The Universal Art of Music, Volume II

Musical notation is shown on a five-line staff, illustrating six bars of a four-note melody, numbered 8, 9, 10, 11, 12, and 13. The clef is a C-clef positioned on the third line, common in Renaissance and Baroque vocal music.

You see in this example that all the voices Voice (vox): In this context, Kircher uses "voice" to mean a specific melodic sequence or arrangement of notes. are changed, while two notes always remain on the same pitch Pitch (chorda): Literally "string," referring to the specific line or space on the musical staff representing a tone.; with the rest changing their starting point. Specifically, in the first 6 permutations, two notes remain on the pitch F (ut), which is also where the melody begins. In the other 6 permutations, three notes always begin on G; and 3 on A, with the remaining two notes always staying on F.

In this way, a melody of eight notes arranged step-by-step through an octave contains 40,320 permutations. If, however, two of the eight notes—arranged in any way—touch the same pitch, that melody allows for 20,160 permutations according to Combinatorial Table II The number 20,160 is half of 40,320, because the two identical notes are interchangeable.. If three notes touch the same pitch, it allows 6,720. If 4 touch the same pitch, 1,680. If 5 touch the same pitch, 336. If 6, then 56. Finally, if 7 touch the same pitch, it will allow only 8 permutations, as the table beautifully demonstrates.

Combinatorial Problem III.

Given a melody of any number of notes, including any number of doubles, triples, or quadruples of notes existing upon the same or different pitches, to find the permutations of the melody.

First, let there be proposed a melody of 5 notes, of which two notes are on A (la, mi, re) and just as many are on B (fa, la, mi), as is clear in the example. The question is: how many times can it be changed since there are two pairs original: "binarij". In combinatorics, these are sets of two identical items. here? You first multiply these by themselves (or square them), and you will have a divisor. By this divisor, the total combination of five things (namely, 120) when divided leaves 30 as the sought-after combination or permutation of the proposed melody Mathematically, Kircher is calculating 5! / (2! * 2!). 120 divided by 4 equals 30.. Indeed, for the understanding of this problem, nothing else is required but a visual inspection and examination of the following example.

A series of musical staves arranged in three horizontal rows displays 18 numbered permutations of a five-note melody. Each row contains six bars, numbered 1 through 18. The notes are a combination of square and diamond shapes on a five-line staff with a C-clef.