G# geometry...from a different angle

Before you waste valuable minutes of your life reading this article I feel I should point out that its intended audience is likely to be somewhat (shall we say) limited. Rather than being one of my usual 'Have a go and do it yourself' articles, it's more of a practical examination of the most efficient design and setup of the saxophone's G# key, along with a bunch of vague theory.
Such things tend to be the domain of those who worry about the molecular structure of the oil they use on keys, and the provenance of the cork they use (third tree on the left, just above the rock shaped like teapot).
You have been warned.

Every once in a while someone throws a question your way that causes you to stop and think about your answer, in spite of being reasonably sure you know what you're talking about.
In this case the question related to the adjustment of the G# mechanism on a sax. As you probably know, the G# key comprises two separate keys that act in tandem. There's the lever key, which is that part that you press down to play a G# - and there's the cup key, which is the part that opens and closes over the G# tonehole.
The lever key is usually a pretty simple design; there's the bit that you press (the touchpiece) and there's the bit that connects to the key cup (the lever arm). The key cup is slightly less simple in that you have the cup on its arm and another arm (the adjuster arm) which connects to the lever arm. On very old horns this was all you got - just two arms, positioned one above the other - but on modern horns the adjuster arm (as its name suggests) it fitted with a moveable stub, and it's on this that the lever arm sits.
There are, of course, variations on this setup. On some horns the adjuster is fixed in position and can't be moved - and if memory serves I believe there are some setups where the adjuster is housed in the lever arm.
Some horns have a completely different arrangement altogether, such as that found on the Conn 6M/10M horns.
For this article we're going to stick with the standard modern arrangement because, as you'll soon see, it's complicated enough without having to take all the variations into account.

The question arose after someone watched a repair video in which it was suggested that it's important that the lever and adjuster arms are set parallel to each other - and the viewer noted that on his sax there was no way these arms would ever be parallel...and so I was asked whether or not this mattered.
My immediate thought was "Well no...because these arms are rarely parallel".
Here's a very simple graphic that shows a parallel setup. You've got the G# key cup in the background, the adjuster arm and stub in the foreground, with the lever arm resting atop it.
But if you have a look at your horn you'll most likely see that it has a completely different arrangement. The adjuster arms may well be more or less in this position, but it's quite likely that the lever arm will be steeply angled. You may also have an angled adjuster arm too.
And you're probably thinking "Well no, there's no way these two arms will ever be parallel" - and you'd be right...and also wrong.

You'd be right if these keys never moved. If they were built parallel and they never move, they'll always be parallel - but they do move...and when they do the relationship they have with each other changes. They might start off parallel, but as soon as you press the G# key down, that's it...they're at an angle. Or they might start off at an angle and end up parallel once the G# key is fully down. But most likely of all (as we'll see) is that they neither start off or end up parallel, but pass though that state at some point in their operation.

Having said as much in my answer to the question, I was then asked what advantages or otherwise there'd be to the various possible setups - and this is what prompted me to build this magnificent device.
This is the Steve-O-Matic Key Combobulator (AKA the Keyomatic) - and as you can see, it's a precision piece of kit, finely crafted from the choicest materials and bristling with space-age technology.
OK, OK, it's a bit of a dog's dinner - but it actually works rather well.
The wooden piece represents the G# lever key - the metal bar the adjuster arm, on which a bearing rests...which is fitted to a plastic stub and held in place on the bar with a small but powerful magnet.
By moving the adjuster back and forth on the bar it's possible to observe the relationship between the two arms.
A couple of marks were made on the board - one denoting the point at which the touchpiece arm touched the body (and thus could not be depressed any further) and another denoting the maximum opening height of the adjuster arm (it's currently hidden behind the lever arm).
In this way it was possible to move the adjuster and note how its position affected the stopping point of each arm. In certain positions the adjuster arm reach full height before the lever arm had been fully depressed, and in other positions the adjuster arm never reached full height before the lever arm had travelled its full distance. This is all pretty basic stuff (to a repairer), and of itself not particularly interesting.

But the use of a bearing made it possible to see the effect of friction between the two arms...and this is where (for repairers, at least) it gets interesting.

I originally took a whole sequence of photos using the 'Keyomatic', but unless you view them at full size they're not especially clear...so I grafted some graphics over them. What follows is a precise representation of what was photographed.

Here's the basic setup at rest with all the various parts labelled.
Note the red and green arcs representing the circles of motion; the red centred on the lever key pivot and the green on the adjuster arm pivot. Note too that while the circle of travel of the adjuster arm will not change (unless you move the adjuster), that of the lever arm will. As the lever arm moves up and down, its distance from the lever arm's pivot will change. This will become important...

For the first demonstration the adjuster was set to the mid point between the pivot point as looking directly down on the pillars, and I've marked the centre point of the adjuster with a vertical line.

