The Two most important Musical Numbers

No, they are not "Happy Birthday" and "New York, New York".
  The first number is 1.059463. If you take the frequency of any note and multiply it by this number, you will get the frequency of the next note a 1/2 step up. The entire equal tempered scale is based on this number which happens to be the 12th root of 2 - that number which when multiplied by itself 12 times, equals 2. This frequency progression is known as a geometric series.

The second most important number is 1.000577789. If you take the frequency of any note and multiply it by this number you will get the frequency of the the pitch that is 1 cent sharper. There are 100 cents in a half-step, therefore 1200 cents in an octave and this number is simply the 1200th root of 2, the number that when multiplied by itself 1200 times equals 2.  This frequency progression is also a geometric series.

So why care? Well, practically speaking, a musician doesn't need to walk around with a calculator, although its handy to be able to figure out the frequency of hot spots or feedback pitches. With A above middle C being 440 Hz it's fairly simple to figure it out especially when you realize that octaves are also a logarithmic series obtained by multiplying the frequency by 2 :  110, 220, 440, 880 etc. 
    More interesting, though, is one result of the math, which is looking at the relationship between CENTS and HERTZ. Hertz refers to the frequency, or the vibrations per second, sometimes called cycles per second. If you multiply 440 by 1.000577789 4 times, you get the frequency of the tone that is 4 cents above 440. It turns out to be 441.01, or virtually 441. This is very handy - the result being that AT A440, 4 CENTS = 1 HERTZ.    So if your piano is tuned to A442, for example, you can have everyone tune their instruments 8 cents sharp, cents being the most common scale on portable electronic tuners and 440 being standard tuning pitch.  
   They will be able to use any note as a tuning reference as long as it is 8 cents sharp and it wouldn't have to be A4 ( A above middle C).  If they are tuning A5, for example, they would still tune it 8 cents sharp, 8 cents at that frequency now represents  2 Hertz instead of 1, which is really just academic but also interesting.
  So here are some examples

If the piano is
          A444           tune   16 cents sharp
          A435           tune    20 cents flat   ( this used to be standard pitch)
          A442           tune     8 cents sharp    ( this is slowly becoming standard pitch )

So now we have made cents out of hertz.  I know math hertz, but it's fun for us geeks.