This system is used by a great number of very successful tuners, and it has but one appreciable disadvantage, which is that involved in changing from fifths upward to fifths downward.  This difficulty is easily overcome, if it were all there is to encounter; but in practice, we find that after tuning the intervals in the above succession down to the last step in the first series, middle C will often have changed pitch somewhat, and the last five tones with their octaves tuned from it will not be in true harmony with the intervals tuned in the first series.  For this reason it is better to go on through, as in our system, tuning by fifths upward, and if there is any change of pitch in the first tones tuned, they may be more easily corrected by going over them in the same way as at the start; also, the amount of difficulty in locating discrepancies is greatly lessened.

SYSTEM C.

The following system is one that is followed by many good tuners of the present day and has many advantages.  To use this system successfully, however, one must be familiar with the number of beats necessary in each interval used.

Take 1F as a standard.

By 1F, tune 2C, fifth above. 
By 1F, tune 1B[b], fourth above. 
By 1F, tune 1A, major third above. 
By 1F, tune 2D, sixth above. 
By 1F, tune 1A[b], minor third above. 
By 1F, tune 2F, octave above. 
By 2C or 2D, tune 1G, fourth or fifth below. 
By 1G, 1A or 2C, tune 2E, sixth, fifth or third above. 
By 1G or 2E, tune 1B, third above or fourth below. 
By 1A or A[#], tune 2C[#], major or minor third above. 
By 1A[b], 1B[b] or 1B, tune 2E[b], fifth, fourth or major third above. 
By 1B[b], 1B, 2C[#] or 2E[b], tune 1F[#], major third, fourth, fifth
   or sixth below.

As each step is taken in this system, the tone tuned is tested with any or all of the tones previously tuned.

You will notice that six tones are tuned by the first standard, F. Therefore, if any error is left in any one of the intervals it exists in this only and is not transmitted to other tones, if corrected before such other tones are used to tune by.

The numerous tests possible, early in the system, and the small compass used, one octave, may be said to be the chief advantages of the system.

The intervals used are the minor and major third, perfect fourth and fifth, and major sixth.  The thirds and sixths beat from about 7 to nearly 12 per second.  The exact number of beats for each step in the system may be calculated from the “Table of Vibration Numbers” in Lesson XIII.  For instance, take middle C (2C) at 256, and its major third, 2E.  The exact third, determined by multiplying 256 by 5/4, is found to be 320.  By reference to the table, we find the tempered third vibrates 322.54.  The difference then is 2.54 vibrations per second, and, knowing that a difference of one vibration from the exact major third produces 4 beats, we simply multiply 2.54 by 4 and we have 10.16, the number of beats we should hear per second when this third is tempered correctly.  Other intervals may be figured out in like manner by reference to the various tables given.