As we have seen in The false piano (1): a tone has a certain number of Hz. Multiply this number by 1,4983 instead of 1,5 in order to reach the fifth of a well-tuned piano. Now consider the 440 Hz tone to be A (the tuning standard). The pure fifth E has the frequency of 1,5 x 440 Hz = 660 Hz. To get the piano well-tuned this means 1,4983 x 440 Hz = 659,251 Hz (rounded off). Listening to the two tones (660 and 659,251 Hz) separately the difference is almost inaudible. The fact that we deal with two different tones becomes clear when the tones are played simultaneously. One can hear the beatings:
Beatings emerge when two tones have close frequencies. Interchange of phasing out and amplifying arises. The false piano (1) says: multiply the number of Hz by 3 to get the tone one octave plus fifth higher. Multiplying by 4 means: 2 octaves higher. And multiplying by 5? This means reaching the tone 2 octaves plus major third higher (see also Pythagoras and strings). For just one major third we should multiply by 5 and then divide by 4 (which means 2 octaves back). So the major third gets the ratio 5/4 = 1,25. The major third ratio in our 'false' piano is 1,26 (rounded off). For C-sharp above A440 this means a frequency of 554,365 Hz instead of 550 Hz. (for the calculation of 554,365 Hz see backgrounds). The 'pure' major third with ratio 5/4 we nowadays experience as too small, as false. The beatings sound like this:
And one octave higher:
But when a piano tuner strikes a tone he will still hear the one tone instead of two. In order to hear beatings two tones are needed! Or is it? Indeed, but now intervals are used to make the beatings audible. For instance, strike a tone and the fifth below. Then the pure fifth sounds in the overtones of the lower tone of the interval (for the explanation of overtones, see Pythagoras and strings).The beatings of the fifth interval are relatively slow because the frequencies are close together. The beatings of the major third interval should be a lot faster.