Bending with Dojo part 3: Slight return

This is the last of three posts about building the string bend calculator and maybe the most interesting to anyone other than me. Part 1 dealt with the basic physics and Part 2 with building the Javascript app. All that's left is to wonder what it all means.



The first thing to say is that when you bend a note the string does stretch and the stretch accounts the distance you have to push the string at the fret. How can I be sure? Looking at the numbers that first come up (for the high E string), the total tension is about 58N (Newtons, about the weight of 6 kilos or 13 pounds).
Since the length does not change very much we have to increase the tension to increase the note. A full tone (2 semitone) bend needs the tension to go up by \(2^{(2/12)} \approx 1.12 \) times, to about 65.1N. Increasing the tension that much causes the string to stretch by a small amount, 1.2mm, and over the scale length of the guitar that small amount allows you to pull it sideways in the middle by a larger amount, 19.7mm (Pythagoras theorem).

Constant stretch

Try scrolling the string gauge up and down. String forces change, but watch the stretch distance: it stays the same. This isn't an error, if you look back at "Stretch it" in part 1:
\[\frac{T}{A}=E\frac{s}{L} \\
(2fl)^2 = \frac{T}{\rho} \]
But \(\rho=Ad\), so A, the part that depends on gauge, could be taken out, and whatever gauge the strings are they will be stretched by the same distance to get up to tuned pitch, and by the same extra distance to get to the bent note. (But this goes wrong for wound strings.)
If you're protesting this is clearly wrong because thicker strings are harder to bend, then of course they are, but it's because the tension they're under is higher, not because you bend them further.

Forcing

The sideways bend force has to balance the sideways component of the string tension. The further you bend the string the more you are pulling "head-on" against the string tension (actually, against the tension at both ends), the standing tension of the strings contributes most of this force. This is why drop-tuning makes so much difference to string bending.

Strings past headstock

The string distance past headstock increases the bend force needed. This is because there's more string stretching, and the slack is pulled between the nut and bridge. You have to bend further and this makes the angle a bit sharper. Since the sideways force you apply balances the sideways force of the string bending further means

Things that are wrong

Like any model things have been simplified a bit, so what's wrong here?

1. Tremolos. If you have a floating tremolo then it provides a bit of give when you apply the extra tension too, so expect more stretch, however the trem springs are quite stiff by comparison to the full length guitar string. The effect is like the 'extra length' string past the headstock. A 'screwed down' tremolo takes some force before it lifts, and until it does start to move it will act like a hard tail (for stretch purposes anyway).
2. String past headstock. We have completely ignored friction at the nut, which will reduce the effect of the extra stretch a little.
3. Density and wound strings. Wound strings are a bit lower density than solid strings, but also the windings don't contribute anything to the string stiffness. Only the core counts for stiffness. It would be possible to have separate gauges for windings and core, instead I've used a modified 'density' number that gives the right tensions for wound strings.

Actually this gives a good insight into why there are wound strings. You want to have the tension about the same in all strings (balanced across the neck), but you also want them to have similar degrees of stretch. This is partly for feel, but mostly because strings stretched to about half their elastic limit will be at their most linear (well-behaved...). We've seen changing gauge has no effect on the actual string stretch needed to get to pitch, but only because the density and stiffness both depend on it. By using a wound core you can break that rule and tweak density and stiffness independently.