by Alan Carruth » Thu Sep 08, 2016 2:11 pm
The 'zip' tone is a longitudinal wave in the string, analogous to a pressure wave in a long thin pipe. The pitch is determined by the effective Young's modulus of the string, the effective density, and the length. So far so simple.
Young's modulus is a measure of how much force it takes to stretch or compress a given size sample of something by a certain amount. The 'effective' Young's modulus of the string is pretty much the Young's modulus of the core material. For steel strings this is more or less invariant with tension: the zip tones of the E and B strings on most steel sets are pretty much the same. Nylon is more complicated. Because of it's chemical structure the Young's modulus goes up as you put on more tension. The change is relative to the ultimate breaking tension of the string. The high E string tends to be under a bit more tension than the G in the D'Addario sets that I have a listing for, so all else equal you'd expect the high E to be closer to it's breaking stress, and have a somewhat higher Young's modulus. Of course, the G is thicker, so it can carry more tension than the E, and that puts it even further from breaking, so the Young's modulus is even lower. Thus the G has a lower zip tone than the E on the classical sets I've checked, and the B is in between. I don't remember the exact numbers, but the table is in the article. That's the plain strings. With wound strings you can more or less ignore the wrap in terms of Young's modulus, but otherwise the relationship between Young's modulus and tension will be similar.
For plain strings the effective density is just the density of the nylon. Wrapped strings are more complicated. The wrap is there to add mass without adding stiffness, so that you get a lower note at the given tension. It also drops the pitch of the zip tone, since the wrap has to move along with the core to at least some extent. In fact, I imagine there's no appreciable 'slippage', but I don't have any data on that.
Since the Young's modulus is only slightly related to the tension on the string, there is no set relationship between the tuning of the string and the zip tone. As it works out, for satisfactory strings the zip tone is usually somewhere between the 7th and 8th partials of the transverse fundamental of the string; the note the tuner says you've got. Thus the zip tone is generally more or less dissonant.
All else equal the longer the scale length you use the lower the pitch of the zip tone. You don't have much choice about the zip tone of the plain strings once you've selected a scale length and string.
Wound strings give you some leeway because you can change the effective density of the winding by twisting or untwisting the string. It's not much but it can be useful.
The reason it's useful is that the zip tone is driven in the vibrating string by the transverse motion and the resulting tension change in the string. It's complicated, but basically when you pluck a string somewhere near one end you actually pull the string a little bit in that direction. That is, when you pluck near the bridge it's as if you grabbed the string and stretched it toward the bridge a little. The closer you play to the bridge the greater the effect. The only way to avoid driving the zip tone is to pluck exactly in the middle of the vibrating length; over the 12th fret for an open string. Nobody does that because it sounds lousy, among other things.
What happens with the zip tone once you've plucked the string gets complicated too. Remember, the zip tone is at a much higher frequency than the fundamental of the string. You can envision it getting a little 'kick' from the string vibration twice for every cycle of the fundamental; once toward the bridge and once toward the nut. If the zip tone pitch is an exact multiple of the frequency you tuned the string to it can build up in amplitude. The more exact the frequency match the greater the effect; the more the amplitude of the zip tone builds. Classical D strings seem to be made this way, for some reason, so they're a problem.
Again, it's complicated, but basically when the zip pitch amplitude is high the two ends of the string are under slightly different tensions at that matching partial. The string wants to make two different pitches near that frequency, and they tend to beat. What you get for sound depends on a whole raft of different variables.
The zip pitch is acting through string tension, and pulling on the bridge top. The taller the bridge the more leverage the string has to rock it, so the guitar produces more of that zip sound, including any problems. So there's one variable for reducing the zip tone. Another, as we've seen, is altering the density on wound strings by tightening or loosening the wrap. That only works for the wound strings, of course. With plain strings your choices are more limited. You can try some different strings to see which ones might work. I suspect that 'carbon' strings might have lower zip pitches, since they're made of a denser material, if I understand it correctly. Otherwise about all you can do is alter the scale length.
One bright spot: since the zip toner is only given a kick every once in a while it ha to match up pretty closely in order to build up enough to be a problem. That means the pitch match has to be really exact. Anything you can do to add some damping to the string will help kill the zip tone too. That may be why playing right on the fret kills it; your finger damps it out.