The good news is yes! There is a predictable relationship for some of the drum sounds. In fact, the relationship is exponential. That means predicting the frequency simply requires discovering two constants, the base and the scale of the exponent. Measuring the pitch to knob value relationship at two points on the scale for each of the 40 algorithms enabled calculation of those constants.
Using those constants, I generated a large, colorful table which prescribes exactly what knob values to dial up in order to get a specified note. You can see that table here.
One thing I found interesting about this little project was that the percent error between the target frequency (from the chromatic scale, based on 440 Hz) and the generated frequency was always very low. In other words, the drum machine makes a very accurate melody generator!
You can also download the spreadsheet I used to create this practical artwork.