Chapter Three

If the quantity of point $π$ original: $π$; used here as a geometric marker on the string is doubled at B-square original: b. quadr.; the square-shaped 'b' used in medieval notation to indicate B-natural, the source of our modern natural $\natural$ and sharp # symbols and Q, let it be marked as B. But again, we shall divide the space between M and B in the middle, and we will mark the middle point of the division with the letter K. If we double the quantity of Z and K, we place C at the end of the doubling. But then G should be situated at equal distances between E and B-square. If, however, we divide G into two equal parts, it will be marked with the letter O. Thus, the entire monochord is divided by a proper partition, as you may recognize in the following figure.

The correct understanding of the given divisions

Here the gut or extended string is tuned

p
o
n
m
l
k
$\natural$ The square 'b' or B-natural
h
g
f
e
d
c
b
a

The tying of the string

A vertical diagram of a monochord string, labeled with alphabetic pitch notation (a through p) to illustrate mathematical divisions and musical intervals. This is a precise technical diagram of a monochord, a fundamental tool in medieval and Renaissance music theory for demonstrating the mathematical ratios of musical intervals.

Alternatively, now that the monochord has been divided, it remains to consider those quantities more deeply and precisely. It must be known, therefore, that the entire distance between A and B is called a tone, and by the Greeks a phtongos original: phtongon; a Greek term for a musical sound or a specific pitch, which among us is interpreted as a "sound." But this will be more easily understood by the following example: let the string be struck while extended to its full length, and let the sound be noted. Then let a finger or some other thinner object, not spread over a great width, be placed under the string; and let the string be struck again. It will happen that it emits a somewhat higher sound. When you wish to compare the higher sound with the lower sound (or the lower with the higher), the distance will be that of a tone. However, what you compare will become a semitone, not because it is half of a tone In the Pythagorean tuning system used in the Middle Ages, a semitone is mathematically slightly less than half of a whole tone, but because the voice does not ascend to the full integrity of a tone; it is called an "imperfect tone." Nevertheless, the practical musician should not be overly concerned with this imperfection of the semitone, since it is a highly theoretical original: speculativa; referring to "musica speculativa," the study of music as a mathematical science matter and removed from practical singers. But it is necessary that the practitioner knows it is not a perfect tone. Indeed, we shall defer subtle disputes of this kind to the second book.

Furthermore, that quantity or distance extended between C and D is a tone, and between D and E is likewise a tone. But between E and F there is a semitone; between F and G a tone; between G and H a tone; between H and I a semitone; between I and B-square a semitone; between B-square and K a semitone; between K and L a tone; between L and M a tone; between M and N a semitone; then N to O a tone; and O to P a tone. And thus the sounds which arise from the striking of related and neighboring divisions, when compared to each other, emit either a tone or a semitone. But the comparison of distant tones produces several types of species. Thus, because the space between C and E consists of two tones, it is called a ditone or bitone equivalent to a major third. However, the distance from A to C is called a semiditonus equivalent to a minor third, because it is formed as an imperfect ditone with half a tone removed. But B to E is a diatessaron a perfect fourth because it is capable of four voices, or it is called a tetrachord from the Greek tetra (four) and chorde (string) which is the division and distance of four strings. For one string or voice is B, another C, the third...