I'm talking about either beats per second of a resulting beat between two strings in the midrange of the piano. My piano seems to vary during different parts of the day, I think because of the temperature variation in the room where it sits.
You take a tuning fork tuned to 440Hz and a second one tuned at 442Hz, the beating will be 2beats/second and the following is true:
440 + 442 = 2beats/sec 440 + 437 = 3beats/sec 880 + 882 = 2beats/sec 220 + 219 = 1beat/sec etc.
So the difference in the frequencies equal the beating per second.
The beating itself is heard as a fluctuation of the volume of the sound (not the pitch)
When two strings a perfectly in tune the attack part of the note will sound harder or tighter. The harmonics (partials) also will be more noticeable thus augmenting the "clarity" or "brightness" of the sound.
As soon as one of the strings are getting "off", the sound become "duller" and less definite. Overall, a not so perfectly tuned piano will sound less satisfying and will loose definition. (definition = every notes can be distinguished clearly)
Even though a normal person can't tell how many beats/sec the piano's unisons are out, this person can still notice quite a little difference in the out of tune unison because of the color change that occurred in the sound. Something like 0.5beat/sec will be noticed because the "quality" of the sound is changed.
Hope that all this make sense.
In general, the beat frequency should be low enough that you can't notice it.
The average lowest range of human hearing is 20Hz, around E0. (Most pianos have A0 as the lowest note, since nearly everyone unimpaired can hear it and it's more "musical"). This means that if the beat frequency is above 20 Hz, you will hear a third note corresponding to that frequency. That's definitely bad.
The "beat" phenomenon still occurs below 20 Hz though, of course. I would approximate one beat every 3 seconds, or a beat frequency of 0.333.... Hz, to be good enough. There's no hard and fast rule, really, since more sensitive people may pick up on a slower beat than others. Once it gets that slow I struggle to notice it, and I certainly wouldn't pick up on it unless I was listening for it. It's harder to get higher strings to stay within 1/3 Hz of each other as well, since tuning adjustments cause a greater frequency change than for lower strings, so you might have to be more lenient with them.
And, unfortunately, you're never going to get around the variation in how your piano sounds. You should try to keep the humidity and temperature consistent (which will help how long the piano stays in tune generally as well), but it will never be perfect.
The Piano Technicians Guild is an association of piano technicians and they have exams to qualify technicians.
The tuning exam has many parts, including a unison part.
In this part, the examinee will tune 24 unisons from C3 to B4.
The examiners measure the difference between all three strings of each unison. If they are out by more than 0.9 cents, (100 cents = a semitone) then the examinee will lose 2 percent. The unison is measured for left to centre, right to centre, and left to right, meaning that an examinee can lose 6% on one note! 80% is a pass, meaning an examinee can not have more than 10 unison pairs greater than 0.9 cents off.
A constant cents offset will result in a different bps depending on where on the piano you are talking about.
Also, humidity fluctuations throughout the day should affect each unison string the same. If you have unions going out during the day, that has to be some other problem.
I think the question relates to how many beats per second (or seconds per beat) is acceptable with unisons to consider them to still be in tune.
We agree that one beat per second indicates that two strings are different by 1 hertz, eg one string is at 440 (A4) and another at 439 and we can hear a slow wow wow wow.
That is a difference of 4 cents. The tolerance in strictest tuning exams happens to be much less eg 1 or 2 cents.
At 220 (a3) compared with 219Hz the difference is 8 cents.
I guess 8 cents on one if not both of the strings is going to make an audible difference in the frequency of beating of the major 3rds, 4ths, 5ths to the trained ear, so at A3 we would not want any of the strings to be as much as 8 cents out.
Maybe I should analyze a few pianos with Tunelab or similar listening devices after they have had professional or paid-for tunings, and find out what is going on.................
It was a valid question although I have never noticed a periodic change like a worsening then an improvement in what I hear in my own piano