De institutione musica II.29: Demonstrationes non esse CCXLIII ad CCLVI toni medietatem.
I prove, therefore, that the distance 243 to 256 is not the whole measure of a half tone. For the difference between 243 and 256 is contained in only 13 units, and 13 is less than the eighteenth part of the lesser number, but more than its nineteenth part. For if you multiply 13 by eighteen, you will have 234, which will in no way equal 243; if you multiply by nineteen, it will exceed it, although every semitone, if it holds a whole half of the tone, ought to be placed between the sixteenth part and the seventeenth part, which will be demonstrated later. Now it will be clear that such a distance of a semitone, doubled upon itself, cannot fill one space of a tone. Come, then, as 256 stands to 243, let us arrange two such proportions continuous with each other according to the rule described above. Let 256 and 243 be multiplied each by itself, and let the greatest term be 65,536. Likewise let 243 grow by its own number, and let the least term be 59,049. Again, let 256 and 243 be multiplied together. The number will therefore be 62,208. Let this be placed in the middle in this way:
256 and 243 are therefore in the same proportion as 65,536 to 62,208. And likewise 62,208 to 59,049. But the greatest of these terms, which is 65,536, will not make one whole tone to the least, which is 59,049. But if the proportion of the first to the second, which is equal to the proportion of the second to the third, were proved to be that of a whole semitone, the two halves joined would necessarily make one tone. Now, however, since the proportion of the extreme terms is not sesquioctave, it is manifest that these two spaces do not properly seem to be halves of tones. For whatever is the half of anything, if it is doubled, makes that of which it is called the half. But if it cannot fill that, the doubled portion is less than the half part; if it overflows and exceeds, it is more than the half part. Besides, it will be proved that 65,536 does not make the sesquioctave proportion if it is compared with 59,049 units, if the eighth part of 59,049 is added to the same according to the rules which were stated in the arithmetical books. Since this does not consist in whole numbers, we leave that same eighth part to be computed by the diligence of readers. It is plain, therefore, that the proportion which is established in 256 and 243 is not a whole half of the tone. Wherefore that which is truly called a semitone is a part of the tone less than the half.