The example below demonstrates that the perception of the tempering of fifths depends critically on beat rates.
Nine different fifths are represented using so-called sawtooth waves, one pure and eight with varying degrees of tempering. Sawtooth waves contain all harmonics.
A second set of related examples uses sine waves. In this case there are no harmonics present.
Beating of a fifth occurs between the third harmonic of the lower note and the second harmonic of the higher one. In the case of sine waves the harmonics are not present. Accordingly no beating occurs. Without a beat rate to guide the ear, the tempering of the fifth cannot be properly assessed.
Sawtooth Waves