The Great Art of Consonance and Dissonance

...positioned. For example: If the number 9,240—representing the completely distinct mutations (mutationes): Different arrangements or permutations of a set of musical notes. of three notes appearing in the previous table—is subtracted from the number of combinations for 3 in this table (which is 10,648), it will leave a remainder of 1,408. These are the sought-after mutations of three notes within a triple octave (trisdiapason): A musical range spanning three octaves, or 22 notes in this system. that are partly similar and partly dissimilar. Likewise, the combination of 4 notes in the previous table, namely 171,560, subtracted from the combination of four notes in this table (234,256), will leave 62,696 mutations of 4 notes that are partly similar and partly different. However, so that you may better grasp the logic behind each, we place three tables here before your eyes, covering only one octave. The first shows the mutations that are entirely different; the second shows those that are partly similar and partly different (containing the results of subtracting the first and third columns from each other); and the third shows mutations that are a mix of different, identical, and similar notes. As follows:

	Multiplier
1	22
2	484
3	10,648
4	234,256
5	5,153,632
6	113,379,904
7	2,494,357,888
8	54,875,873,536

And so on into infinity.

First Table, containing distinct mutations of notes.	Second Table, of mutations partly the same and partly different; this is the difference between columns I and III.	Third Table, of mutations partly different, partly the same, and partly different and similar.
1. 22	1. 0	1. 22
2. 462	2. 22	2. 484
3. 9,240	3. 1,408	3. 10,648
4. 171,560	4. 62,696	4. 234,256
5. 4,316,008	5. 837,624	5. 5,153,632
6. 53,721,360	6. 60,669,544	6. 113,379,904
7. 85,941,760	7. 2,408,416,128	7. 2,494,357,888
8. 12,893,126,400	8. 42,382,746,936	8. 54,875,873,536

I shall add another kind of combination here: namely, the mutations that can be made from 22 notes where 21 are placed upon the same pitch (chordam): Literally 'string,' referring to a specific note or line on the musical staff. and one is placed on a different one. Second, how many mutations can be made from 22 notes where 20 always remain on the same pitch and 2 are on different ones; and so on until 11 against 11. From this point, the numbers will decrease in the same proportion that they grew. To explain this, nothing more seems to be required than examples (Paradigmata): Illustrative models or diagrams. of each.

These 22 notes allow no mutation.

This illustration shows a single musical staff containing twenty-two identical notes, represented as white diamond-shaped marks. They are all placed on the middle line of the staff and numbered 1 through 22. Because every note is the same, no reordering or 'mutation' is possible.

Series

The illustration displays a musical staff with twenty-two notes. The first twenty-one notes are all placed on the middle line, while the twenty-second note is raised to the space above it. This represents a sequence where only one element varies.

Mutations.
21 notes and 1 note are varied 22 times.