now think back to the voltage divider of length , out of which we want to make a resistor piano using a frequency that is linear in the voltage. if the frequency range of the resistor piano spans two full octaves, such that , then . convince yourself that this makes sense, since the frequency doubles with every octave. if we wanted the frequency at the midpoint to have increased by one octave, such that , how should the resistance of the paper from to compare to the total resistance of the paper? in other words, determine .