Equal Temperament.—­Equal temperament is one in which the twelve fixed tones of the chromatic scale[D] are equidistant.  Any chord will be as harmonious in one key as in another.

[D] The chromatic scale is a succession of all the half steps in the compass of one octave.  Counting the octave tone, it contains thirteen tones, but we speak of twelve, as there are only twelve which differ in name.

Unequal Temperament.—­Unequal temperament was practiced in olden times when music did not wander far from a few keys which were favored in the tuning.  You will see, presently, how a temperament could be set in such a way as to favor a certain key (family of tones) and also those keys which are nearly related to it; but, that in favoring these keys, our scale must be constructed greatly to the detriment of the “remote” keys.  While a chord or progression of chords would sound extremely harmonious in the favored keys, they would be so unbalanced in the remote keys as to render them extremely unpleasant and almost unfit to be used.  In this day, when piano and organ music is written and played in all the keys, the unequal temperament is, of course, out of the question.  But, strange to say, it is only within the last half century that the system of equal temperament has been universally adopted, and some tuners, even now, will try to favor the flat keys because they are used more by the mass of players who play little but popular music, which is mostly written in keys having flats in the signature.

Upon the system table you will notice that the first five tones tuned (not counting the octaves) are C, G, D, A and E; it being necessary to go over these fifths before we can make any tests of the complete major chord or even the major third.  Now, just for a proof of what has been said about the necessity of flattening the fifths, try tuning all these fifths perfect.  Tune them so that there are absolutely no waves in any of them and you will find that, on trying the chord G-C-E, or the major third C-E, the E will be very much too sharp.  Now, let your E down until perfect with C, all waves disappearing.  You now have the most perfect, sweetest harmony in the chord of C (G, C, E) that can be produced; all its members being absolutely perfect; not a wave to mar its serene purity.  But, now, upon sounding this E with the A below it, you will find it so flat that the dissonance is unbearable.  Try the minor chord of A (A-C-E) and you will hear the rasping, throbbing beats of the too greatly flattened fifth.

So, you see, we are confronted with a difficulty.  If we tune our fifths perfect (in which case our fourths would also be perfect), our thirds are so sharp that the ear will not tolerate them; and, if we tune our thirds low enough to banish all beats, our fifths are intolerably flat.

The experiment above shows us beautifully the prominent inconsistency of our scale.  We have demonstrated, that if we tune the members of the chord of C so as to get absolutely pure harmony, we could not use the chord of A on account of the flat fifth E, which did duty so perfectly as third in the chord of C.