The Great Art of Consonance and Dissonance

...permutations, the total of which is so great that if someone were to write down only a thousand of these same changes every day, he would labor in describing them for 22,260,896,143 years. This is a thing altogether incredible to those who do not know the power of numbers and the nature of combination, which are nevertheless easily demonstrated from the preceding arguments. From this it is also clear how an inconceivable multitude of permutations can emanate from any motet Motet (Motecta): A polyphonic musical composition, usually on a sacred text. consisting thereafter of 50, 100, or 200 notes. Certainly, if the entire machine of the world were filled with paper, it would not suffice to contain all the permutations even in a combination of 30 notes.

Problem V. Universal.

This proposition has something wonderful about it, from which it seemed best to begin more deeply; and so that we may more easily explain our meaning, we shall begin from the very example of the proposition. Let someone from among the Music-makers Music-makers (Musurgis): A term Kircher uses for composers or scholars of the musical arts. propose this combinatoric riddle: having set in order 22 notes, first all different, ascending through a triple octave Triple Octave (trisdiapason): A musical interval spanning three full octaves., exactly as many as there are notes in the musical scale of Guido The "Guidonian Scale" refers to the system of hexachords and the "Guidonian Hand" developed by Guido of Arezzo, which formed the basis of Western music theory in Kircher's time., as appears in this present diagram.

It is asked first, how many times those notes can be simply changed, and we shall immediately satisfy this question through Lemma I and Problem I. Since all are different, they will admit the customary combination of the number 22, as Table I shows, namely: 1,124,000,042,171,709,440,000. This massive number represents 22 factorial (22!), the total possible arrangements of 22 unique notes.

It is asked therefore secondly, how many times 2 notes can be changed within the space of a triple octave, and how many times again 3, 4, 5, 6, 7, 8, 9, and so on up to the number 22 exclusive—how many times, I say, within a determined triple octave interval a voice of however many notes assumed from the 22 notes is changeable? In order that this may be known, do thus: Since 22 notes have been assumed, multiply 22 by 21 (that is, by the next smaller number), and 462 will be produced. And so many times can two notes be arranged within a triple octave interval, always differing. The reason for this is that since all 22 notes are different and ascending by degrees, it follows consequently that the order can change twenty-one times. For the first change occurs through 21 degrees. The second change will occur when the notes are arranged thus, or ascend through a third. The third change of notes will occur through a fourth. The fourth through a fifth. The fifth change through a sixth, and so on for the others up to 22, which is the triple octave. Since, therefore, in 22 notes there are 21 different changes, these consequently multiplied by 22 will produce the proposed number 462. So many times, therefore, can two notes be changed within a triple octave, and no more, as the proposition requires.

All of which things must be understood regarding the 462 different permutations of two notes variously arranged within the interval of a triple octave, as is clear in the attached examples. From this, if you desire to know how many times three notes within the interval...

A musical diagram displays a vertical scale of 23 notes on a staff. To the left, the notes are numbered 1 through 23. To the right, brackets identify various musical intervals: "Triple Octave" (Tridiapason) spans the whole scale; "Double Octave" (Disdiapason); "Octave with a Fifth" (Diapa. cum Diape.); "Fifth" (Diapente); "Octave" (Diapason); and "Fourth" (Diatessaron). Diamond-shaped notes ascend the staff from bottom to top.