On Music.

...consequently consonant, because the interval that has the first in the lower part is a minor Tone, as is also the one found in the second part between F F faut and G G solreut. I also wish to advise you that if you consider the excess by which the larger term of the ratio containing the ancient Ditone: An interval of two major tones (81:64), creating a sharp major third compared to the "pure" version. (which is 81) exceeds the smaller (which is 64); you will find—according to the rule given to you a little above—that by exceeding its smaller term (64) by nearly four times The author means the difference, 17, is slightly more than a quarter of 64, which is 16., it gives a clear indication that it advances beyond the true nature of the Sesquiquarta: The ratio 5:4, which produces a pure or "just" major third.. I say the Diateffaron: The ancient Greek name for the interval of a fourth. is that middle interval which arises from the Sesquiterza: The ratio 4:3, which produces a perfect fourth. ratio, contained in its lowest terms by these numbers: 4 and 3.

Musical notation with two short staves. Each staff shows two notes, labelled 'f' and 'g', representing the interval of a minor tone.

The Diateffaron is a middle interval between consonance and dissonance.

Nor should it seem inconvenient to you that I have mentioned the Fourth under the name of a "middle interval," seeing as it is the third among five consonant intervals (the forms of which are found between the number one and six) In the "Senario" system of Zarlino, the numbers 1 through 6 generate all primary harmonies: Octave (2:1), Fifth (3:2), Fourth (4:3), Major Third (5:4), and Minor Third (6:5). and the other numbers placed between them, arranged in their natural order. It has two on the lower side, which are the octave and the fifth, and two others on the high side, which are the major and minor Thirds. Furthermore, the sound that issues from its extremes is, in its simplicity of nature, truly such that it does not offend the hearing as dissonances do; yet it does not have the quality of delighting it like the other consonances. Thus, it can deservedly be called "middle" between the two. Nor should one wonder at this at all, since nature never uses its things to pass from one extreme to another without touching the middle. And just as the smallest of the perfect intervals, being furthest from perfection, is less consonant than the others; so likewise the smaller imperfect consonances resonate more than the larger ones, as they are further from imperfection. This is exactly the opposite of what would happen with dissonances, if one were to apply the sevenths to perfect consonances and the seconds to imperfect ones, as we are clearly going to demonstrate in the proper place. I return to speaking of the Fourth, noting that it principally contains within itself a major Semitone, a major Tone, and a minor Tone. It can be considered as being composed of the minor Third and the minor Tone; which we shall see, according to the example, from the product yielded by adding their lowest terms together.

Two staves of musical notation with four notes each. To the right of the staves are mathematical ratios grouped by curly braces as transcribed below.

{ 6 : 5 Ratio original: "Forma" of the minor Third.
10 : 9 Ratio of the minor Tone.
{ 60 : 45 Ratio of the Fourth, outside of its root terms.
4 : 3 In its root terms.

One can also obtain the Fourth of the same ratio and measure from the product yielded by the ratios of the Major Third and the Major Semitone; which we shall see in the following example, where their ratios are added together.

Two staves of musical notation with four notes each. To the right of the staves are mathematical ratios grouped by curly braces as transcribed below.

{ 5 : 4 Ratio of the Major Third.
16 : 15 Ratio of the Major Semitone.
{ 80 : 60 Ratio of the Fourth, outside of its lowest numbers.
4 : 3 In its lowest numbers.

Since the Fourth (being consonant and in its true ratio) must contain what I have said and proved, it follows necessarily—contrary to the opinion of the simple practitioner—that the intervals placed below are not at all consonant Fourths, nor of the same ratio as the first ones. This agrees with what Zarlino: Gioseffo Zarlino (1517–1590), the most influential music theorist of the Renaissance, whose theories Galilei often critiqued. says in proposal 28 of the second discussion of his Demonstrations. This happens because each of them contains two major Tones and one major Semitone separating them. When the numbers of the ratios containing these intervals are added together, we will have from their product a ratio that exceeds the Sesquiterza (4:3) by a Comma: A tiny musical interval (81:80), representing the mathematical "error" between different tuning systems., and the extremes will necessarily be dissonant. Even the practitioner can see with his own eyes that the Tones are such as I have described through the examples given; however, in the example that follows, I will prove it to you as a Theorist, by means of the mathematical science original: "Arittmetica facultà".

Referencing proposal 28 of the second discussion.

The Fourth original: "Diateffaron" is not found between these strings.

Musical notation with two staves showing multiple notes. Beneath the lower staff are the numbers 1, 2, 3, and 4 indicating specific notes.