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You need the gain-setting resistors to cause a gain of slightly less than 3 so the diodes can adjust the output level.
In your circuit with a minimum gain of 3 the opamp output will reach clipping and the diodes will not do anything.
Yes.does that mean after diode adjustments, the gain will be set back to a maximum value of 3?
Yes.
You want the diodes to conduct only a little and only at the peaks of the sine-wave for the lowest amount of clipping distortion from the diodes.
In this case, when the signal reaches a cetain value, the diodes short out R6, and now the gain is:
1 + (15K + 0/10K) = 2.5. that will stop the signal from growing. In that way, the tops of the signal aren't clipped by hitting the power rail.
The gain is always 3. It takes a little time for the oscillations to build up to the clipping value, but it takes a gain of 3 or more to even get the oscillations started. The reason your circuit worked with R3 = 10k, R4= 15K and R6=5K is because the gain equation for your circuit is: 1 + (15K + 0/10K) = 2.5. that will stop the signal from growing. In that way, the tops of the signal aren't clipped by hitting the power rail.