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.)

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:

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?