De institutione musica III.12: In qua numerorum proportione sit comma et quoniam in ea, quae maior sit quam LXXV ad LXXIIII, minor quam LXXIIII ad LXXIII.
First, then, I say that the numbers which contain the comma hold between themselves a greater proportion than 75 to 74 and less than 74 to 73. This will be demonstrated as follows. And first it must be recalled that six tones exceed the octave by a comma. Let A be 262,144, and B the number holding the consonance of the octave to it, set in the double, 524,288; and let C be distant from the number A by six tones, and let it be 531,441, all of which are to be gathered from the arrangement of the tones in the second book. Between B and C, then, the proportion of the comma is contained. I take away the number B from the number C; there remains D, set in 7,153 units, and this number D is less than the seventy-third part of the number B, but greater than its seventy-fourth part. For if I multiply the same number D, which is 7,153, seventy-three times, I get a number E set at 522,169 units; if I multiply it seventy-four times, a number F, 529,322. Of these, E, which is increased by seventy-three times, is less than the number B, and F, increased by seventy-four times, is greater than the number B. It was rightly said, therefore, that D is less than the seventy-third part of B, and greater than its seventy-fourth. Wherefore the number C exceeds the number B by a part of B less than a seventy-third, but greater than a seventy-fourth. The proportion of C to B is therefore greater than 75 to 74 and less than 74 to 73. For in the former the unit is a seventy-fourth part of the lesser, in the latter the same unit is a seventy-third part.
The same may be explained otherwise, taking first this for granted: that if the proper difference of numbers is added equally to the numbers in a proportion, the proportion contained between those that result after the addition will be less than that between the former, which before any addition were distant by some proportion; as six and four, if to both their difference, that is two, is added, make 8 and 6; but between 6 and 4 the sesquialter proportion is contained, between 8 and 6 the sesquitertian; and the sesquitertian proportion is less than the sesquialter. This having been so premised, let the numbers above which contained the proportion of the comma be set down, that is 531,441, and let it be A. Let B also be 524,288. Let their difference be C, 7,153. Let the number C measure the greater number, which is A, seventy-five times. If then I multiply the number C seventy-five times, I get the number D, which is 536,475. The number D therefore precedes the number A, which is E, by 5,034. Again let the number C measure the number which is B seventy-four times, and let it be multiplied. The number F will therefore be 529,322, which is greater than the number B by the same E.
The number D exceeds A by the number 5,034, and the number B is exceeded by F by the same number E. If, then, we add the number E to the number A, D will result; if we add the same E to the number B, F will result. But D has been increased seventy-five times, that is, by C multiplied, and F by C multiplied seventy-four times. D and F therefore hold between them the proportion which 75 has to 74. But D and F are A and B with E added to them. A greater proportion must therefore be contained between A and B than between D and F. For D and F were made from the numbers A and B with E added. The proportion between D and F is therefore less than between A and B. But between D and F is the same proportion as between 75 and 74. Between A and B, therefore, the proportion is greater than between 75 and 74. And A and B contain the comma; the proportion of the comma is therefore greater than 75 to 74.
Since, then, we have shown that the proportion of the comma is greater than that which the numbers 75 and 74 contain when compared, we must now show how the numbers containing the space of the comma hold between themselves a proportion less than 74 compared with 73. This will be shown in the following way. We must first recall what we said in the second book, when we spoke of the measure of the difference. For if from any proportion we take away the difference of the numbers that contain it, those that are left will hold a greater proportion than the numbers that were there before the diminution of the difference. Let there be 8 and 6. From these I take away their own difference, that is 2; there are 6 and 4. But in the former the sesquitertian proportion is contained, in this the sesquialter. The sesquialter proportion is greater than the sesquitertian. Let there be, then, the same A and B, which are described above, and let their difference be C. I multiply the difference C by seventy-four; I get the number F, namely 529,322, which, compared with the number A, is exceeded by the number G, namely 2,119. Again let the same C be multiplied seventy-three times; they will make the number K, that is 522,169, which, compared with B, is exceeded by the same G, by the same 2,119. When G is taken away from the numbers A and B, F and K result. A and B will therefore hold a less proportion than F and K. But F and K hold the proportion which 74 has to 73. For they were made by multiplying these by C. The proportion of A and B, the numbers containing the comma, is therefore less than 74 to 73. But it was shown a little before that the same proportion of the comma is greater than 75 to 74. It has therefore been shown that the numbers which contain the comma have a greater proportion between themselves than 75 to 74, and a less than 74 to 73, which was to be shown.