On the Institution of Music (De institutione musica)

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How Plato says consonances are made. Chapter [130].

Plato says that a consonance original: consonantia; a sounding together of two different notes that is pleasing to the ear is created in the ear in this way: It is necessary, he says, that a sharper sound be faster. Thus, when it strikes a low sound, it enters the ear quickly. Having struck the farthest part of that body, it is returned, as if by a second pulse, in a repeated motion. But it is then slower and not as fast as when it was first sent out by the initial impact. For this reason, it is also lower. Therefore, when it returns—now being "younger" lower in frequency due to losing energy—it meets the incoming low sound for the first time; they mingle together and, as he says, produce one consonance.

What Nicomachus thinks against Plato’s view. Chapter [131].

But Nicomachus Nicomachus of Gerasa, a mathematician whose work heavily influenced this text does not think this was truly stated. For he argues that a consonance is not made of similar things, but rather of dissimilar things coming into one and the same concord. If a low sound were mixed with another low sound, no consonance would occur, since similarity does not produce this musical concord, but rather dissimilarity. While they are distinct as individual voices, they are joined together when mixed.

But Nicomachus thinks a consonance is created in this way: He says there is not just one single pulse that emits a simple mode of sound. Instead, a single strike of one string hits the air many times and creates many voices. But because the speed of this occurrence is so great, one sound strikes against another, as it were. The gap between them is not perceived, and it comes to the ears as if it were a single voice. If, therefore, the occurrences of lower sounds are commensurable mathematically proportional with the occurrences of sharper sounds—according to the ratios we mentioned above—there is no doubt that this very commensuration mixes them together and creates a single voice, which is a consonance.

Which consonance excels another by merit. Chapter [132].

Among all the consonances we have mentioned, a judgment must be made, both by the ear and by reason,

as to which of them should be considered better. For the ear is affected by sounds—or the eye by sight—in the same way that the mind judges numbers or continuous quantity. If a line is set before us, nothing is easier for the eye or the mind than to perceive its double. Likewise, after the judgment of the double, the triple follows. In the same way, the parts of the triple are divided into the quadruple. Because the description of the double is easier, Nicomachus thinks the diapason term: diapason; the interval of an octave, based on the 2:1 ratio is the best consonance. After this comes the diapente term: diapente; the interval of a perfect fifth, based on the 3:2 ratio, which holds the middle, and then the diapente and diapason an octave plus a fifth, or a compound fifth, based on the 3:1 ratio, which is a triple ratio. He judges the rest according to this same mode and form. However, Ptolemy Claudius Ptolemy, the famous astronomer and music theorist does not view this in the same way; I will explain his entire opinion later.

How the things spoken should be understood. Chapter [133].

Nevertheless, we are attempting to treat everything that must be carefully explained hereafter in a summary and brief fashion. This is so that, in the meantime, the reader's mind may become accustomed to these ideas through a certain anticipation, before descending into a more vigorous and detailed treatment later.

For it was the custom among the Pythagoreans that when anything was said by their master Pythagoras, they did not dare to ask for a reason; for them, the authority of the teacher was his word. This was done so that, for a long time, the student's mind—strengthened by solid doctrine—might eventually find the reason for those same things for himself, even with no one teaching him. So also now, let us entrust to the reader's faith the things we have proposed: that he should believe the diapason exists in a double ratio, the diapente in a sesquialtera the ratio 3:2, the diatesseron term: diatesseron; the interval of a perfect fourth, based on the 4:3 ratio in a sesquitertia the ratio 4:3, the diapente and diapason in a triple ratio, and the bis diapason two octaves in a quadruple ratio. Later, this reasoning will be explained more diligently, as well as the ways in which musical consonances are gathered by the judgment of the mind, and all other things mentioned above will be built up in a more ample tradition.