The Universal Art of Music, Volume II (1650) - Page 24

720 divided by 36 original: "per 36", which is the square of the combination of three In this context, Kircher uses "combination" to mean factorial. The combination of 3 is 3! (3 × 2 × 1 = 6). The square of 6 is 36. (that is, 6), and the result will be 20, the sought number of permutations original: "mutationum"; for the stated melody can be changed that many times, as is evident in the given example.

Problem IV. Combinatorics.

Given a melody Melody (vox): A sequence of notes. of any number of notes, of which two are the same on one pitch Pitch (chorda): Literally "string," referring to the specific line or space on the musical staff. and three are the same on another pitch; to find the combination, that is, the number of permutations of the melodies.

Let there be a certain melody of nine notes, of which two are the same on pitch A, three are the same on pitch C, and two others are again on pitch A, as is evident in the present example; what is the permutation of the proposed melody? Proceed thus: multiply the combination of three (which is 6) by the square of the combination of two things (which is 4). You will have a divisor of 24. Divide the combination of nine 9 factorial (9!)., namely 362,880, by this, and 15,120 will result, which are the sought permutations in the given melody.

A short musical staff displays nine notes on five lines using a C-clef on the third line. The notes are grouped to illustrate the math: two notes on the second line (labeled A), three notes on the second space (labeled C), then two more on C and two more on A. The letters A A C C C C C A A appear below to guide the reader.

The combination of 4 4 factorial (4!). is 24, the square of which is 576.

If a melody of 22 notes should occur, of which two notes are repeated twice, and one note is repeated thrice, and two others are repeated four times, it is asked how many times the melody can be changed? The process is as follows: you will square the combination of 2 and you will have 4; you will multiply this by 6 (the combination of three) so that you have 24; then you will multiply this again by the square of the combination of 4, which is 576; and the product, 13,824, will give the divisor. By this, the combination of 22—that is, 1,124,000,727,777,607,680,000 This is the value of 22 factorial (22!).—when divided, will give 8,130,792,301,632,000, which is the number of permutations of the sought song.

Various Examples.

In a melody of nine notes.

Whenever all notes are in different positions, the melody will have 362,880 permutations, as shown.

Whenever two notes out of the nine have the same position, the melody will have 181,440 permutations.

Whenever three out of the nine are on the same pitch, the melody will have 60,480 permutations.

Whenever seven out of the nine occupy the same pitch, the melody will have 72 permutations.

Whenever 2 and 2 out of the nine are the same, the melody will have 90,720 permutations.

But when 2, 2, and 2 out of the nine are of the same position, the melody will be changed 45,360 times.

Diverse melodies of 9 notes based on position alone.

Six separate musical examples are shown on five-line staves with C-clefs. These provide visual proof for the calculations on the left:
1. A melody where every note is on a different line or space.
2. A melody where two notes share the same line.
3. A melody where three notes share the same space.
4. A melody where seven notes are all placed on the same line.
5. A melody with two pairs of repeating notes.
6. A melody with three pairs of repeating notes.

Glossary of Terms: Combinatorics (Combinatorium), Factorial (Combinatio), Permutations (Mutationes), Nine notes (Nouem notarum), Pitch/String (Chorda), Square (Quadratum), Number of permutation (Numerus mutationis).