The gap is necessary to keep the core from saturating when it is used with a direct current. The material should be non-ferromagnetic with low permeability since it is supposed to interrupt the flux path. Traditionally some form of paper or pressed paper is used. I like plastic sheets since you can buy them in very specific fine gauges, which allows to fine tune the gap for best performance at a given current. The downside to all of this is that it greatly reduces the inductance, hence gapped transformers will always be bigger than their ungapped counterparts for the same power.
Yes, it just means winding two or three wires at the same time, but there can be different reasons to do so. In the case of switching them parallel, the reason isn't really to lower the resistance. If there is so much space on your bobbin that you can just double up one of the windings you should rethink your design... The two wires in parallel are used instead of one thicker wire of the equivalent resistance. The reason to do so is that thinner wire is easier to wind, will stack better and waste less space and makes for a more compact winding overall. The other reason can be seen in the API output transformer, here the primary and secondary winding are wound together. This ensures better coupling between the two and reduces leakage inductance. Another possible scenario would be where you want perfectly symmetrical windings, for example a tube heater supply. Obviously not recommended for high voltage applications where a big voltage differential between the two wires could occur.
You're absolutely correct about gapped transformers taking a serious drop in inductance. But I should point out that the lower inductance translates to lower maximum drive levels at low frequencies (say 20 or 30 Hz). By the time frequency gets to 1 kHz or higher, there's enough primary inductance with or without a gap. Because low phase (deviation from linear phase) distortion requires that the -3 dB point for LF response be about 1 Hz or less, it becomes very, very difficult to include a gap in its design. All Jensen designs had this (considered by many to be extreme) extended LF response. LF phase distortion (not shift, but non-linear phase shift) can have a very audible effect on instruments like kick drums. Jensen therefore never made transformers with gaps. Had we done so, they would have been extremely large and prohibitively expensive. The hard work for any transformer is at low frequencies. As I explain in the transformer chapter I wrote, for a constant-voltage drive, the flux density in the core of a transformer increases as frequency decreases. The real test for a transformer handling music (where bass dominates signal levels) is how much level can it handle (for say 1% THD at 20 Hz, as was the Jensen practice). Or you can make smaller transformers look better by rating them at 50 Hz ... or more.
Bifilar windings are generally used to reduce leakage inductance. Leakage inductance is very important if the transformer secondary drives a highly capacitive load (like long output cables). That capacitance and the leakage inductance form a 2-pole low-pass filter - and it's best if the cutoff frequency of that filter is well beyond 20 kHz. Therefore, nearly all Jensen output models are bifilar, trifilar, or quadfilar wound. However, the tradeoff for multifilar windings is coupling capacitance (often in the 10 nF to 30 nF range). If you use a bifilar output transformer driven by a single-ended output stage (as in consumer gear, for example), you will notice that the signal on the transformer secondary is not symmetrical with respect to ground (and become less symmetrical as frequency increases). Mind you, this is
not a problem, but those who believe the pervasive myth about equal and opposite voltage swings on balanced lines imagine that it is! In fact, the
impedances of the two output lines with respect to ground is very well matched - making it a truly balanced output. This assumes, of course, that the capacitance is uniformly distributed in the transformer winding - which in Jensen designs it is within ±2%.
Because input transformers generally have low capacitance loads (input of an amplifier, for example), they can tolerate much higher leakage inductance and still have high bandwidth. This is a good thing because most good input transformers insert a Faraday shield between primary and secondary to stop the aforementioned capacitive coupling. But, because it physically separates primary and secondary windings, there's a big increase in leakage inductance. How that tradeoff is optimized is what separates good transformers from excellent ones.
