The quality of tone has little to do with a false string. Strings on a violin are like glasses for the eyes. Some violins need different strings than others to sound their best. You can ruin a good sounding violin with a wrong string, even if it was expensive. Of course a bad sounding violin will stay bad no matter what you put on. A chain is as strong as the weakest link.
Wolf notes are not connected with a string but with a particular resonating frequency. This frequency varies from instrument to instrument. A wolf may go unnoticed since most players instinctively adjust their bowing to avoid it becoming too obvious. In fact, most instruments have a wolf note of some strength. A wolf tone will be less strong if you use a less powerful string. But that does not mean that the stronger string is false. In general the best set up for the strongest sound will create a stronger wolf tone. Making changes to the lower bass area of the top can usually influence this wolf tone. This is the area near the chinrest on violins and violas. You can check this by putting a finger on that area while you play the wolf tone. Most wolf tones will disappear. You can increase the tension on this area of the top by using a side-mounted chinrest. The net pressure on the top will similar as that of a center-mounted chinrest. But with two clamps on the bass side you increase the stiffness more in that area. The center-mounted chinrests will increase the tension a little on both sides and have less influence on the bass side of the top. Some instrument repairmen also move the sound post to a tighter spot to weaken the wolf.
A side-mounted chin rest might dampen a wolf, but over tightening can also cause damage. The lower block inside the violin is in the center and does not withstand excessive pressure of the side mounted chinrest clamps. The ribs can start to buckle if you tighten the chinrest too much.
3. Perceived Pitch Changes:
Sometimes plucking or playing an open string with varying pressure will seem to generate changing pitch. The string often gets wrongly blamed of being “false.” If you play a solid core metal string on a violin it will raise in pitch if you play forte. When bowing with more pressure, the string is actually pushed from its normal position, which results in an increased tension. The vibrations will consequently create a higher frequency/pitch. A string with a gut or nylon core can minimize the tension increase of the vibrating string. Since a cello string is much longer there is less change in pitch when playing forte so they don’t create the same problem.