrate in my post on balance, intending to use weight through the finger often leaves it with no choice but to work considerably harder, compared to when the finger itself is proactive about connecting out of free will. As for the claim about sensitivity, I'm simply baffled. Let's introduce an excellent analogy that I heard pianist and teacher David Kuijken use. If you want to scratch your nose, do you do so by moving the whole arm back and forth from your shoulder, or do you move the finger itself? Well obviously the arm carries the finger to your nose. But I certainly don't get as much feedback or sensitivity when I try to scratch by moving the whole arm, while the finger only seeks to maintain "structure" (rather than get on with performing the scratch for itself). By extrapolating further on this idea, it also gives an excellent representation of how narrow it is to offer but two polarised options of "finger technique alone" or weight. If I lie on my back and involve arm weight in the scratching, it's the last thing that could add to either control or sensitivity of feedback about the movement. So, I certainly don't use any weight and neither does it help in either respect. But do I scratch my nose with the finger "alone"?
Well, it depends what you mean. The arm always plays a role of some kind, whether here or at the piano. Just not one of having to either generate the basic movement or apply notable weight. My arm is entirely responsive to the finger movement, but it just doesn't need to take an active role in performing the movement. It merely aligns and supports. Likewise at the piano. If "finger alone" means locking the arm stiffly, it's a bad idea whether you're scratching your nose or playing the piano. But a responsive arm is something else altogether than a weighted one. To imply just two options doesn't bring us even close to the reality. I get the most feedback when my finger strokes the key and the arm is made light enough to respond freely to resulting reactions. It's not finger technique alone. But neither is it anything to do with weight-based technique. If I weigh down notably on every finger, my arm stifles my freedom to perform that finger movement, and is not free to "breathe" with the reactions. With notable weight, things tighten up in the forearm when you try to move the finger with anything resembling a deliberate quality- because the application of weight starts to create a blockage against the reaction, thus working muscles harder. It's not unmanageable for a single note, but it's not half as sensitive as what I experience by stroking the finger and keeping the arm light enough to respond freely.
See the difference in practise at the piano here:
Which looks more sensitive to you? In the first execution, my specific goal is to use "weight-in-motion" to move the keys. So my arm thus sinks down while moving each individual key, before having to lift back up to repeat. Otherwise weight is not "in motion" And as Polishook also claims "Again, no need to use strength – arm weight alone is sufficient". Sufficient to move a piano key, yes. Sufficient to give the finest control over every sound, or to sculpt a longer phrase though? No. Perhaps the author will say I have followed his instructions wrongly in some way. However, I would welcome him to clarify precisely which points I am disobeying, if so. The reason this style of playing doesn't work well for a phrase is specifically because I followed the explicit reference to "weight alone"- and thus ended up producing sound by plopping the weight of my arm down on each key, without moving the finger. Not only was there no word from the author on the need for to do so, but we were also directly told that that the weight would be sufficient. I'm not trying to play Devil's advocate here. This is the genuine mistake I made in the past, as a result of trusting such one-sided descriptions as an accurate description of the process. I'd be glad if he were to clarify whether I have done as he intended, or whether he feels I've gone against something he states. But seeing as we're being told that weight is supposed to be in motion, that's exactly what I am indeed doing in the first example (while I instead keep it balanced in the second, and produce the tone with the most wilful motion of my fingers). It seems clear that the description itself is wrong, rather than my application.
Also, which looks closer to the fluid energy of tai-chi, visually? Bobbing around to set weight into motion, or stroking the finger through the key until it gradually evolves into a "standing" position that is still brimming with life and vitality? Which looks more like smooth walking? Dropping down through individual notes is a good exercise for a very tense student to loosen their arms up, but to literally produce sounds one by one with a series of individual arm motions is but an occasional tool in wider technique. This is absolutely not the fundamental building block of normal playing. Today, there are occasions when I use that type of motion, deliberately. But I generally find far more sensitivity and control when my standard intention is to stroke the keys via generous finger motions, while the arm follows the phrase horizontally and without any intention to weigh down on each key. The arm responds freely to every finger action- meaning that I am neither using "finger technique alone" (the arm is abundantly involved through the phrase) nor employing either gravitational weight nor arm momentum to provide the basic motion of the key.
