Equal Temperament
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The common intonation of our day is the equal-tempered scale.  Equal temperament is the idea of equally distributing the intervals of each semitone (a half step) in the scale.  Like all the other systems, this system too has its advantages and disadvantages.

Equal temperament follows the well-temperament era in history, but it was not developed as a reaction or improvement of its predecessor.  Rather, it was a response to the popularization of fretted stringed instruments in the 17th and early 18th centuries.  The underlying characteristic of this intonation is that all keys sound the same (due to the equal intervals).  Thus, modulation is fully allowed without loss of quality.

Since the semitone interval is constant throughout, this implies that all whole steps, 3rds, 4ths, 5ths, etc. are also constant intervals – despite what is chosen as a root.  Thus, when a key modulates or is transposed, it will always sound as equally good or bad as the original.

12-tone Equal-Temperament

Today, we use a 12-tone equal-tempered scale.  This is why there are seven white keys and five black keys on a keyboard.  The octave is the only pure interval retained in this system.  Here is why.  Mathematically, 12-tone equal-temperament is defined as 

  (the intervals a are equal)

and 

Thus,

Solving for a (the semitone),

a is an irrational number, and it is impossible to create pure intervals other than the octave.  So in essence, every note with the exception of the root and its octave intervals is “out of tune”.

The 12-tone equal-temperament is a well-temperament, but not the only well-temperament.  This system meets the two criteria.  It is unique in that it is the only regular well-temperament.  In comparison to the well-tempered intonation Bach used, the equal-tempered system is more out of tune in most keys.  So why is it used over well-temperament?  Reasons include the convenience of modulation without retuning, the ease of transposing into a vocalist’s range, and the addition of more fretted instruments.

Equal-temperament can be adapted to make a scale with any natural number of tones in it, and for any desired root.  The equation

      

where n is the desired # of tones, will determine the ratio of the smallest interval.  Once the interval is found, the remainder of the scale has also been determined.