The removal of the difficulty had to be sought through some system of 'temperament,' that is, by deliberately falsifying some tones by a very small amount so that practical effects might either be truer or that the error might be so distributed as to be unnoticeable. Two principal systems of temperament or practical tuning have had historic importance, the 'mean-tone' system, which was in general use through the 18th century, and often held to, especially for organs, till much later, and the 'equal' system, which was first suggested just before 1700, came into more or less use during the 18th century, though not for the organ, and has now become universal. The former system sought to make certain selected keys or tonalities as good as possible at the expense of certain others which were outlawed. The latter system seeks to make all keys alike and therefore equally usable, but in doing so is forced to make all of them equally incorrect by small amounts.
The 'mean-tone' system is so called because it assumed that the interval
of a 'tone' (as C-D) is in tuning to be made halfway (or a 'mean')
between the larger or Pythagorean 'tone' (9/8) and the smaller (10/9), or,
in other words, that the major thirds were as far as possible to be true,
but the fifths a quarter-comma flat. The result of this, if C is taken as
the starting-point, is that E and Ab are correct, C#, F and A a quarter-comma
sharp, Eb, G and B a quarter-comma flat, D, F# and Bb exactly
midway between their two possible values, and Db, D#, Gb, G# and
A# more or less unusable. (By a slightly different application of the system,
however, G# might be made correct, but Ab unusable.) The deviations
of the worst intervals approached a half-semitone, and these collectively
were known as the 'wolf,' to 'drive' which 'out' of the other intervals and
into them was the object of tuning. This system was admirable so long as
composers confined themselves to certain scales or keys. The major keys
of C, G, D, F, Bb, Eb and the minor keys of C, G, D, were very good,
but all others bad (if G# were favored instead of Ab, A major would take
the place of Eb major and A minor that of C minor). Free modulation
was impossible, especially in the use of minor keys. On the other hand,
the sweetness of the major thirds in the 'good' keys was a decided
advantage.
The 'equal' system is so called because it assumes that the octave is to be divided into twelve exactly equal semitones, making every interval in every scale or key equal to the same interval in all other scales or keys. The result of this, if C is taken as the starting-point, is that F and G are almost exactly correct, D and Bb very good in certain relations, but the one sharp and the other flat in others, C#, D#, G# and A# decidedly sharper than they should be, Db, Eb and Ab rather flatter than they should be, and F# and Gb either moderately sharp or flat in some relations or
