nd. Young tuners sometimes get
confused and accept one beat as being two, taking the period of
augmentation for one beat and likewise the period of diminution. This
is most likely to occur in the lower fifths of the temperament where
the beats are very slow.

Two strings struck at the same time, one tuned an octave higher than
the other, will vibrate in the ratio of 2 to 1. If these two strings
vary from this ratio to the amount of _one_ vibration, they will
produce _two_ beats. Two strings sounding an interval of the fifth
vibrate in the ratio of 3 to 2. If they vary from this ratio to the
amount of _one_ vibration, there will occur _three_ beats per second.
In the case of the major third, there will occur _four_ beats per
second to a variation of _one_ vibration from the true ratio of 5 to
4. You should bear this in mind in considering the proper number of
beats for an interval, the vibration number being known.

It will be seen, from the above facts in connection with the study of
the table of vibration numbers in Lesson XIII, that all fifths do not
beat alike. The lower the vibration number, the slower the beats. If,
at a certain point, a fifth beats once per second, the fifth taken an
octave higher will beat twice; and the intervening fifths will beat
from a little more than once, up to nearly twice per second, as they
approach the higher fifth. Vibrations per second double with each
octave, and so do beats.

By referring to the table in Lesson XIII, above referred to, the exact
beating of any fifth may be ascertained as follows:--

Ascertain what the vibration number of the _exact_ fifth would be,
according to the instructions given beneath the table; find the
difference between this and the _tempered_ fifth given in the table.
Multiply this difference by 3, and the result will be the number of
beats or fraction thereof, of the tempered fifth. The reason we
multiply by 3 is because, as above stated, a variation of one
vibration per second in the fifth causes three beats per second.

_Example._--Take the first fifth in the table, C-128 to G-191.78, and
by the proper calculation (see example, page 147, Lesson XIII) we find
the exact fifth to this C would be 192. The difference, then, found by
subtracting the smaller from the greater, is .22 (22/100). Multiply
.22 by 3 and the result is .66, or about two-thirds of a beat per
second.

By these calculations we learn that the fifth, C-256 to G-383.57,
should hav