The comparison, then, of these ratios of the minor third, D-F, and the fifth, D-A, with the perfect ratios of these intervals, shows that each is too small by the ratio expressed by the figures 80 to 81.  This is called, by mathematicians, the syntonic comma.

As experience teaches us that the ear cannot endure such deviation as a whole comma in any fifth, it is easy to see that some tempering must take place even in such a simple and limited number of sounds as the above series of eight tones.

The necessity of temperament becomes still more apparent when it is proposed to combine every sound used in music into a connected system, such that each individual sound shall not only form practical intervals with all the other sounds, but also that each sound may be employed as the root of its own major or minor key; and that all the tones necessary to form its scale shall stand in such relation to each other as to satisfy the ear.

The chief requisites of any system of musical temperament adapted to the purposes of modern music are:—­

    1.  That all octaves must remain perfect, each being divided into
    twelve semitones.

    2.  That each sound of the system may be employed as the root of a
    major or minor scale, without increasing the number of sounds in
    the system.

3.  That each consonant interval, according to its degree of consonance, shall lose as little of its original purity as possible; so that the ear may still acknowledge it as a perfect or imperfect consonance.

Several ways of adjusting such a system of temperament have been proposed, all of which may be classed under either the head of equal or of unequal temperament.

The principles set forth in the following propositions clearly demonstrate the reasons for tempering, and the whole rationale of the system of equal temperament, which is that in general use, and which is invariably sought and practiced by tuners of the present.

PROPOSITION I.

If we divide an octave, as from middle C to 3C, into three major thirds, each in the perfect ratio of 5 to 4, as C-E, E-G[#] (A[b]), A[b]-C, then the C obtained from the last third, A[b]-C, will be too flat to form a perfect octave by a small quantity, called in the theory of harmonics a diesis, which is expressed by the ratio 128 to 125.

EXPLANATION.—­The length of the string sounding the tone C is represented by unity or 1.  Now, as we have shown, the major third to that C, which is E, is produced by 4/5 of its length.

In like manner, G[#], the major third to E, will be produced by 4/5 of that segment of the string which sounds the tone E; that is, G[#] will be produced by 4/5 of 4/5 (4/5 multiplied by 4/5) which equals 16/25 of the entire length of the string sounding the tone C.