Anyway, to see whether it is fair to assert that there is more sensitivity through a weight-based technique, try the exercises within this video, away from the piano. Start with the hand in something resembling a playing position and either rest down on it, or drop down from a small height. Then start again with the hand resting down only very gently, fairly flat, and lighten the arm all the further still while the fingers and thumb gently draw in together, to open the arch. Do you experience more sensation in the contact of your fingertips when resting weight down into the hand (or to put it another way- when the fingers are having to resist the weight that you are applying to them)? Or when gently stroking the fingers inwards while the arm is light enough to drift up and away with the reaction? I definitely favour the latter. If I should switch to wishing to rest weight down, after having finished stroking the fingers in, the loss of sensitivity that occurs can be absolutely phenomenal! Just the most gentle connection is enough, when the hand finds a living balance through a gradual path of unrestricted movement. When feeling the soft fur of a cat, would consciously striving to rest the weight of your arm upon the cat offer more quality of sensation than a lighter and more gentle contact?
It's worth noting that if you also should attempt the exercise while starting from significant weight (I mean the prior one to that- leave the poor cat alone please), you'll experience just how stifling weight can be upon the hand's ability to expand safely. Can you spot on which occasions I was moving freely and easily and when I was having to apply a little strength, to perform the movement against the resistance of extra weight? Find out for yourself both what gives you the most feedback and what feels the most effortless, when you play around with the options. Finally, try experimenting with a similar style of recitative passage at the piano, as I demonstrated earlier (I used a passage from Tchaikovsky's Romance op 5, if you want to try the same one).
"So finger strength isn’t the issue at all. Using the resources of the body efficiently – the weight of the whole mechanism all connected together as it is – that’s the issue. Athletes use whole-body weight-based technique all the time. Which is why tennis players talk about getting body weight behind the racquet. Same in baseball with bats or really in any sport where a stick or a bat hits a ball. Weight-based technique is why balls get “hit out of the park” so to speak.(...)
The advantages of weight-based technique are why boxers put their body into a punch."
I barely know where to get started here, as it's not only remarkably one sided but gives us no significant clue as to what the word "weight" is even supposed to mean to the reader. To illustrate this, let's think what weight can add to a punch in the only situation where it's notably literal, firstly. Sit on a bed or soft chair and put your hand in the air. Then relax your arm and allow your hand to fall freely down. Okay, you probably wouldn't want your youngest daughter to have been lying right underneath that, but would you say that you could have knocked out Mike Tyson, had you managed to channel that into a punch? After all, we all remember that scene in Rocky II in which he raised his fist in the air, let it fall on top of Apollo Creed's head and knocked him out cold (or am I misremembering?). Gravity is really not all that quick to generate speed. Drop a ball from a height and see if you can then outrun it with your hand to catch it again. Not exactly an astounding feat, because the earth's g-force is nothing terribly fancy as far as rates of acceleration go. It's especially lame compared to the levels of muscle generated acceleration that boxers put into a genuinely powerful punch. And that's to say nothing of the fact that it's impossible to even send any notable weight into generating the vast hand speed associated with a horizontal punch.
Jack Dempsey's diagram of a sled shows how we can redirect acceleration via weight to work in a different direction, supposedly illustrating that it can also be directed to a punch. Well, Jack, I'm afraid you're no heavy-weight when it comes to intellect. Now, I suppose you could grease up a slide and rest your fist on it. At least I have no alternative idea how else we might redirect energy of a descending fist into a horizontal path- and he doesn't seem to have even tried to conjure up any theory of his own. You could let your relaxed fist slip down it and then see how much speed it generates before contacting a stack of bricks at the bottom, although I'm less than convinced that you'd impress sufficiently within the local dojo to trigger an emergency black belt award ceremony there and then. While I probably wouldn't fancy my chances against Dempsey in a back alley, in a one-on-one bout of bare-knuckle science he'd probably have been lucky to see the end of the first round against an average high schooler, if he thought a punch might in any way be linked to either falling babies or speeding sleds. The fist can neither fall for long enough to gather the kind of speed that either a baby or sled builds up (not even if dropping vertically) nor is there anything to redirect the utterly unimpressive speed into a horizontal path. In fact, to put these issues squarely into perspective, the fastest recorded punch is 44 miles per hour- which would require a fall of some 130 or so feet, for such a speed to be reached via falling weight!!! So what is all this talk of "weight" supposed to mean- given that gravity was so bad at the task of generating a serious punch, even in the ridiculously contrived situations where it could make a true contribution?
