18. This being thus demonstrated, let us see whether the sesquioctave will admit a division into two equal parts; which if it will not do, neither will a tone. However, in regard that 9 and 8, which make the first sesquioctave, have no middle interval, but both being doubled, the space that falls between causes two intervals, thence it is apparent that, if those distances were equal, the sesquioctave also might be divided into equal parts. Now the double of 9 is 18, that of 8 is 16, the intermedium 17; by which means one of the intervals becomes larger, the other lesser; for the first is that of 18 to 17, the second that of 17 to 16. Thus the sesquioctave proportion not being to be otherwise than unequally divided, consequently neither will the tone admit of an equal division. So that neither of these two sections of a divided tone is to be called a semitone, but according as the mathematicians name it, the remainder. And this is that which Plato means, when he says, that God, having filled up the sesquiterces with sesquioctaves, left a part of each; of which the proportion is the same as of 256 to 243. For admit a diatessaron in two numbers comprehending sesquiterce proportion, that is to say, in 256 and 192; of which two numbers, let the lesser 192 be applied to the lowermost extreme, and the bigger number 256 to the uppermost extreme of the tetrachord. Whence we shall demonstrate that, this space being filled up by two sesquioctaves, such an interval remains as lies between the numbers 256 and 243. For the lower string being forced a full tone upward, which is a sesquioctave, it makes 216; and being screwed another tone upward it makes 243. Which 243 exceeds 216 by 27, and 216 exceeds 192 by 24. And then again of these two numbers, 27 is the eighth of 216, and 24 the eighth of 192. So the biggest of these two numbers is a sesquioctave to the middle, and the middle to the least; and the distance from the least to the biggest, that is from 192 to 243, consists of two tones filled up with two sesquioctaves. Which being subtracted, the remaining interval of the whole between 243 and 256 is 13, for which reason they called this number the remainder. And thus I am apt to believe the meaning and opinion of Plato to be most exactly explained in these numbers.