If your crossover expects a 4 Ohm tweeter, then in order for it to work correctly, you need to add a 6 Ohm resistor in parallel with the tweeter. This provides a 4 ohm total impedance, which will make the crossover point correct. Otherwise, the tweeter will produce sound at a much lower frequency than expected.
FREQUENCY?? No, not possible. Yes, the tweeter will get the wrong amount of power, and possibly the balance of sound would be wrong. I'd suspect the tweeter would be not producing much output, but the LC components in the crosover could do just about anything with this mismatch.
Oh, maybe I get your meaning, the tweeter would start getting power from the crossover at a lower frequency than intended. Yes, could be, assuming the crossover just uses a series cap for the tweeter. But, with the 12 Ohm impedance, it might never get much output, compared to the other speakers.
Jon
The power in the tweeter at any given high frequency (well above the crossover frequency) is the same, with or without the resistor! The problem is that the crossover frequency is shifted down because the resistance of the speaker coil is part of the filter itself. Higher R means lower Fc.
\$F_{c}=\frac{1}{2\pi RC}\$
The crossover described by the OP is a 1st order one, and for the tweeter, is just a simple capacitor. It, together with the resistance of the speaker/tweeter (12 ohms) forms a one pole high pass filter. At very high frequencies, the capacitor is low impedance and the speaker sees the full voltage (very little voltage dropped across the cap). At very low frequencies, the capacitor is high impedance and the speaker sees very little voltage (it is almost all dropped across the cap).
The shift in the frequency of the crossover point means that the tweeter will be exposed to more musical power and might burn out or be physically damaged by large excursion.
(edit) Even if it isn't damaged, there will likely be a hump in the frequency response of up to +6 dB due to the overlap of the mid and tweeter spanning about 1.5 octaves.
(/edit)FYI, the woofer has a coil in series with it forming a LR lowpass filter, and the midrange has a coil and a capacitor in series with it, forming a RLC bandpass filter. (A much better 2nd-order crossover would add shunt elements as well, but the OP said his had 2 coils and 2 caps, so it can only be 1st order).
Since the OP said the crossover was designed for 4 ohms, it needs a 4 ohm load for the crossover frequencies to be correct. That is why the 6 ohm resistor is added in parallel to the 12 ohm tweeter.
(edit:) It does not attenuate the tweeter response at all (a series resistor would). The 12 ohm tweeter might or might not be efficient enough to provide a flat(ish) response in combination with the other drivers. Randomly picking a woofer, mid, and tweeter will not usually yield a 3-way speaker with a good flat response. Usually, one or two drivers need to be padded (attenuated).