Arithmetic, Geometry, and Music

Arithmetica.

the third part of which quaternary cannot be found, and therefore it lacks the epitritus sesquitertius ratio. The second, however, which is nine, has the number twelve as its sesquitertius. The twelve, however, because it has a third part, is compared in a sesquitertius proportion to the number sixteen, which is dissolved by the section of the third part. Twenty-seven, however, because it is the third triple, has thirty-seven and so on as its sesquitertius, and again it is compared to forty-eight in the same proportion. If sixty-four were placed there, they would again fulfill the same power of proportion. These sixty-four could not be adapted again to any sesquitertius because they are not contained by a third part. Thus, this is found in all triples: that the number at the end of that same proportion has as many preceding it as its first departed from unity. And whoever has as many numbers of the same proportion above himself as the first of them lies from unity, the part by which the number compared to them could effect the same proportion is denied to be found. And this is the description of the triple.

g

And the description of the quadruple is according to this form, to which whoever approaches, having been instructed by the previous ones, will tremble for no reason; and he will note the same correspondence for the other multiples as well.

b

In which it is also clear, as was shown before, that the multiples are the first of the superparticulars. For doubles create sesquialters, triples create sesquitertii, and all multiples create all superparticulars in order. There is also something wonderful in this. For where the first width is double, and those which are placed alternately against the continuous ones are under them, they will be doubles according to the series of width. But if they are triples, the lower orders will surpass their own terms by triple multiplication. But in the quadruple, by the quadruple; and this does not fail in infinite speculation. But the angular ones of all of them must necessarily happen to be multiples. But there will be triples of doubles, quadruples of triples, and quintuples of quadruples; and all things will agree with one another according to the same immutable ratio of order. With these explained, let the discussion be turned to the following series of the work. Concerning from which superparticulars a multiple interval is made when a mediation is placed, and the rule for finding it. Chapter 3.

s

If, therefore, the first two superparticular species are joined, the first species of multiplicity arises. For the double is composed from the sesquialter and the sesquitertius, and every sesquialter and sesquitertius join to form a double. For three is the sesquialter of two, and four is the sesquitertius of three; but four is the double of two.

A diagram depicts a row of three cells containing the numbers 2, 3, and 4. A large arc labeled "double" spans from 2 to 4. Below the arc, lines connect 2 to 3 and 3 to 4.

Thus, therefore, the sesquialter and sesquitertius compose one double. But indeed, if there were a mediation and a double, between the double and the mediation one such mediation can be found that is sesquialter to one extreme and sesquitertius to the other. For if six and three are placed on opposite sides—that is, the double and the mediation—if four is placed in the middle, it contains the sesquitertius ratio to the number three, and the sesquialter to the six.

A diagram with three cells containing numbers 3, 4, and 6. A large arc above labeled "double" connects 3 and 6. Two smaller arcs below connect 3 to 4 (labeled "sesquitertius") and 4 to 6 (labeled "sesquialter").

s

Behold, therefore, it has been said: that the double is joined from the sesquialter and the sesquitertius, and these two species of superparticulars procreate the double, that is, the first species of multiple quantity. Again, from the first species of multiple—that is, from the double—and the first superparticular—that is, the sesquialter—the species of the multiple containing it, that is, the triple, is joined. For twelve is the double of the number six, and eighteen is the sesquialter to the twelve, which is triple to the number six.

A diagram with three cells containing numbers 6, 12, and 18. A large arc above labeled "triple" connects 6 and 18. Two smaller arcs below connect 6 to 12 (labeled "double") and 12 to 18 (labeled "sesquialter").

e

And with the same 6, 9, 18 placed, let nine be placed in the mediation. It will be sesquialter to the six, which is sub-double to 18, and 18 is triple to the six.

A diagram with three cells containing numbers 6, 9, and 18. A large arc above labeled "triple" connects 6 and 18. Two smaller arcs below connect 6 to 9 (labeled "sesquialter") and 9 to 18 (labeled "double").

e

From the double, therefore, and the sesquialter, the triple ratio of proportion arises, and when a resolution is made, it is called back into them again. But if here—that is, the triple number, which is the second species of the multiple—a second species of the superparticular is fitted, the form of the quadruple is continuously woven, and it will be dissolved into the same parts by natural partition, according to the mode which we demonstrated above.