(C 256 — E 322 5/10) (E 322 5/10 — G[#] 406 4/10) (G[#] 406 4/10 — C 512)
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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 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
    A, by 5/6 of 25/36 of entire string, which equals 125/216
    3C, by 5/6 of 125/216 of entire string, which equals 625/1296