A PROFESSIONAL PIANO TUNE-UP MUST BE DONE BY EAR
If you tune up a piano like it should be mathematically perfect, it is not going to sound good (for example, like if we tune the A's at 55, 110, 220, 440, 880).
To make a piano sounds good we need to stretch the octaves. Problem is, there is not an exact sequence to follow. It depends on each piano.
Most of the piano tuners do their job utilizing a digital device, but machines don't hear the same way we, humans, do. There are some specialized devices to tune up pianos that mimic the way the human ear perceives the sound. Certainly, these devices work better than regular chromatic tuners, but still, they are not enough because each piano has different acoustics and the machine is not able to adjust itself to each different situation.
If you are a professional pianist or a high-level amateur, you will immediately notice the difference when a piano is tuned by ear.
However, there are some situations where we need to assist ourselves with some electronic device. Old pianos, bad quality pianos (like many Chinese and Korean pianos), or pianos that hasn't been tuned in a long time are very difficult to stabilize, so we need to keep an eye on the machine to know if the strings that we already tuned hasn't gone out already when we get back to them making the job impossible. In those cases, the piano never is going to sound good at the first tune-up. If the piano is in good condition and it just has been neglected, it will take a few tune-ups before it finally stabilizes. Cheap pianos or in bad condition may never get good enough. Spinets are especially bad pianos, not just because of their poor design, but because of the too short length of their bass strings.
A helpful video from Living Pianos