We come, now, to the last third, G[#] (A[b]) to C, which completes the interval of the octave, middle C to 3C.  This last C, being the major third from the A[b], will be produced as before, by 4/5 of that segment of the string which sounds A[b]; that is, by 4/5 of 16/25, which equals 64/125 of the entire length of the string.  Keep this last fraction, 64/125, in mind, and remember it as representing the segment of the entire string, which produces the upper C by the succession of three perfectly tuned major thirds.

Now, let us refer to the law which says that a perfect octave is obtained from the exact half of the length of any string.  Is 64/125 an exact half?  No; using the same numerator, an exact half would be 64/128.

Hence, it is clear that the octave obtained by the succession of perfect major thirds will differ from the true octave by the ratio of 128 to 125.  The fraction, 64/125, representing a longer segment of the string than 64/128 (1/2), it would produce a flatter tone than the exact half.

It is evident, therefore, that all major thirds must be tuned somewhat sharper than perfect in a system of equal temperament.

The ratio which expresses the value of the diesis is that of 128 to 125.  If, therefore, the octaves are to remain perfect, which they must do, each major third must be tuned sharper than perfect by one-third part of the diesis.

The foregoing demonstration may be made still clearer by the following diagram which represents the length of string necessary to produce these tones. (This diagram is exact in the various proportional lengths, being about one twenty-fifth the actual length represented.)

Middle C (2C) 60 inches.
--------------------------------------------------
O                                                O

E (4/5 of 60) 48 inches.
--------------------------------------------
O                                          O

G[#] (A[b]) (4/5 of 48) 38-2/5 inches.
--------------------------------------
O                                    O

3C (4/5 of 38-2/5) 30-18/25 inches.
--------------------------------
O                              O

This diagram clearly demonstrates that the last C obtained by the succession of thirds covers a segment of the string which is 18/25 longer than an exact half; nearly three-fourths of an inch too long, 30 inches being the exact half.

To make this proposition still better understood, we give the comparison of the actual vibration numbers as follows:—­

Perfect thirds in ratio 4/5 have these vibration numbers:  =

1st third            2d third                3d third
(C 256 — E 320)     (E 320 — G[#] 400)     (G[#] 400 — C 500)
---------------     ------------------     ------------------
no beats            no beats                no beats

Tempered thirds qualified to produce true octave:  =