The lowest C on a piano is called C1 in scientific pitch notation. The C one octave up is C2 and so forth. Middle C is C4.
The frequency of Cn is approximately 2n+4 Hz. This would be exact if C0 were 16 Hz, but it’s a little flat. In order to make A4 have frequency 440 Hz, CO must have frequency 16.3516.
Notes other than C take their number from the nearest C below. So A4 is the A above C4, middle C. The lowest note on a standard piano, the A below C1, is A0.
At one point in time C0 was defined to be exactly 16 Hz. The frequencies of notes have been defined slightly differently across time and location.
Mathematically perfect octaves, however, don’t sound quite right. The highest notes on a piano would sound flat if every octave were exactly twice the frequency of the previous octave. So we tune the lowest notes a little lower than the math would say to, and the high notes higher.
In the original thread I said that C0 was the lowest C on a piano when I should have said C1.
No discussion of mathematics and piano tuning would be complete without mentioning Fermi problems. As I discuss here,
These problems are named after Enrico Fermi, someone who was known for being able to make rough estimates with little or no data.
A famous example of a Fermi problem is “How many piano tuners are there in New York?” I don’t know whether this goes back to Fermi himself, but it’s the kind of question he would ask. Of course nobody knows exactly how many piano tuners there are in New York, but you could guess about how many piano owners there are, how often a piano needs to be tuned, and how many tuners it would take to service this demand.