The G# key has been depressed about halfway, the touchpiece stop is approaching the body and both the lever and adjuster arms have raised - and the G# pad is half open (I've removed the key cup etc. from the graphic for clarity).

The first thing to note is the centre line on the adjuster - it's rotated clockwise to a position just shy of two o'clock.
The adjuster is not usually able to rotate (though this has been a feature on some horns), and so this movement of the adjuster would be translated into friction as the adjuster moved up the lever arm towards the lever arm's pivot.
The second thing to note is how the circles of movement have changed - or one of them anyway. The adjuster arm circle remains the same, but because the adjuster has moved relative to the lever arm's pivot, the circles of movement are barely intersecting (as they were in the previous graphic).
The mechanism has entered the 'sweet spot', and for a time there'll be little or no movement of the adjuster...and thus little or no friction. Remember the point about parallel arms? Well, this is it - this is around the point at which the key arms are mathematically parallel, and the mechanism is working with the least amount of friction.

The G# key is now almost fully depressed and the G# key cup is now fully open - with the top of the key cup just contacting the closing adjuster that sits above it. As is good practice, there's just a hint more travel available on the G# touchpiece - and pressing it down that last millimetre or so will raise the lever arm off the key cup adjuster. This extra travel ensures that the bell keys (which are linked to the G# touchpiece) are fully able to close.

The adjuster has moved very slightly from its last position, and taking the total movement of the G# touchpiece into account you could say that for close on half its throw, there was little to no friction on the adjuster. Conversely you could say that for just over half its throw there was a fair amount of friction on the adjuster.

But something very curious is about to occur. If you fiddle around with the Keyomatic (a popular pastime in these parts, and one they can't arrest you for...yet) and take it beyond the G# key's possible travel, the circles of motion continue their divergence - and look what happens.
The centre line on the adjuster has moved back towards vertical, which means the adjuster is now moving away from the lever arm pivot.
On paper, admittedly, it looks like a wholly irrelevant point...you simply couldn't operate the mechanism in this manner - but what is relevant is that this reversal of direction occurs earlier the nearer the adjuster is to the lever arm's pivot point.
I started off with the adjuster at the mid point (looking downwards) between the lever arm and adjuster arm pillars - but by moving the adjuster just another few millimetres closer to the lever arm, it was possible to observe the reversal effect occurring before the G# key had reached its full length of travel. What this means is that there'll be friction on the adjuster as the lever arm rises, a short period of no friction followed by another period of friction...but in the opposite direction.
Naturally, this isn't of much concern when the lever arm is being raised...but the whole process reverses as the lever arm comes down, and that's really when you want the very least amount of friction.

So what's the answer?
Well, what will help is moving the adjuster nearer the adjuster arm pivot.
In this position the G# key cup is fully open, and you'll note that the G# touchpiece still has some way to go before the stop hits the body.
This points up the inevitable compromises when trying to adjust the mech for best performance. What you get with this setup is virtually no movement of the adjuster. It enters a state of equilibrium soon after it rises, and remains there for the rest of its travel. It's a big win, but it comes at the cost of far more motion on the G# key cup as compared to the touchpiece - which means that when the G# key is barely halfway down, the G# cup is fully open...and any more movement on the G# touchpiece is just wasted. And the leverages are less than ideal - because the closer the adjuster is to the lever's pivot point, the more closing force the lever arm can exert on the adjuster arm. You might be able to dial some of this out by tweaking the tension of the lever arm spring - but dealing with the travel on the touchpiece is a trickier business.

So what does it all mean?
Not a lot, really. That's to say that it used to mean a lot - back in the days when horns were hewn from solid brass, and players had a grip that could bend steel - but someone, somewhere, built a keyomatic and discovered that there's this 'sweet spot'...and that there are compromises to be evened out. And I say 'evened out' because you can't eliminate them completely without some major design headaches (not to mention the expense and impracticability).

And so you're more than likely to encounter the design on the right.
The adjuster is moveable within a small area (the sweet spot area), and the geometry of the key has been arranged in such a fashion that despite the adjuster arm being at an angle to the key cup arm, the relationship between it and the lever arm is more or less parallel.

Should you be concerned? No, not really - not unless your day-to-day business is that of tweaking horns to get the very best out of them...and even then you have to be aware of the rule of diminishing returns. Striving for absolute perfection in just one area of a mechanism doesn't mean that the rest of it will follow suit - and you might find that you get better real-world results by accepting a degree of friction (which can be somewhat negated by the use of appropriate buffers) if the overall feel of the mech is improved.
As is so often the case with instruments - which are relatively crudely built in engineering terms - first you set them up 'by the book'...and then you throw the book away and rely on a combination of feel and experience.