The bridge location can either be measured or tuned electronically for the exact location. I will discuss both.
Hopefully, you have read the previous articles about setting up the string action for the best height for you. You will also need the banjo tuned to correct pitch for best results. To place the bridge correctly on the banjo head, you will need a ruler. Measure the distance from inside the top nut (bone) at the peghead to the 12th fret. Now, measure the distance from the 12th fret to the bridge. The measurements must be the same. If the measurements are not the same, move the bridge carefully until both measurements are the same. Retune the strings then check the measurements again. Caution; Be careful when you move the bridge. Make sure it is standing up straight and at right angles to the strings. If the bridge falls over, it makes a frightening noise or you may break the bridge. Move it a little at a time.
The second method of correctly locating the banjo bridge is to use a tuning meter. With the banjo in tune, play the 12th fret harmonic note ** on the middle G string. Check with the meter, it should read a perfect G. Now, play the fretted note on the 12th fret of the same G string. Check with the meter. If the fretted note is higher or sharper than G on the meter, the bridge is too close. The bridge must be moved further back toward the tailpiece. Or, if the note you played at the 12th fret is too low or flat on the meter, the bridge must be moved closer to the fretboard. Move the bridge a little at a time. Retune the strings and do the procedure again until both the harmonic note and the 12th fret note are exactly the same pitch. Now the bridge is in the correct position for all of the fretboard notes to be accurate even up to the last fret.
If you have mastered this bridge location procedure using the 12th fret, you can also experiment using the 19th fret. That is, the harmonic note at the 19th fret and the fretted note on the 19th fret. These notes are also exactly the same pitch. This is even more accurate than using the 12th fret. This tuning procedure can be used for all instruments that have frets and adjustable bridges. I’ve used it on banjos, mandolins, electric guitars, and electric basses. It does not work on violins or dobros (no frets). Even acoustic guitars can be checked to see if the manufacturer has located the bridge properly. To adjust a fixed bridge guitar, some professional has to change the bridge saddle.
More technical information on the science and mathematics of music can be found in books or on the web. Hermann Helmholtz was a German scientist in the 1850’s. He did all the math on things like note pitch, resonance, law of the length of strings, harmony, etc.
** To play a harmonic note, touch the string very lightly over the 12 fret and play the string. This should give you a note that is one octave above the open string tuning. There are also harmonic notes over the 5th, 7th, 19th, and 24th fret positions.