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TEMPERAMENT.
243
On comparing the Equal with the Mean Tone Temperament, we see that the former has its Fifth better in tune than the latter, but that it is inferior in all its other intervals, especially in the Major Thirds. On the other hand it must be recollected that in Mean Tone Temperament, it is only possible to play in a limited number of keys, while in Equal Temperament all keys are equally good or equally bad. If it be desired therefore to play in all keys, the Equal Temperament is decidedly the better ; in fact, the only one possible under these circumstances.
As Equal Temperament is the one now universally employed on instruments with fixed keys, it will be of advantage to be able to compare its intervals with those of just Intonation, without the necessity of using the somewhat cumbrous vibration ratios. We can do this by employing a method devised by Mr. Ellis. Suppose a piano to be accurately tuned in Equal Temperament, the Octave being divided into twelve exactly equal parts. Further suppose each of these twelve equal semitones to be accurately divided into 100 equal parts; each of these minute intervals, Mr. Ellis has termed a Cent, so that there are 1,200 equal Cents in the Octave. Fig. 90 shows the magnitudes of the intervals in Equal and in True Intonation, expressed in these Cents.
The evils of Equal Temperament arise chiefly, of course, from the fact that overtones, which should be coincident, are not so, but produce audible beats. In addition to this, the Differentials, except in the case of the Octave, do not exactly correspond with any tones of the scale, and may generate beats with some adjacent tone, if this latter be sounding at the time. In the case of the Fifths and Fourths, these beats, being very slow, do not produce any very bad effects: for example, with the Fifth from C = 264 we have only one beat per second, thus:
792------------791
264 Differential Tone = 395-5 264 = 131-5 C, = 132. With the other intervals the case is different, more rapid beats being generated. The 2nd column of the following table shows the number of beats per second produced between partials which are