are studied over carefully a few times. If everything is not
clear at the first reading, go over it several times, as this matter
is of value to you.

QUESTIONS ON LESSON XII.

1. Why is the pitch, C-256, adopted for scientific discussion, and
what is this pitch called?

2. The tone G forms the root (1) in the key of G. What does it
form in the key of C? What in F? What in D?

3. What tone is produced by a 2/3 segment of a string? What by a
1/2 segment? What by a 4/5 segment?

4. (a) What intervals must be tuned absolutely perfect?

(b) In the two intervals that must be tempered, the third and
the fifth, which will bear the greater deviation?

5. What would be the result if we should tune from 2C to 3C by a
succession of perfect thirds?

6. Do you understand the facts set forth in Proposition I, in this
lesson?

LESSON XIII.

~RATIONALE OF THE TEMPERAMENT.~ (Concluded from Lesson XII.)

PROPOSITION II.

That the student of scientific scale building may understand fully the
reasons why the tempered scale is at constant variance with exact
mathematical ratios, we continue this discussion through two more
propositions, No. II, following, demonstrating the result of dividing
the octave into four minor thirds, and Proposition III, demonstrating
the result of twelve perfect fifths. The matter in Lesson XII, if
properly mastered, has given a thorough insight into the principal
features of the subject in question; so the following demonstration
will be made as brief as possible, consistent with clearness.

Let us figure the result of dividing an octave into four minor thirds.
The ratio of the length of string sounding a fundamental, to the
length necessary to sound its minor third, is that of 6 to 5. In other
words, 5/6 of any string sounds a tone which is an exact minor third
above that of the whole string.

Now, suppose we select, as before, a string sounding middle C, as the
fundamental tone. We now ascend by minor thirds until we reach the C,
octave above middle C, which we call 3C, as follows:

Middle C-E[b]; E[b]-F[#]; F[#]-A; A-3C.

Demonstrate by figures as follows:--Let the whole length of string
sounding middle C be represented by unity or 1.

E[b] will be sounded by 5/6 of the string 5/6
F[#], by 5/6 of the E[b] segment; that is, by 5/6 of
5/6 of the entire string, which equals 25/36