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 have 1.29 beats:  nearly one and a third per second, and that the highest fifth of the temperament, F-341.72 to C-512, should be 1.74, or nearly one and three-quarters.  By remembering these figures, and endeavoring to temper as nearly according to them as possible, the tuner will find that his temperament will come up most beautifully.  This is one of the features that is overlooked or entirely unknown to many fairly good tuners; their aim being to get all fifths the same.

Finishing up the Temperament.—­If your last trial, F-C, does not prove a correct fifth, you must consider how best to rectify.  The following are the causes which result in improper temperament: