Piano Key Frequency and Micro Tuning
on a piano, each adjacent key (color doesn't matter), from left to right, their frequencies forms a geometric sequence. The scaling factor is 2^(1/12)≈1.05946.
here's a video of micro tuning.
the white keys are labeled A B C to G. Starting with C on the left most white key of the 2 black keys group.
black keys don't have a name by themselfs, but is named relative to adjacent white keys. A black key to the right of a white key (say C) is called C♯ (sharp), or D♭ (flat). Basically, ♯ is used to jump to the right.
move hand to somewhere in the middle and find the closest C key, that key is called Middle C, and find the A key above it. That A, is called Concert A. By standard tuning, that A key, has a frequency of 440Hz.
now, the western music scale is such that, every 12 intervals, the frequency doubles. So, when you find the middle A key (which has frequency 440Hz), the next A to the right, has frequency 880Hz. True for any key.
since Bach's The Well-Tempered Clavier, we have “equal temperament” tuning. Adjacent keys frequency differ by a constant factor. e.g. A has frequency 440Hz. The key right of it has 440*c, where c is a constant. then 440*c*c, 440*c*c*c, till 12 keys after it we have 440*c^12, and should be 880Hz. 440*c^12==880. So c = 2^(1/12)
the original video retweet in this thread, is not standard tuning. It's a micro tuning. i.e. you have to go more than 12 intervals for frequency to double.
Here's the standard tuning:
on a piano, you might wonder why the black keys? why not all just white keys? well, lots history, but basically, the function of the current irregular shape (the 2/3 groups of black keys jutting out) is so that it's easier for fingers to find keys by touch.
on a piano, since key's frequencies is regular by a factor, why are they named irregularly C C# D D# E F F# G G# A A# B C. (those with # are black) Why not A to L? again, history. Like programing language syntax and math notations, lots are result of hundreds years of evolution, habit.
See also: How to Read Music Sheet