I am simply looking for ways in which Tesla might have incorporated the numeric values of 3, 6, and 9 into the actual functioning of his TC's. For example, as a relationship between the resonant frequency and other design features. Obviously, the options for doing so would be limited by the way in which a TC works. My lack of knowledge of these options is the reason for my question.
actually, if you look at the US power grid, there are some features of what you are talking about. AC power in the USA uses a line frequency of 60Hz, it is distributed in 3 phases 120 degrees apart, the nominal outlet voltages are 120V, 240V, 480V. 3-phase power is the minimum number of phases needed to insure that the simplest motors always start turning in the same direction. in fact, servo motors also use 3 phases for the same reason, to insure the slave device follows the motion of the master device without ambiguity. the choice of 60hz as the frequency was chosen by Tesla, who, after building the first AC station at Niagara Falls, saw that Westinghouse's suggested line frequency of 25Hz caused flickering in various types of lamps (and so did 30Hz), so Tesla built the second power station using 60Hz. it actually worked out better than 50Hz (which was adopted in europe) because the amount of copper and iron in generation and transmission hardware saved was substantial (because of the higher frequency, the amount of copper required was less because the inductance needed was smaller, and the amount of iron for a 60Hz transformer core is also a bit less). as for the outlet voltages being 120V, 240V, and 480V, that has actually bounced around a bit, and has been changed as time has passed, going through slight increases: 100, 105, 110, 115, 120, and 125. i'm not really sure what continuously drives the change, most likely improvements in the transmission hardware. officially, it's 120, and has been since the last DC systems were phased out in the late 50s-early 60s (interestingly, there were still "hot-chassis" AC/DC powered radios and TVs manufactured into the late 1970s. but, there's really nothing "special" about 60hz power, or the outlet voltages. if we really wanted to update the power system, and make it more efficient for the amounts of steel and copper used in distribution, a good start would be to switch to something like the military 400Hz systems. a 100W transformer at 400Hz, requires less than 1/4 the amount of steel and copper to produce than a 100W transformer for 60Hz. Tesla may have used ratios of 3's in a lot of his coil construction, but much of it would have simply been that's what he preferred to use. sometimes the math is easier too when you do something like that.