3.  In this system of muting, state definitely which string is
    tuned first after the continuous mute is removed.  Which second? 
    Which third?

    4.  After the unisons are finished in the temperament, which string
    is tuned next, if we go immediately from the temperament to the
    over-strung bass?  Which second?  Which third?

    5.  Upon beginning to tune the treble beyond the temperament, which
    string is tuned first?  Which second?  Which third?

    6. (a) How many mutes are used in tuning outside the strings of
       the temperament?

       (b) In what proportion 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.)