17. But first of all, we shall better understand what this leimma or remainder is and what was the opinion of Plato, if we do but call to mind what was frequently bandied in the Pythagorean schools. For interval in music is all that space which is comprehended by two sounds varied in pitch. Of which intervals, that which is called a tone is the full excess of diapente above diatessaron; and this being divided into two parts, according to the opinion of the musicians, makes two intervals, both which they call a semitone. But the Pythagoreans, despairing to divide a tone into equal parts, and therefore perceiving the two divisions to be unequal, called the lesser leimma (or defect), as being lesser than the half. Therefore some there are who make the diatessaron, which is one of the concords, to consist of two tones and a half; others, of two tones and leimma. In which case sense seems to govern the musicians, and demonstration the mathematicians. The proof by demonstration is thus made out. For it is certain from the observation of instruments that the diapason has double proportion, the diapente a sesquialter, the diatessaron a sesquiterce, and the tone a sesquioctave proportion. Now the truth of this will easily appear upon examination, by hanging two weights double in proportion to two strings, or by making two pipes of equal hollowness double in length, the one to the other. For the bigger of the pipes will yield the deep sound, as hypate to nete; and of the two strings, that which is extended by the double weight will be acuter than the other, as nete to hypate; and this is a diapason. In the same manner two longitudes or ponderosities, being taken in the proportion of 3: 2, will produce a diapente; and three to four will yield a diatessaron; of which the latter carries a sesquiterce, the former a sesquialter proportion. But if the same inequality of weight or length be so ordered as nine to eight, it will produce a tonic interval, no perfect concord, but harmonical enough; in regard the strings being struck one after another will yield so many musical and pleasing sounds, but all together a dull and ungrateful noise. But if they are touched in consort, either single or together, thence a delightful melody will charm the ear. Nor is all this less demonstrable by reason. For in music, the diapason is composed of the diapente and diatessaron. But in numbers, the duple is compounded of the sesquialter and sesquiterce. For 12 is a sesquiterce to 9, but a sesquialter to 8, and a duple to 6. Therefore is the duple proportion composed of the sesquialter and sesquiterce, as the diapason of the diapente and diatessaron. For here the diapente exceeds the diatessaron by a tone; there the sesquialter exceeds the sesquiterce by a sesquioctave. Whence it is apparent that the diapason carries a double proportion, the diapente a sesquialter, the diatessaron a sesquiterce, and the tone a sesquioctave.