Thus you see that six things, all different, can be rearranged 720 times. If, however, two of them were the same, there are 360 variations; if 3 are the same, 120; if 4 are the same, 25 original: "25". Mathematically, the permutation of 6 items where 4 are identical should be 30 (720 ÷ 24), but the author is following the values printed in Table II on the previous page, which contains several calculation errors.; if 5 are the same, they can be changed six times; and if all 6 things are the same, there is 0 In modern combinatorics, we would say there is only 1 way to arrange identical items, but the author likely means there are zero additional variations possible. variation. You will proceed no differently regarding any other number of things contained within ten. We have composed this table according to the rules delivered a little before, so that it would not be difficult to extend it to any number whatsoever. However, we declare the use of this table most extensively in our Combinatorial Art original: "Arte nostra Combinatoria". This refers to Kircher’s broader work on the "Art of Knowledge," where he explores how all information can be categorized and combined., where we apply it to the principles of all arts and sciences; the curious reader will be able to return to that work in due time.

CHAPTER I.

On the Combinations of Notes.

Through several Problems, we will teach here the method and reasoning by which one may determine how, and how many times, any number of musical notes Notes (Notæ): The individual symbols representing musical pitches on a staff. may be varied and combined among themselves.

Combinatorial Problem I.

Let there first be a melody Melody (Vox): Literally "voice," but in this context referring to a specific sequence of musical notes. of three notes which ascends through a third, from C through D to E. The question is how many times it can be rearranged within this interval so that no note is placed twice on the same degree of the scale, but each is unique within that same interval? I answer that this can be done six times and no more; for since, according to Lemma I, three diverse things can be changed six times, and here there are three notes placed on three different degrees, it follows necessarily that they can also be changed six times within the interval of a single third. Indeed, for a further demonstration of this, nothing else seems to be required except a visual proof, which follows:

The illustration displays musical notation on a five-line staff with a C-clef. It shows six variations of a three-note melody (the notes C, D, and E) using square and diamond-shaped notes typical of the period. Each variation is numbered 1 through 6 beneath the staff to show the different possible permutations of the three pitches.

You see, therefore, in the example, how the proposed melody is changed six times so that no note is placed twice on any one degree, but each is different within the same interval; we have therefore achieved what was proposed.
No differently, a melody of four notes ascending step-by-step from C to F through the interval of a fourth can be changed 24 times according to Lemma I. For since the 4 places in the interval of a fourth are all different, the melody can be changed six times starting from the note Note (Chorda): Literally "string," a term used here to refer to a specific pitch or "chord" in the sense of a scale degree. C; and six times starting from note D; six times starting from note E; and finally six times starting the melodic permutations from note F. Moreover, four multiplied by six gives the total number of the proposed combination [24]. As is evident in the following example.

The illustration shows musical notation on a five-line staff with a C-clef, depicting sequences of four notes (C, D, E, and F) in various permutations. Six variations are shown in the first section, numbered 1 through 6 below the staff, all of which begin with the note C.