uce 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: =

(C 256 - E 322 5/10) (E 322 5/10 - G[#] 406 4/10) (G[#] 406 4/10 - C 512)
-------------------- ---------------------------- -----------------------
10 beats 13-1/10 beats 16 beats

We think the foregoing elucidation of Proposition I sufficient to
establish a thorough understanding of the facts set forth therein, if
they