I need to edit my 555 timer in astable mode to get at least an interval change of 60% for the duty cycle, for example varying from 40 to 100% and so on. I am using a 10k potentiometer and 1 K resistor as R1 and R2 so I cannot go below 52 %. Any idea

 

Lets have a circuit diagram first , then we can see what you need. You can do it with an hysterical circuit using diodes.

 

resistor as R1 and R2 so I cannot go below 52 %. A

Have you tried to half R7 so that it's time constants with C3//Cr is less than R1?

 

You can do it with an hysterical circuit

 

Below is a 555 astable design whose duty-cycle variation is close to 0-100% with pot position.
(Shown for pot positions of 1% and 99%.)
Edit: Either the OUT pin or DIS pin (with pullup) can be used as the output.
The frequency is approximately 1/(0.73*Rpot*C2)

 

 

Use Monostable design, you need good R (1% preferably) and Cs. I had a tone generator project and the best site out there is http://555-timer-circuits.uk/ avoid all the messy calculations.

alternative: http://www.ohmslawcalculator.com/555-astable-calculator