I present here the easiest, approximate, equal temperament scheme. More accurate algorithms can be found in the literature (Reblitz, Jorgensen). No self-respecting professional tuner would use this scheme; however, when you get good at it, you can produce a decent equal temperament. For the beginner, the more complete and precise schemes will not necessarily give better results. With the more complex methods, a beginner can quickly get confused without any idea of what he did wrong. With the method shown here, you can quickly develop the ability to find out what you did wrong.
Mute the side strings from G3 to C#5. Tune A4 to the A440 fork. Tune A3 to A4. Then tune up in contracted 5ths from A3 until you cannot go up any more without leaving the muted range, then tune one octave down, and repeat this up-in-5ths and down-one-octave procedure until you get to A4. For example, you will start with a contracted A3-E4, then a contracted E4-B4. The next 5th will take you above the highest muted note, C#4, so you tune one octave down, B4-B3. All octaves are, of course, just. The contracted 5ths should beat a little under 1 Hz at the bottom of the muted range and about 1.5 Hz near the top. The beat frequencies of the 5ths between these highest and lowest limits should increase smoothly with increasing pitch.
When going up in 5ths, you tune flat from just to create a contracted 5th. Therefore you can start from just and tune flat in order to increase the beat frequency to the desired value and set the pin correctly at the same time. If you had done everything perfectly, the last D4-A4 should be a contracted 5th with a beat frequency of 1 Hz without any tuning. Then, you are done. You have just done a "circle of 5ths". The miracle of the circle of 5ths is that it tunes every note once, without skipping any within the A3-A4 octave!
If the final D4-A4 is not correct, you made some errors somewhere. In that case, reverse the procedure, starting from A4, going down in contracted 5ths and up in octaves, until you reach A3, where the final A3-E4 should be a contracted 5th with a beat frequency slightly under 1 Hz. For going down in 5ths, you create a contracted 5th by tuning sharp from just. However, this tuning action will not set the pin. Therefore, in order to set the pin correctly, you must first go too sharp, and then decrease the beat frequency to the desired value. Therefore, going down in 5ths is a more difficult operation than going up in 5ths.
An alternative method is to start with A and tune to C by going up in 5ths, and checking this C with a tuning fork. If your C is too sharp, your 5ths were not sufficiently contracted, and vice versa. Another variation is to tune up in 5ths from A3 a little over half way, and then tune down from A4 to the last note that you tuned coming up.
Once the bearings are set, continue as described in the Kirnberger section above.