Musical Note Frequencies

I was thinking about how doubling the frequency of a note gives you a note one octave higher. Case in point: A 220Hz--> A440Hz, contrast this with the next octave A 440Hz --> A 880Hz and we see that the span (in Hz) of each successive octave is greater than the previous. The frequency of ascending notes has an exponential relationship that goes like 2^x. I was specifically thinking about this because the only musical note and corresponding frequency I knew was A440Hz and I wanted to know the frequency of any note I could think of. Knowing this single reference frequency and the fact stated in the first sentence of this post I found a formula that will tell the frequency of any note, I figured I would reference the formula to the 88 keys of the piano for convenience where n=1 is the lowest note and n=88 is the highest.

This graph of the above formula shows the exponential curve nicely and I've added the piano keyboard as the x-axis which corresponds to the domain of the function, integers 1 through 88. I have no immediate use for this information but I just feel better knowing that I now have an easy way to know the frequency of all the notes in western music. The formula actually works for n values less than 1 and above 88 but those probably won't get used too much.

One last thing I wanted to incorporate before I could rest was the phenomenon familiar to piano tuners called "stretch". It so happens that there is inharmonicity in pianos, especially in smaller scale and poorer made pianos but even to some extent in the finest pianos ever made. One result of this inharmonicity is that harmonics (overtones) will occur not perfectly at integer multiples of the fundamental frequency. This causes some notes to sound bad in the extreme ranges of the piano when the string is theoretically in tune to the frequencies in my little formula above. Stretching the notes sharp in the treble and flat in the bass is typically done to compensate. Additionally, I think we like to hear the high notes a little on top of the pitch anyway just to avoid the appearance of being flat. The amount of stretch depends on the piano and the mood of the piano tuner during the tuning. I have modified the formula to accommodate stretch. The .1 value can be increased or decreased for more or less stretch. All in all I think it's a decent model.