Dialogue on Ancient and Modern Music

On Music.

The Octave is said to be the Queen of consonances.

BAR. Everything shall be granted to you. It is necessary, first of all, to commit well to memory (following, however, the Syntonic of Ptolemy Syntonic of Ptolemy: A tuning system, also known as Just Intonation, that uses simple ratios for thirds and sixths, unlike the older Pythagorean system. as I have said) those numbers in their lowest terms that separately contain each of the intervals found within the Queen of Consonances The Octave.. These intervals, according to the opinion of the authors of these matters, do not exceed fifteen in number. Starting from the smallest, I say that the Comma is contained in its radical terms by the proportion called Sesquioctogena original: "Sesquiottantesima"; the ratio 81:80., between these numbers ———— 81. 80.

Musical intervals of the Syntonic [system] and the numbers that contain them. Why the Octave is called Diapason. Corrupted names of the intervals.

To which the Greeks perhaps gave the name Diapason original: "Diapaſon"; from the Greek "dia pason chordon," meaning "through all the strings.", because it contains within itself (according to its meaning) each of the named intervals, and it alone stands for every other. But it should be noted that these intervals—aside from the Fourth, Fifth, Octave, and the Sesquioctave or Tone—were never understood by such names by any of the ancient or modern musicians, except by the authors of these matters and their followers. Their corruption (due to the deformity they have when compared with the things already understood and known by those names) generates no small confusion and disturbance in the minds of those who read their writings. They have borrowed these names from the Diatonic Ditonic Diatonic Ditonic: The Pythagorean tuning system used in the Middle Ages, which results in "harsh" major thirds (81:64) compared to the "sweet" thirds (5:4) of the Syntonic system. and masked Ptolemy’s Syntonic with them to color certain designs of theirs, as you will better understand. We shall see clearly in the proper place how little this "garment" fits and how unsuitable it is for that system.

STR. For what reason did Pythagoras establish the Fifth between three and two (assigned to us as the terms of the Diapente The Fifth.), or the Fourth between four and three (in which he established the Diatessaron The Fourth.), rather than the Octave?

BAR. This was not a human work or invention, but one of nature. It is quite true, however, that Pythagoras (as I have said) was the first to consider it.

STR. By what was he led, I pray?

A way to hear any desired interval in its proportion.

BAR. Listen: stretch two strings of the same length, thickness, and quality in unison upon a flat surface. Then, with a compass, divide one of them into as many equal parts as there are units in the larger number of the interval you wish to hear from those strings. Then, by means of a fixed bridge bridge: original "scannello"; a small block used to stop a string at a specific point to change its pitch., deprive one of the strings of as many parts as the larger number exceeds the smaller. When the two strings are then struck together, you will hear the whole (representing the larger number) and from the parts (representing the smaller) the dissonance or consonance contained in the proportion thus applied to those strings. And if one wished to hear any desired interval on a single string, they could do so in this other manner:

Another way.

First, divide the string into as many equal parts as there are units contained in the smaller numbers of its proportion when summed together. Then place the bridge as a divider between one term and the other number. In striking the two parts of the whole string (divided by the bridge in this way) separately or at the same time, you will hear the sought-after interval. For greater understanding, here is a practical example.

Example.

Let us suppose we wish to hear, in this second manner, the Diatessaron (the Fourth), which is contained in its true form by numbers having a Sesquitertial The ratio 4:3. proportion between them. Add together its smaller terms, which are, as I have shown you, 4 and 3. These, added together, make seven. Then divide the proposed string into that many equal parts, and place the bridge over the point that separates the four parts from the three. When these parts are struck together, or one after the other, you will hear the consonance of the Diatessaron sounding between them.

There are seven species of the Diapason.

And to reveal to you in this regard some natural effects of the proportions of numbers, I want to place here on the side, in their smallest terms, each of the consonant intervals contained within the Diapason The Octave., so they can be heard in both manners. From these, as simple intervals, the composite ones are born. By considering them diligently, they will give you light and knowledge of many things pertaining to our discussions. You will find, furthermore, that those intervals within the Diapason—both consonant and dissonant (yet suitable for singing)—do not, if considered soundly, exceed the number of its species.

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