Well, some serious clarification is due, if anyone is to make sense of such a colossal discrepancy between the myth and scientific reality of "weight". Here, I'd like to expand on a style of analysis I've previously used at the piano regarding relative speeds- although I'll illustrate it through a practical exercise. Firstly hold your elbow slightly bent and swing your hips to the side, while keeping the arm in a held position. Here we used a turn of the whole body to accelerate the fist. A formidable blow? Hardly. The fist speed was limited to the speed generated by the body, which was very small. Turning the whole body through the strike is almost certainly what people mean when using "weight" in this particular context. The word tends to be used in a colloquial fashion, as a substitute for the more accurate term "momentum". Polishook may have intended this colloquial meaning in much of his article- although it's very confusing if so, as the passages he quotes from Dempsey are about literal weight in relation to gravity. That is an altogether different issue. Literal weight can create acceleration instead of muscles (although we've seen how dubious this was). Colloquial weight, (technically momentum) in boxing, tennis or baseball is generated almost exclusively by muscular acceleration of the body and thus has zero in common with his pianistic idea of using weight "instead of" muscle, for energy. As far as I can gather, he appears to be switching freely between the two different interpretations of the word "weight", as if they were just a single concept, rather than entirely different ones (if not, I'd warmly invite him to clarify what he does actually mean, as it's not at all clear to me).
Anyway, as we've seen, using the body "weight" (or rather, momentum) to generate the fist movement achieves next to nothing on its own. We accelerated the fist to a small speed, but divorced from more important movements it doesn't even register as a real punch. So, this time turn your hips and also throw your shoulder forward around your torso. On top of the speed achieved by the hips, we have added some speed to the fist. The fist is now separating out from the body in what I call "positive movement" and thus reaching a greater speed than that of the body itself. Although we didn't really add much yet. So this time, add the extension of the arm via the elbow. Finally, we have a REAL punch. The arm adds a truly notable level of acceleration to the fist, away from the shoulder. This speed is now being added to speed generated by the body. The most important ingredient in accelerating the fist was by far the arm motion. Clearly it makes more sense to build up a good jab first (a punch that all boxers learn). The body can then be added, to supplement momentum behind what is already a good fist speed. But if the fist itself is not accelerating out and away from the moving body (with plenty of speed of its own) the body has been conclusively demonstrated as incapable of doing the arm's work.
Try applying the same concepts to a tennis forehand. Once again the body produces pitiful results, when merely coupled with a static arm. If the arm itself is swinging freely and easily, you can turn the body through, to achieve some extra racquet head speed, plus considerable extra momentum that supports through the contact. But you cannot possibly accomplish results by pretending that the arm is only there to make a structure that will pass on the power from the hips. No, it's there to perform the stroke and it must do so without fail. The turn of the torso can again add extra power, but the arm must always individuate out from the body to generate the bulk of the racquet head speed. It's not about unifying the body into one piece that merely transmits motion from the start of the mechanism to the very end. It is a matter of parts both receiving speed from the prior link in the chain AND individuating out to generate independent movement. Successive parts need to make their own meaningful contributions to motion, or the whole thing fails badly.
Now, in either of these scenarios, I see no problem from thinking about weight (whether in the colloquial or literal sense). It's abundantly obvious that you won't achieve results from locking up your arm and then turning the hips. It's equally obvious that drooping limp body parts towards the floor is not going to generate a whole lot of horizontal motion. The imagery of gravity doing work can loosen up muscles in readiness for unrestricted freedom of movement, without creating a genuine distraction from key elements. In no way would pretending that gravity is responsible discourage those individuations of body parts, while generating acceleration.
But let's return to pianism for a while here, in an entirely comparable fashion. When a pianist who hasn't first learned to move his fingers properly thinks of using arm weight to set the keys in motion, it's like telling a tennis player who turns his body (without swinging the arm a jot) that he needs to forget his arm and use his torso more. Okay, the tennis situation is ludicrous and unlikely to ever occur, but the pianistic equivalent to such a situation is exceedingly common. Instead of necessarily freeing the mechanism in readiness for essential hand movements (which is what I believe happens under the surface, within success stories), weight ideals at least risk causing huge distraction from them. At worst they may even gradually program an impressionable mind to actively intend to avoid finger movements (after all, remember how strenuous it was to perform notable finger movement against armweight, in the earlier exercise!). I suffered badly from this and I've encountered numerous students who can sink weight, yet have rigid/collapsing hands that have never learned to apply the necessary baseline of movement to a key. Under arm weight, a hand that has inadequate capacity for movement will buckle under the weight into what I call negative movement.
Think back to a decent punch. The aspect that people almost always stress is the role of the body. However, the fist is carrying the greatest speed found anywhere in the mechanism, when it individuates out and away from that turning body. It could not correctly be seen as something that merely transmits the lower speed of the body. It must not only add extra movement on top of that from the body- but a truly significant amount. Likewise in tennis, the fastest speed generated must be passed specifically to the racquet head, when the arm individuates out from the turning motion of the body. None of these examples are about fusing parts into a structure, but about separating them out from each other during the action. In the diagram above, the knuckle ends up travelling faster than the fingertip (due to the finger being compressed from both ends, between the force of arm-weight and the key's resistance). Thus the fastest speed peaks at the knuckle. Not only does the fingertip fail to individuate out with a higher speed, but it doesn't even manage to move as fast as the knuckle (which is gaining on the tip and thus certain to be moving faster than it). The extra energy carried in this speeding knuckle joint merely crashes into the keybed as an aftershock, without reaching either fingertip or key.
