ortion is the number of times the mute is
changed to the number of strings tuned?

7. (a) How many mutes are used in tuning the treble beyond the
temperament?

(b) In what proportion is the number of times the mute is
changed to the number of strings tuned?

8. Which pairs of pins are marked in the square piano to guide the
tuner in placing his hammer? Also, how are they marked?

9. Having marked your pins as instructed, how would you find the
pins belonging to a pair of strings struck by F on key-board? How
those struck by G[#]?

10. Tell what you can of the requirements necessary to insure that
a piano will stand in tune.

LESSON XII.

=MATHEMATICS OF THE TEMPERED SCALE.~

One of the first questions that arises in the mind of the thinking
young tuner is: Why is it necessary to temper certain intervals in
tuning? We cannot answer this question in a few words; but you have
seen, if you have tried the experiments laid down in previous lessons,
that such deviation is inevitable. You know that practical scale
making will permit but two pure intervals (unison and octave), but you
have yet to learn the scientific reasons why this is so. To do this,
requires a little mathematical reasoning.

In this lesson we shall demonstrate the principles of this complex
subject in a clear and comprehensive way, and if you will study it
carefully you may master it thoroughly, which will place you in
possession of a knowledge of the art of which few tuners of the
present can boast.

In the following demonstrations of relative pitch numbers, we adopt a
pitch in which middle C has 256 vibrations per second. This is not a
pitch which is used in actual practice, as it is even below
international (middle C 258.65); but is chosen on account of the fact
that the various relative pitch numbers work out more favorably, and
hence, it is called the "Philosophical Standard." Below are the actual
vibration numbers of the two pitches in vogue; so you can see that
neither of these pitches would be so favorable to deal with
mathematically.

International--3C--517.3. Concert--3C--540.

(Let us state here that the difference in these pitches is less than a
half-step, but is so near that it is generally spoken of as being just
a half-step.)

Temperament denotes the arrangement of a system of musical sounds in
which _each one_ will form a serviceable interval with _any one_ o