Indeed. In addition to @John_Harshman’s reply, I would add this off-the-top-of-my-head list of ways to alter overall mRNA levels (which is probably what @Geremy means by “gene transcription”) via very modest changes (often no more than one base):
Changing the stabilities of the mRNAs, by slightly changing motifs recognized by RNA binding proteins that mediate turnover.
Changing the stabilities of the mRNAs, by slightly changing motifs that are targeted by small RNAs (microRNAs).
The origination of new microRNAs (as explained nicely by @NLENTS).
Changing the concentrations of one or more transcription factors (or other regulatory proteins), by changing their rates of synthesis or turnover. (Mechanisms alluded to above apply here as well.)