The Universal Art of Music, Volume II (1650) - Page 27

Book VIII. The Miraculous Art of Music

...the interval of a triple octave Triple Octave (trisdiapason): A musical span covering three full octaves. can be changed in such a way that they are never the same, or rather, so that the same note is never placed twice upon the same degree. Multiply 20 by 462 (which was the sum produced a little earlier from 20 times 22), and 9,240 will be produced—the number of mutations for three notes variously placed within a triple octave. If, indeed, you multiply this number 9,240 by 19, you will have 175,560, the number of mutations for 4 notes variously placed within a triple octave; and thus you will proceed in order all the way to 22, which will be the ordinary combination of the number 22. Truly, all these things become clearly known in the following diagram, in which you see three orders of numbers. The first 22 notes running in order and by degrees through the triple octave denote the others, which we call "assumed notes," signifying how many assumed notes from the 22 can be changed within that triple octave.

A vertical diagram and table for calculating musical combinations. On the left is a vertical musical stave spanning three octaves (trisdiapason), containing 22 notes. Next to it are several columns of indices: Roman numerals I-XXII and Arabic numerals 1-22, labeled at the bottom as "Notes of the triple octave" and "Assumed notes". To the right of these is a column of multipliers (labeled "prefix") starting at 22 and descending to 1, followed by a column of resulting large numbers representing the total possible mutations for that number of notes.

Notes (n)	Multiplier	Calculation Result
I	22	22
II	21	462
III	20	9,240
IV	19	175,560
V	18	3,160,080
VI	17	53,721,360
VII	16	859,541,760 The original OCR reads 85941760, but based on the progression 53,721,360 × 16, the result is 859,541,760.
VIII	15	12,893,126,400
IX	14	180,503,769,600 The printed digits in the source often vary from modern precise calculation due to historical typesetting or calculation errors; these figures represent the exponential growth of permutations.
X	13	2,346,549,004,800
XI	12	28,158,588,057,600
XII	11	309,744,468,633,600
XIII	10	3,097,444,686,336,000
XIV	9	27,877,002,177,024,000
XV	8	223,016,017,416,192,000
XVI	7	1,561,112,121,913,344,000
XVII	6	9,366,672,731,480,064,000
XVIII	5	46,833,363,657,400,320,000
XIX	4	187,333,454,629,601,280,000
XX	3	562,000,363,888,803,840,000
XXI	2	1,124,000,727,777,607,680,000
XXII	1	1,124,000,727,777,607,680,000

The true numbers ascribed to each degree of notes mark the number of mutations which any number of the assumed consonance original: "assumptæ consonantiæ"; referring to the group of notes chosen to be arranged. obtains within the triple octave. Thus, within a double octave...