De institutione musica III.1: Adversum Aristoxenum demonstratio superparticularem proportionem dividi in aequa non posse atque ideo nec tonum.
In the preceding book it was demonstrated that the consonance of the fourth is joined from two tones and a semitone, the fifth from three tones and a semitone, but that these semitones cannot make up a whole half of a tone, if they are considered and treated singly, and that therefore the octave in no way reaches six tones. But since Aristoxenus the musician, entrusting everything to the judgment of the ears, does not hold that these semitones are, as the Pythagoreans say, narrower than a half, but that, as they are called semitones, so they are halves of tones, we must dispute the matter again for a little while, and first demonstrate that no superparticular relation can be divided by a whole mean in a known number. For between two numbers containing a superparticular proportion, whether they are the first ones, whose difference is unity, or later ones, no middle number can be placed so that the proportion which the least holds to the middle, the middle holds to the extreme, as in geometric proportion. But a middle number placed between the same terms can either make the differences equal, so that there is equality according to the arithmetic mean, or will make the harmonic mean, or any other of those which we mentioned in the arithmetical books. If this is demonstrated, it cannot stand that the sesquioctave proportion, which is the tone, can be separated into halves, since every sesquioctave consists in the superparticular kind of inequality. This will be shown better by induction. For if, by examining the several proportions, namely the superparticular, none at all occurs which is divided into equal proportions by the interposition of a middle term, there is no doubt that a superparticular comparison cannot be divided into equal parts. But if something sung seems to the ears consonant when, to any voice, a voice distant by two tones and a whole semitone is compared, nature shows that this is not consonant; but since every sense is unable to grasp the least things, the sense of hearing cannot distinguish this difference, which goes beyond the consonant, but it will be detected if such a small portion grows through the same errors very frequently. For what is not clearly seen in the smallest, composed and joined, is seen when it has already begun to be great. From what proportion, then, must we begin? Shall we give a shortcut to the inquiry if we begin with that about which we inquire? That is, whether the tone can be divided into two equal parts or not. Now, therefore, we must treat of the tone and demonstrate how it cannot be divided into two equal parts. If anyone transfers this demonstration to the remaining superparticular comparisons, it will be demonstrated similarly that a superparticular cannot be separated into equal parts in a known and whole number. The first numbers containing a tone are 8 and 9. But since these follow each other so naturally that there is no middle number between them, I multiply them by two, the least number I can. They make 16 and 18. Between these falls the natural number 17. Therefore 18 to 16 is a tone, but 18 compared with 17 has it whole and a seventeenth part of it. The seventeenth part is naturally less than the sixteenth; the proportion contained under the numbers 16 and 17 is therefore greater than that contained under 17 and 18. Let these be set down in this way, and let there be A, B, C. A whole half of the tone will in no way fall between C and B. For the proportion CB is less than the proportion CA. The half must therefore be placed toward the greater part. Let the half be D. Since, then, the proportion DB, which is the half of the tone, is greater than the proportion CB, which is the lesser part of the tone, and the proportion AC, which is the greater part of the tone, is greater than the proportion AD, which is the half of the tone, and the proportion AC is sesquisextadecimal and CB sesquiseptendecimal, there is no doubt that the whole half falls between the sesquisextadecimal and the sesquiseptendecimal. But this can in no way be found in a whole number.
But since the number 17, compared with the number 16, has the proportion of a sesquisextadecimal, if we seek the sixteenth part of that number 17, it will be a unit and a sixteenth part of a unit. If we join this to the same number 17, we shall have 18 and a sixteenth part. If, then, 18 and a sixteenth part is compared with 16, it will rightly seem to exceed the measure of the tone, since only the number 18 keeps the sesquioctave proportion to it. Hence it comes about that, since the supersesquisextadecimal proportion, increased twice, exceeds the tone, it is not a whole half of the tone. For whatever by being taken twice exceeds something will be seen to be more than the half of that which it exceeds. Therefore the supersesquisextadecimal will not be the half of the tone. And for this reason no other proportion greater than the sesquisextadecimal can be the half of the tone, since the sesquisextadecimal itself is greater than a whole half of the tone. But since the sesquiseptendecimal proportion follows the sesquisextadecimal continuously, let us see whether that, multiplied twice, does not fill the tone. The term 18 holds the sesquiseptendecimal part of the number 17. If in the same proportion we compare another number with 18, it will be 19 and a seventeenth part. If we compare with the term 17, placed in the sesquioctave proportion, we shall have 19 and an eighth part. But the eighth part is greater than the seventeenth part; the proportion of the numbers 17 and 19 and an eighth is therefore greater than that contained in 17 and 19 and a seventeenth part, which are, that is, two sesquiseptendecimal proportions. Two sesquiseptendecimals do not, therefore, seem to fill one tone. The sesquiseptendecimal is therefore not the half of the tone, since those which when doubled do not fill the whole do not hold the half. For the half, doubled, is always equal to that of which it is the half.