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