Dialogue by Galilei

the lowest terms of these according to this example; which will also make known to you by how much the minor Sixth exceeds the fifth.

Musical staff showing two diamond-shaped notes representing a Fifth (Quinta).

Musical staff showing two diamond-shaped notes representing a major Semitone (Semituono maggiore).

A large curly brace with the number 6 in the center, grouping the previous two intervals to show their summation into a minor Sixth.

Where the Tritone: An interval of three whole tones, known for its harsh, dissonant sound. occurs, the minor Sixth cannot be obtained except by means of more than two intervals, or by some unusual interval, and therefore I will remain silent about them. I want us to now see the same occurrences in the Major Sixth, of which the practitioners say its lowest terms are between these numbers 5 and 3; and that it consists of two major Tones, two minor Tones, and one major Semitone. It can first be considered as being composed of the Fourth and the Major Third; so that the vacuum of its terms may be entirely filled by such intervals. This truth will be felt by hand every time the numbers of their proportions are added together, and the product they yield is contained by the same numbers that contain that [Sixth]; as the following example manifests to us.

Musical staff showing two diamond-shaped notes representing a Fourth (Quarta).

Musical staff showing two diamond-shaped notes representing a major Third (Terza maggiore).

A large curly brace with the number 4 in the center, grouping the previous two intervals to show their summation into a major Sixth.

The Major Sixth is not found between these notes.

From such a product, the Major Sixth has been obtained in its true form; but it will not be obtained as such when considered in the same apparent intervals between the string of F faut and that of d la sol re F and D in the Guidonian hexachord system of note naming.. This is because the Fourth found between them in the high register does not fall under the form of the first one shown, but under this other one: 27 to 20. This ratio, as was proven in its proper place, is superfluous by one Comma A very small interval representing the difference between two tuning systems.. And whatever proportion we shall have from such intervals, I will make it known to you with this example, in which their lowest terms are added together.

Musical staff showing two diamond-shaped notes representing the ratio 27:20.

Musical staff showing two diamond-shaped notes representing a major Third (Terza maggiore).

A large curly brace with the number 5 in the center, grouping the previous two intervals.

One has obtained, as you see, the Superelevenpartient 16th [27:16], which is the true form of the Major Hexachord of the Diatonic Ditonic The Pythagorean tuning system.; and because it exceeds the Major Sixth of the Syntonon of Ptolemy Ptolemy’s "Just" tuning system, which uses the 5:3 ratio. by one comma, it is necessarily dissonant. For no other reason does Zarlino Gioseffo Zarlino (1517–1590), the most famous music theorist of the age and Galilei's former teacher turned rival. say in proposition 35 of his Demonstrations in the second discourse: that by adding the Major Tone to the Fifth, no consonance can be born. And that the Major Hexachord exceeds the Major Sixth by such an amount (as I have said) can be seen sensibly by subtracting the one from the other. One will indeed have a consonant Major Sixth from the product yielded by the Fifth joined to the Minor Tone; as can be sensibly seen by adding together the numbers that contain them in the following example.

Musical staff showing two diamond-shaped notes representing a Fifth (Quinta).

Musical staff showing two diamond-shaped notes representing a minor Tone (Tuono minore).<-

A large curly brace with the number 6 in the center, grouping the previous two intervals.

One can also have this imperfect consonance in its true form from the minor Sixth and the minor Semitone, with the help, however, of the present sign ♭ The flat sign, which lowers a pitch.; and between the strings of the example that follows below;