De institutione musica II.31: Quibus proportionibus diapente ac diapason constent et quoniam diapason sex tonis non constet.
But since we have spoken at greater length of the consonance of the fourth, let us discuss the consonances of the octave and fifth more briefly and almost in bare numbers. The fifth consists of three tones and a semitone, that is, of the fourth and a tone. Let the numbers be set down which the description above comprehended: 192, 216, 243, 256. In this arrangement, therefore, the first term to the second and the second to the third hold the proportions of tones, but the third to the fourth that of the lesser semitone, as was shown above. If, then, the eighth part of 256 is added to it, there will be 288, which, compared with 192, makes the sesquialter space of proportion. Therefore there are three tones, if the first is compared with the second, the second with the third, and the fifth with the fourth. The comparison of the third to the fourth term holds the lesser semitone. But if the fourth is of two tones and the lesser semitone, and the fifth of three tones and the lesser semitone, and the fourth and fifth joined seem to make one octave, there will be five tones and two spaces of lesser semitones, which do not seem to fill one tone. The octave, therefore, is not composed of six tones, as Aristoxenus thinks. This appears plainly when it is set out in numbers. For let six tones be arranged in order, constituted, that is, in sesquioctave proportions. But six sesquioctave proportions are produced from the sixth octuple. Let six octuples be set down in this way:
From this last number, then, let the six tones constituted in sesquioctave proportion be placed in this way, the octuple terms being first set down, so that the eighth parts of the terms are joined at the sides of the terms themselves. Let the description be as follows:
The reason for this arrangement is as follows. The continuous line, which is called the limit, holds the octuple numbers. From the sixth octuple the sesquioctave proportions are drawn. Where we have written the eighth parts, they are the eighth parts of those numbers beside which they lie. If these are added to the numbers beside which they lie, they produce the later numbers.
For example, in the first, which is 262,144, its eighth is 32,768. If these are joined to each other, they make the later number, which is 294,912. And the same is found in the rest. If, then, the last number, which is 531,441, were the double of the first number, which is 262,144, the octave would rightly seem to consist of six tones. But now, if we seek the double of the least number, that is, of the first, it will be less than the number which is the greatest and highest. For the double of the number 262,144, which holds the consonance of the octave to it, is 524,288. This is therefore less than the number which holds the sixth tone, that is, 531,441. The consonance of the octave is therefore less than six tones. And that by which six tones exceed the consonance of the octave I call the comma, which consists in the least numbers 524,288 and 531,441. But what Aristoxenus thinks of these matters, who gave all judgment to the ears, I shall relate elsewhere. Now, to avoid tedium, I shall condense the course of the book.