Aaron Parecki

When we tune an instrument, we would like for all our octaves and fifths to be perfect. One way to tune an instrument would be to start with a pitch and start working out the fifths above and below it it. We start with some frequency that we call C. Then 3/2 times that frequency is G, 9/4 times that frequency is D (an octave and a step above our original C), and so on. If you learned about the “circle of fifths” at some point in your musical life, then you know that if we keep going up by fifths, we’ll eventually land back on something we’d like to call C. It takes a total of 12 steps, and so if we keep all our fifths perfect, the frequency of the C we get at the end is 312/212, or 531441/4096, times the frequency of the C we had at the beginning. You might notice that 531441/4096 is not an integer, much less a power of 2, so our ears would not perceive the C at the end as being in tune with the C at the beginning. (531441/4096 is about 130, which is 2 more than a power of 2, so we would hear the C at the top as being sharp.) And it’s not a problem with the assumption that it takes 12 fifths to get from C to shining C. We can never get perfect octaves from a stack of fifths because no power of 3/2 will ever give us a power of 2.