Navy Electricity and Electronics Training Series (NEETS)Follow the link, select Module 2 (Introduction to Alternating Current and Transformers), read Chapter 1. Learn about how any current in any conductor creates a magnetic field (see Module 1 for info on how that applies to direct current); learn about how any magnetic field moving through a conductor induces a current in the conductor; learn how a coil can focus the field for a conductor carrying alternating current.
I read the initial comments in this thread, but not all. I got lost in comments and guesses (no offense is intended here).
Hum can get radiated by heater wiring in the form of the field set up around the conductor, and it can be picked up by nearby conductors in accordance with the principles outlined in NEETS Module 2. But a loop of wiring carrying a current that's folded in half (like the 2 legs of the heater wiring) has essentially equal & opposing currents (it's the same 1 current in 2 places, opposite because one leg is the return path for the other leg). If those wires were placed in close proximity, the fields might cancel and eliminate the risk of hum radiation into nearby wiring.
Twisting the wire places the centers of each wire as close together as possible, which makes the resulting field around the wires as small as possible.
Some wiring (like telephone twisted pair) is attached to a differential circuit; what matters is the difference in signals between the two conductors. External noise fields cutting across the conductors could induce an interference signal into the wires unless the
exact same signal is induced in the differential pair of wires. There must be some difference in the induced signals unless the centers of the 2 wires occupy the exact same place in space (impossible), so the wires are twisted to bring their centers as close together as possible.
Twisted pair in many signal-carrying applications is about common-mode noise rejection where the intended signal is differential.
Sometimes you have to kill interference, but differential (or balanced) wiring doesn't apply to your circuit;
co-axial wires were developed to address this need, as it places the centers of the 2 conductors (hot/ground) in the
exact same spot (the center of the center hot conductor) because the outer shield is on the same axis as the center conductor.
Do you have to twist heaters to kill hum? Maybe not, but it sure wouldn't hurt. The law says you have to put on a seat belt every time you drive a car but you don't always need it (you don't crash every time you drive). You can find vintage examples of products that don't use twisted heaters (or modern ones like the Soldano pictured; the Soldano uses a.c. heaters).
But if you wind up getting hum due to untwisted heaters, it's a b!tch to fix after the fact, or replace a socket as Sluckey pointed out (to be fair, replacing a socket is almost never consistent with a pretty final result). But some amps with heater traces on the p.c. board are quiet (some have a hum that can never be eliminated).
I'm intrigued by the ease of assembling the straight socket wires. I twist heater wiring in every amp I make (and consider its proximity/positioning to other wiring) as a way of guaranteeing I won't have hum
due to that cause. I don't do it because I believe untwisted heaters will certainly result in hum.
To a different point/question raised, most people copy what they see without knowing why it was done that way. Late tweed Fenders onward had twisted heaters, so that became proclaimed by some as "the only right way" to do it. Some of the Soldano clone folks insist on the straight parallel wires because it's in the original amp. Some know either approach is optional, and might produce a good result.