Turn gate on and off while on rising edge of a signal
Perhaps you can use a window comparator along with a delayed signal slope comparator as shown below.



The top two comparators set your 1V to 6V range, the bottom one compares the input signal with a delayed version of the input. Depending on your frequency some adjustment to R4 and C1 may be required.

If all three conditions are met the output will be pulled high by R5.

I used LT1018s but LM339s will work just as well if your frequency is not too high. Comparators need to be open-collector or open-drain though.



HOWEVER

The issue with the above circuit is any noise in the signal can cause extra edges at the transition points 6V and 1V.

As such it is prudent to add some hysteresis.



Ultimately though, it would be better to use the edges separately to clock and reset a D-Type.