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Dargon Magazine Article
Re the Dragon article you mention: Do you remember whether it only adjusted melee weapon initiative for weapon type, or actual attack frequency as well? The latter would be the kind of speed adjustments I'm talking about, which seem to be very rare in RPG melee rules.
Why rely on mere memory when I can just look through the stacks (and stacks, and...) of old magazines? I knew I kept them for a reason. Well, a reason other than making my wife crazy.
Dragon Magazine, issue 71 (March 1983), "Who gets the first swing?" by Ronald Hall.
It's largely a modification of initiative for AD&D 1e, centered around determining a new set of modifiers to the traditional d6 initiative roll. You factor in weapon length, dexterity, size, experience, and strength to get modifiers for closing and in-range initiative rolls, as well as rules for charging, fending, retreating, and pressing.
However, more to your point, there was also a system for replacing the standard 'multiple attacks based on level' that AD&D had.
If the figure's AP number [modified initiative roll] is between 1 and 6 inclusive, one attack is allowed at the specified time (with results of 6 attacking first, 5 second, etc.).
If an AP number is greater than 6, a second attack by that character or monster is possible. The first attack is executed ahead of any "1-6" results (see the preceeding paragraph). Then 12 is subtracted from the initial roll, yielding a result of -5 or greater, and another d6 is rolled. If the sum of those two results is greater than zero, the number is then treated like any other AP roll and given its place in the sequence of attacks during that round. With this system, a third attack is theoretically possible for those with high AP modifiers (+7 or greater).