This time, the finger is generating at least some motion of its own (just as the arm had to be in the punch/tennis stroke), meaning that the fastest speed is unquestionably channelled specifically into the key itself. The fingertip is now gaining further away from the knuckle- just as the fist was gaining away from the body in a suitable punch. Even if we are intent on playing via individual arm movements, the full potential won't ever see the business end of events unless the hand can make at least some small positive contribution to slice through the key resistance. Oh, and in case you're wondering why this matters (given that pianism isn't all about generating the fastest possible key speed) this post explains why directing your greatest acceleration towards the point of contact has as much to do with precise control over the very softest sounds, as it does to with producing the loudest ones efficiently. Although, for a more a concise example, picture aiming a ball at a near target. It's every bit as important that your hand is accelerating out from your body for a gentle but precise throw, as it is when aiming for raw distance.
Moving on, if we still insisted on using the weight to generate the key motion, how fast could weight generate repetition? Firstly, lets look again at how effective gravity really is as an accelerator. Throw a ball in the air to around eye level and watch as it changes direction. Do you notice how it seemingly "hangs" in the air for a while?
Okay, it's not quite as extreme as Wile E Coyote, in the seconds before he looks down and finally plunges into the gorge, but using weight to produce movement is not any basis for fast reversals of motion. Consider also that a piano key moves but a cm or so. How much speed was generated in the first cm of the drop? Very little at all, even in free fall (and remember that a piano key will also be putting up notable resistance that lowers the rate of acceleration). The falling baby of Dempsey's diagram could scarcely be less relevant, because in even moderately fast pianism we don't have time for long falls, over which to gather acceleration. In slower and particularly in chordal pianism it's indeed possible to gather speed over a bigger drop, as Rubinstein famously took to matters of extreme. But it's objectively inaccurate to attribute the possibility of achieving more than around three (probably a very generous estimate) or so key movements per second to free-falling weight. Even if you pull back very sharply after each drop, we're looking at six individual reversals of direction per second. Take a small object and drop it from a cm or so then pick it up and repeat, as fast as you can. Even if you should choose a bouncy object, it won't be very fast. This should give you a pretty good idea of how relatively slowly weight creates speed, in the early moments of a fall- and of how utterly implausible it is to attribute key movements to weight, during even a moderately brisk scale.
"So, again, no need to use strength – arm weight alone is sufficient. Which doesn’t mean muscles aren’t necessary. Because they are. The role of the muscles is to help the body maintain an efficient structure – so we can use and apply body-weight-in-motion."
To say that the above is a case of the most extraordinarily one-sided wording would be generous. I'm afraid it's not technically even correct, unless we're talking about an extraordinarily narrow style of limited piano technique, that would rule out all manner of possibilities. As we saw in boxing, weight in motion cannot possibly explain the world record punch speed. Given that it wasn't generated by a boxer having plunged from the necessary 130 ft in the air, the likelihood that his muscles created the acceleration is a safe enough bet. Far from being about "maintaining" a static structure, it's all about separating out the parts of the structure into motion. Neither can weight come close to explaining many of the most routine ingredients of pianism. Aside from "structure", muscles have an unequivocally important role of generating the movement that causes keys to go down. I've already illustrated how few repetitions are possible per second, by setting weight in motion. If the author didn't intend the literal interpretation of the term that he himself chose to employ, I could only invite him to consider alternative wording- as I don't know how myself or any other reader might be able to deduce what he really is alluding to. But the statement, in the form he asserts, it is incorrect beyond any reasonable doubt.
It's interesting that the video selection is of Errol Garner- who certainly does depend on plenty of literal use of falling weight to move keys. Now, I take no issue with Garner's phenomenal musical talent, but does his fondness for bouncing the arm represent everything you need in pianism? In case you're second guessing, this is certainly not a jazz v classical issue, mind. I've posted it before, but I'll post it again here:
These kinds of runs likely approach some 12 or more notes per second. The arm is neither falling 12 times per second, nor being picked up ready to fall again on an additional 12 occasions within the very same second. As I said, 6 reversals per second is probably a rather generous estimate for what is possible from weight in motion. 24 per second would be plain ludicrous. Muscles generate motion in his fingers, beyond question (regardless of whether it is the ones in the forearm, or the intrinsic muscles in the palm) while the arm moves at a perfectly leisurely rate. Yes, there are places where he does use dropping gestures- but exclusively in places where he has the time to do so, and where it's conducive to the type of sound he wants. For smoothness and for speed, it's abundantly visible that results are produced by moving the arm laterally, while the fingers produce the motion of the keys (without letting the arm's weight collapse down, other than as the exception). This very simple style of motion is the foundation of piano technique. It works every bit as well for Tatum style licks, legato melodies in Chopin, or passagework in Mozart concertos.
