A simple but flawed idea to make a fixed frequency variable duty cycle oscillator based on a 555 timer and using the push-pull output to drive the RC timing through two routing diodes, a pot and a series resistor to limit the min/max duty cycle to something sensible at around 9%/91%.
If the diode drops were negligible and the 555 timer output pulled fully up to VCC and down to ground then the sum of RC time constant for the rising part of the sawtooth through one side of the pot plus the series resistor plus the RC time constant for the falling part of the sawtooth through the other side of the pot plus the series resistor would be a constant and so although the rise and fall times would be adjustable, the overall period would be fixed.
However, in the real circuit, the diode drops are not negligible: they still constitute about 10% of the peak to peak swing of the V(SAW) for VCC = 15V (and much more for lower values of VCC).
Also, the real bipolar 555 timer output swing has different voltage drops with respect to ground and below VCC output which means the high and low levels do not swing fully between ground and VCC. This effectively makes the rising edge time constant different from the falling edge time constant. This also causes different currents to flow in the routing diodes and so they present different drops for the rising and falling edges of V(SAW).
These effects combined mean that the frequency varies significantly at different duty cycle settings.