Now, as a matter of fact, in a scale which is equally tempered, no two
fifths beat exactly alike, as the lower a fifth, the slower it should
beat, and thus the fifths in the bass are hardly perceptibly flat,
while those in the treble beat more rapidly. For example, if a certain
fifth beat once a second, the fifth an octave higher will beat twice a
second, and one that is two octaves higher will beat four times a
second, and so on, doubling the number of beats with each ascending
octave.

In a subsequent lesson, in which we give the mathematics of the
temperament, these various ratios will be found accurately figured
out; but for the present let us notice the difference between the
actual tempered scale and the exact mathematical scale in the point of
the flattening of the fifth. Take for example 1C, and for convenience
of figuring, say it vibrates 128 per second. The relation of a
fundamental to its fifth is that of 2 to 3. So if 128 is represented
as 2, we think of it as 2 times 64. Then with another 64 added, we
have 192, which represents 3. In other words, a fundamental has just
two-thirds of the number of vibrations per second that its fifth has,
in the exact scale. This would mean a fifth in which there would be no
beats. Now in the tempered scale we find that G vibrates 191.78
instead of 192; so we can easily see how much variation from the
mathematical standard there is in this portion of the instrument. It
is only about a fourth of a vibration. This would mean that, in this
fifth we would hear the beats a little slower than one per second.
Take the same fifth an octave higher and take 2C as fundamental, which
has 256 for its vibration number. The G, fifth above, should vibrate
384, but in the tempered scale it beats but 383.57, almost half a
vibration flat. This would give nearly 2 beats in 3 seconds.