Quick mathematical summary of the problem:<p>An octave is a 2:1 ratio between frequencies, so 880 hz is one octave above 440 hz. A perfect fifth is a 3:2 ratio between frequencies, so 660 hz is a perfect fifth above 440.<p>In the modern western system of music, twelve perfect fifths is harmonically equal to seven octaves. In other words,<p>(2/1)<i></i>7 == (3/2)<i></i>12<p>Unfortunately, we know this is mathematically untrue.<p>Furthermore, three major thirds is harmonically equal to one octave:<p>(2/1) == (5/4)<i></i>3<p>This also is mathematically untrue.<p>Hilarity ensues.