What my father told me back in the stone age, when distributors had "points", was that the wider the gap, the better. Would burn more fuel. A lot of old-timey ignition systems weren't up to the task, however. Claimed that the big advantage of electronic ignition systems was that they would provide a hotter spark, allowing said wider gaps. Typically, a wider gap is specified on modern cars (.040" to .060") compared to older model Flintstone-era cars (.025" to .035"), so there may be something to what he said...
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Dick Naugle says: 1. Prepare food fresh. 2. Serve customers fast. 3. Keep place clean.
In general, wider gap = more spark duration, smaller gap = hotter, more intense spark.
You might partially compensate for over-retarded factory "emissions" timing in midrange by having a smaller gap, due to it sparking milliseconds sooner.
Expect higher plug erosion with smaller gaps. Since plugs have become something of a "fit and forget" thing with 60K - 100K expected service life these days, it's important to remember to check them regularly if you're running shorter gaps.
Older motors with poorly homogenised charges, particularly carbed and TBI cars might benefit more from the longer spark with a wider gap, to keep "relighting" the mixture a bit if it has large droplets, or is poorly mixed. So if you shorten your gap and HC readings go up, ensure better fuel atomisation and tumble/swirl or put it back to stock.
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Regarding the timing, the "advance" is extremely minimal, but a smaller gap will require less energy to jump the gap so the spark plug will "spark" earlier...though most likely only fractions of a degree earlier...not enough to make a noticable difference. Smaller gaps are "usually" better for engines that will be operated at higher rpms and do not have the ignition system robustness for a larger gap at those high rpms.
In terms of larger than stock gaps; the larger gap has traditionally been known to help low rpm fuel burn, and low rpm torque....just the areas most hypermilers would like help with. The downside to a larger gap is that under higher rpm, higher load operations you may experience a few misses as the gap is harder to jump under those conditions.
I gapped mine up 0.010 over stock and it has helped cold starting and low rpm lugging....it seems to pull smoother when I'm trying to accelerate from 1000rpm in a higher gear. I also advanced my cam (SOHC) by 9 degrees and that combination with the plug gap has very noticably increased low rpm torque....torque from 800-2000rpm is definitely higher than when the car was "stock".
About 9 years ago, I upgraded the ignition system on my truck with a Jacobs ignition kit. It worked off the exisiting coil but directed a stronger spark to the distributor via a bigger add on coil. In any event, the instructions said to increase the spark gap initially by .010" over (or by a chart they included I can't remember) as a baseline starting point. From there keep adjusting wider to find a good spot for it. Well, I can tell you from that, as I kept going .005" over from a previous gap, you could notice the top end power falling off at lower and lower rpms as the gaps kept getting bigger and bigger. Anyone who has driven with worn spark plugs that are highly eroded over stock gap will also notice this same thing. The wider gap does open it up to expose it better to the mixture BUT at higher rpms the higher cylinder pressures work against the ignition system and actually creates additional resistance to the spark jumping the gap. For this very principle, that would be the reason why engines running forced induction typically run smaller gaps.