Book VIII. Miraculous Music-making

...can be changed within a fifth (diapente): An interval spanning five notes of the scale, such as from C to G. so that all the intervals are different. Proceed as follows: take the number 20, which corresponds to 2 notes (marked as II) in the table, and divide it by the ordinary combination of two—that is, divide by 2—and 10 will result. These are exactly how many distinct mutations (mutationes): Different arrangements or permutations of a set of musical notes. two notes placed within a fifth allow. Similarly, 3 notes placed within a fifth allow the same number, and 4 notes within a fifth also allow the same number. This is true if you divide the combination number in this table—specifically 20 by 2, 60 by the combination of three (which is 6), and 120 by the combination of four (which is 24). Kircher is applying the mathematical rule for permutations where the order matters but certain constraints are applied to the positions of the notes. Truly, here are examples in which it clearly appears that each mutation is so distinct that no two notes occupy the same degree or space.
Another enigma.

I. Example of 3 notes within a fifth.

A series of ten musical fragments on a five-line staff, each containing three notes in white mensural notation (rhomboid-shaped notes) arranged in different intervals within the span of a fifth, numbered 1 through 10 below the staff.

II. Example of 4 notes within a fifth.

A series of ten musical fragments on a five-line staff, each containing four notes in white mensural notation arranged in different intervals within the span of a fifth, numbered 1 through 10 below the staff.

Problem VI. A most universal combinatory rule.

Given any number of notes, how many mutations can be made from any number of notes chosen from them, whether those notes are different, the same, or similar, or a mixture of similar and different.

In the preceding section, we showed the method by which, given any number of notes, we can derive different mutations from any chosen number of those notes. Now it must also be shown how mutations of notes can be made from any number assumed within a certain interval, whether those notes be different, similar, or partly similar and partly different.

Proceed in this way. Let the Great System (Systema maximum): The full range of the musical scale used in this period, typically spanning three octaves or 22 notes. of the musical scale consist of 22 notes, for example. Suppose someone desires to know how many times each musical interval assumed within it allows for precise mutations. For example, I desire to see how many 3 notes, variously arranged within the musical scale, undergo mutations. Act thus: multiply 22 notes by itself, and 484 will be produced; these are the mutations that 2 notes, variously arranged within the whole system, can undergo. Furthermore, multiply this number 484 again by 22, and you will have 10,648, the sought-after mutations of three notes within the said scale.

In this manner, 22 multiplied by 10,648 will produce 234,256, the number of mutations of four notes variously arranged within the scale. Thus, 22 multiplied by the recently found number 234,256 will give the mutations of [five] notes variously placed within the given System. By always multiplying the last product by the notes of the entire scale (that is, by 22), the number of all mutations for that number of notes which follows next in natural order will be produced. If you truly wish to know the number of notes that are partly similar and partly different, subtract the numbers of the preceding table from the numbers of this table, and the remainder will give the desired result.