by Alan Carruth » Tue Jul 02, 2019 2:39 pm
"Would 6 degrees really be enough downward force to achieve enough volume? "
Actually, as far as I can tell, break angle has no effect on volume, once you get to 'enough'. And you'll know it if you don't have enough. I've done a fair amount of work on this, and talked about it at some length on line, but I'll try to give a decent short summary.
First, the actual forces the string exerts on the top of the saddle are well defined, and depend only on the displacement angle of the string, the tension, and some of the characteristics of the string itself. The main force driving the top is the 'transverse' force: if the string is moving 'up and down' relative to the plane of the top it's pulling the top along with it. If the string is moving 'side to side' it's also pulling the top that way, but, for the most part, a guitar top can't move in that direction. There is also a twice-per-cycle tension change signal, but that's not a factor on archtops, and, besides, is much weaker than the transverse force for most strings (that's one place where the characteristics of the string come in).
So, suppose you pluck by pushing 'down' on the string somewhere near the bridge and then releasing it. The string bounces 'up' as soon as you let it go, but it can't make an 'upward' angle greater than the 'downward' angle it started out with. The closer to the bridge you pluck the larger the angle you'll get from a given displacement, but it will take a larger force to get a given displacement. If you push the string down too far it hits the fretboard, so that sets another limit. As it turns out, it's pretty hard to get an angle anywhere near six degrees, so the string will probably never make an 'up' angle greater than that at the saddle. This means that so long as the break angle exceeds that the string will always stay in contact with the saddle throughout it's whole vibration cycle. So long as it does that all of the force will be transmitted to the top of the saddle, and thence to the soundboard.
What would happen if the pluck angle did exceed the break angle? The string would hop off the top of the saddle, probably briefly, once per cycle. For the time when it was out of contact the vibrating length of the string would be the distance from the nut to the tailpiece, rather than the nut to the saddle. This would introduce all sorts of 'interesting' new frequencies into the sound, and you'd know it.
Of course, you never only push the string straight down; there's always going to be some sideways motion too. This can cause the string to roll across the saddle top, which also introduces some spurious frequencies into the sound. A shallow string nocth usually keeps thing in line.
I performed a rather lengthy experiment some years ago on this. I used a mechanical plucker to drive the strings of a Classical guitar 'vertically' at a known point with a known amplitude, and recorded the sound. I 'plucked' each open string six times so I could make sure they were all the same. This was done in my 'semi-anechoic closet', and the sounds were recorded in the same way for each pluck on my computer.
The variables in the experiment were the break angle and height of the saddle off the top. I was able to tie the strings differently to get two different break angles with the 'standard' saddle height of 11mm off the top. One tie gave a 'normal' break of ~25 degrees, while the other gave a break of only ~6 degrees. Then a taller saddle was put in that gave a ~25 degree break with the low-angle tie, but a string height off the top of ~18mm (don't try this at home!). Thus I had two setups with the same string height and different break angles, and two with the same break angle and different heights.
The recorded sounds of the strings were analysed. Changing the break angle without altering the string height off the top made no difference in either the maximum amplitude or the duration of the sound, nor was there any measurable difference in the overtone series of any string. When the height off the top was increased the power was the same, as near as I could tell, but the overtone series changed.
On an arch top guitar the twice-per-cycle tension change does not seem to drive the top usefully, since the tension is actually taken up by the tailpiece. On a flat top or classical guitar the tension change tends to rock the top of the bridge toward the nut, and produce some sound. The force is not large, and it's not driving the top in an efficient way to produce sound, but it's there. The taller saddle fed a bit of energy into the top at even multiples of the fundamental of the string, so there was a bit more energy in the second and fourth partials of the string in particular when the strings were higher off the top. This would not happen on an arch top.
I also ran listening tests, using a 'synthetic strum' for each case. Again, nobody could reliably tell the difference when the string height off the top remained the same, but the break angle was changed. When the string height changed almost everybody could hear the difference.
The bottom line, as far as I can tell from the experiments that I've done, is that break angle is the only thing that matters on an archtop. You need to have 'enough', and six degrees or a bit more should do it so long as the strings don't skate off across the bridge top as they're being played. Too much break angle and down bearing force does seem to be a problem. I can't say what 'too much' is in any general sense; it probably varies depending on the way you made the guitar. The 'pop up' model of the string tension pushing up and balancing the down bearing is just one plausible way of looking at it, and almost certainly leaves a lot to be desired, if it's even remotely 'right'. So long as you're within the range between 'enough' and 'too much' I don't think the exact angle matters. That's a reasonably broad range in my experience.