What Tatum exemplifies is how truly simple means can produce the most extraordinary results. The only thing we should regard as "remarkable" about his finger technique is the sheer speed and control. But far from involving any kind of radical motions, he simply mastered elements that any pianist can already be learning in their earliest lessons. Weight in motion can be simple too, in the right context, but there are places where it precludes results, if it's your basic foundation. Arms don't move that fast and Tatum is clear evidence that they have no need to do so. The extraordinary things in Tatum couldn't arise by using falling weight to replace the role of muscle motions. Garner achieves some of his most outstanding results via a baseline of dropping weight and I'll certainly not argue otherwise. I doubt if his sense of timing could easily have been achieved in any other way. However, while his runs may not have been quite as super smooth and light as Tatum's notorious virtuosity, there are plenty of moments in the video where he too uses the slower gliding horizontal arm motion, while the fingers individuate to produce the key motions. No matter how many moments of authentic weight in motion are indeed on display, he certainly wasn't some one trick pony who only knew how to drop weight.
I'll stress again that I sincerely have no ill will towards the author. However, I've "pulled no punches" in these honest responses to his assertions, because I write as one of the many pianists who has previously been duped into assuming that such phrases as "arm weight alone is sufficient" exist in reality, rather than purely in some pianists' extremely vivid imaginations. Such ideas seem so enticing, that's it's hard not to be swayed by a picture of having to use very little effort. However, owing to my deficient ability to apply finger movement, attempting to use weight to achieve the impossible ultimately made me work harder and more strenuously. For all I know, the author may teach far more about cultivation of finger movements, when teaching in person. However, by writing an article without a single word to acknowledge that that fingers and muscles must generate not only "structure" but also movement of keys (just as the arms must generate the majority of the fist/racquet head speed in punches or tennis strokes) he paints a severely flawed picture. Not only is it objectively flawed on an analytical level, but it is potentially misleading to anyone who should take the ideas at face value, without having first learned extremely advanced finger agility.
I'll finish by giving a brief taster of a concept that will likely be fleshed out more in an upcoming core post on the basics of generating sound. Alan Fraser uses Tai-Chi walking as an excellent analogy for the action of fingers in pianistic legato. However, I wonder if a step machine is an even better one still- given that it introduces something analogous to the actual depression of the piano keys? If you've ever used one, consider the style of motion on a medium to higher setting (so the pedals don't put up a huge resistance to being moved). Do we simply stand on them and allow weight to push the pedals down while the leg keeps structure? No. Do we try to lock the torso into a fixed position and shove down with the legs, as if they were a whole separate entity from a rigidly fixed body? No. Like the title of the machine suggests, we are supposed to imagine that we are trying to walk UP some stairs. By trying to push the body up and away over each step, we actually tend to balance the torso into a pretty steady equilibrium. It doesn't collapse notably down on top of each foot and neither is it weight that drives the pedals. The legs move the mechanism via muscular engagement and there's no question that they could be seen as a mere "structure" to pass on weight in motion. Although a conscious effort to weigh the upper body down through the legs only provides greater burden (just the same as such a mindset only burdens when climbing real stairs) there should be a clear feeling of responses in the torso, to the action of the legs. What if we take a similar image of leading with motions of the fingers, that try to push up at responsive knuckles and arms (which end up nicely balanced like the body), when seated at the piano?
EDIT- I initially neglected to deal with most remarkable analogy of all, namely the tomato. Firstly, after all the talk of body weight in motion, it's a curious turnaround, if he suddenly wants us to hold back and keep the weight extremely subtle. However, according to his follow-up comment, it turns out that the intended point was to show how gently weight can be applied. But it's beyond me why we're even told to apply any- regardless of how gently. It take quite some religious devotion to armweight, to spy the problem of pushing through a fragile tomato without bursting it- and then start with the assumption that we should look to rest armweight upon it. Hell, why not go back to his original ideas and rest the body weight in too, if we're going down such a remarkable path? Gentle or not, I really don't recommend that you start from such an irrational approach to the problem as to reach straight for the big guns, in the expectation that they will offer the greatest precision.