In getting young people to engage in a serious study of music we are giving them an 
 opportunity to know themselves better by becoming more precisely aware of the 
 amazing power that music has over them. Also, as we shall see, we are giving them 
 an opprotunity to deepen their knowledge of the natural world-and of our 

 connection to it-by becoming more aware of the mathematical order that underlies 
 music.
 The connection between music and mathematics was established by the legendary 
 Greek, Pythagoras. Pythagoras discovered that the most commonly used (and 
 most singable) musical intervals had intelligible mathematical counterparts.  
 Let's use the octave as an example. To the musician, notes that are one octave 
 apart sound alike-the only difference is that one is higher, or lower, than the 

 other. Modern science tells us that an octave is amusical interval in which one note 
 has either double or half the frequency of another note-if one note has the 

  frequency of 400 Hz (hertz or cycles per second), the note an octave above it 
 has a frequency of 800 Hz and the note an octave below has a frequency of 

 200 Hz. So, the ratio for an octave is 2:1.
 Pythagoras discovered this connection without the knowledge of frequencies: he 
 simply divided a string in half and, to his utter amazement, heard that this division 
 produced the octave. Likewise, he discovered that when one string is 

 two-thirds the lenght of another, it will produce a highter note that fits another 
 common musical interval, a perfect fifth (the first melodic interval in 

 "Twinkle, Twinkle, Little Star").  This discovery-that notes that sound good 

 together can be represented mathematically with ratios of small whole 

 numbers-was far 

 reaching; it suggested that great music was grounded in the very nature of the 
 physical universe-which explains why humans respond to it.      
 From "The Neglected Muse" by Peter Kalkavage which he wrote in American Educator Fall 2006.