Tempering is simply the spacing of notes. Actually all 88 piano keys are tempered. The temperament as most tuners know it is the spacing of the notes in an octave. As you know, modern tuning is called equal temperament based on the 12th root of 2. A mathematical equation to equally space the notes. Organs indeed are tempered with the same formula.
Pianos however have harmonics that are out of tune, referred to as inharmonicity, Out of tune harmonics are often referred to
as partials (part of the whole).
This inharmonicity causes a deviation of the actual fundamental pitch from being a perfect match to the mathematical formula for equal temperament.
For instance of A4 is vibrating at 440 cycles per second, the partial created by the string being divided by 2 (the second partial) will likely vibrate faster than 880. It might be 880.1 or 880.2 but it could be anywhere. So to tune A5 we might need to tune it to 880.2 to have it sound, or be in tune. These are call coincidental partials. I explain to customers (who are interested) that this is simplifying tuning an octave in what tuners call a 2:1 ration for an octave. We can also match other partials like 4:2, 6:3 etc.
And I explain that the same thing goes on with every interval we tune.
And figuring out this puzzle is different in every piano, which is what makes tuning so much fun for me!.
Don't organs and harpsichords also need tempering?
Maria G. Pollock
Registered Piano Technician
773 307 5207
The best way to get things done is do them.