The current paradigm of isolated attosecond pulse production requires a few-cycle pulse as the driver for high-harmonic generation that has a cosine-like electric field stabilized with respect to the peak of the pulse envelope. Here, we present simulations and experimental evidence that the production of high-harmonic light can be restricted to one or a few cycles on the leading edge of a laser pulse by a gating mechanism that employs time-dependent ionization of the conversion medium. This scheme enables the generation of broadband and tunable attosecond pulses. Instead of fixing the carrier-envelope phase to produce a cosine driver pulse, the phase becomes a control parameter for the center frequency of the attosecond pulse. A method to assess the multiplicity of attosecond pulses in the pulse train is also presented. The results of our study suggest an avenue towards relaxing the requirement of few-cycle pulses for isolated attosecond pulse generation.