Dialogo del Galilei

4. 3. Form of the Fourth.
10. 9. Form of the Minor Tone.
40. 27. Form of the Super 13-part-of-27: The ratio 40:27, which represents a musical interval slightly smaller than a perfect fifth.

The interval obtained by adding together the Fourth and the Minor Tone is the Super 13-part-of-27; this consists of a Fifth diminished by a Comma: A tiny musical interval, specifically the difference between a major and minor tone.. One could be further assured of this truth by adding it to itself; for from both one would obtain the Sesquialtera: The ratio 3:2, which is the mathematical form of the perfect Fifth (Diapente). from both. One might doubt, based on what we have said, whether the following interval—considered as a Fourth in the high part and a Tone in the low part—was a Fifth of the first or the second species; to which I respond by saying that it is of the first and not of the second species in any way.

A musical notation example on two five-line staves. The upper staff shows the notes C and F followed by a G. The lower staff shows G and C followed by a D. Brackets and text labels indicate the Fourth and the Tone intervals mentioned in the text.

And if it were argued against me that the Tone it has in the lower part is a minor one—which (according to the way I have shown it to be composed of such intervals) ought to be a major tone—I would respond that this is very true, and that for this reason it becomes consonant in that place. This is because the Fourth it has in the high part is "superfluous" original: "soperflua," meaning augmented or having an excess by exactly as much as is missing from the minor Tone in the low part to make it equal to a major tone. In such a way, from that superfluous interval and from this diminished one—since the superfluity of the one is equal to what is lacking in the other—one obtains from them the Fifth in its true form and proportion; as anyone can sensibly see by adding together their lowest terms. It will also be found in its true form from the Tritone and the major Semitone; but certainly not from the Semidiapente: A diminished fifth. and the minor Semitone, as the following example makes manifest.

64. 45. Form of the Semidiapente.
25. 24. Form of the minor Semitone.
40 } 1600. 1080. Form of the Super 13-part-of-27, outside of its lowest numbers.
40. 27. In its lowest numbers.

From such a sum, we have obtained the Super 13-part-of-27, which (as has been said) consists of a Fifth diminished by a Comma; so that from the Semidiapente and the minor Semitone, one does not get (as the practical musician believes) the same product as from the major semitone and the Tritone. We must not leave behind the knowledge of how much the Fifth exceeds the Tone, and how much it exceeds the Semidiapente; the knowledge of which the two following examples will give us entirely.

3. 2. Form of the Fifth.
X
45. 32. Form of the Tritone.
6 } 96. 90. Form of the major Semitone, outside of its lowest numbers.
16. 15. In its lowest numbers;

The Tritone is surpassed by the Fifth by a major Semitone; the proof of which would be to add the Tritone and the major Semitone together, from which one would undoubtedly obtain the Fifth. But this is not necessary, as the thing is clear in itself; however, we will come to the following example to demonstrate by how much the Semidiapente is surpassed by the Fifth.

3. 2. Form of the Fifth.
X
64. 45. Form of the Semidiapente.
135. 128. Form of the Super 7-part-of-128: The ratio 135:128, a microtonal interval known as the "major chroma" or "limma."

The minor Semitone in this place exceeds the Comma of the Sesquiottuagetresima: Likely referring to the ratio 81:80 (the syntonic comma), though the term literally suggests a ratio of 84:83 or similar..

The Semidiapente is surpassed by the Fifth by the Super 7-part-of-128, which (as I proved to you above when examining the major Tone) consists of a minor Semitone plus one Comma. And so that the practical musician (along with his wondering) does not say to me that I want the same interval to contain now this and now that form, I say that above I considered the present Semitone as contained by the Sesquiuentisimaquarta: The ratio 25:24, a common minor semitone., and lately as contained by the Super 7-part-of-128. In this position, it is truly capable of what I have said: to which I add that the interval which is enclosed within the boundaries of the Super 7-part-of-128 is the true form of the minor Semitone of the major Tone; as I am about to prove to you shortly below.

A small musical notation snippet on a five-line staff at the bottom-left margin, showing notes with flat signs (B-flat) to illustrate the minor semitone discussed.

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