## Saturday, 24 August 2013

### 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.