Dialogue of Galilei

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From this sum, we have obtained the "super 7-parting-20" This ratio, 27:20, represents an augmented fourth found in certain tuning calculations., which (as I told you) consists of a Fourth increased by a Comma: A tiny musical interval representing the minute difference between two notes that are theoretically the same but arrived at through different mathematical paths.; this can be clearly perceived by subtracting the comma from it. Furthermore, if it were otherwise, it would follow that if the lower part descended by a tone of the quality found between the strings G G solreut and A alamire, the extremes would not resonate as a perfect Diapente: The ancient Greek term for the interval of a Fifth.. This fact, as will be seen in its proper place, is not true at all.

I want us to see now whether the intervals considered below—a Third in the upper part and a Tone in the lower—are complete Sesquiterze: The ratio 4:3, which defines the Perfect Fourth. or not, and how the theorist considers them and why. I raised this doubt because of my desire to settle every difficulty and doubt that might arise for you concerning this important head of our discussion. A little above, regarding the Minor Third, I proved to you that in ascending between D D solre and F F faut, it was not actually that imperfect consonance contained by the Sesquiquinta: The ratio 6:5, which produces a "pure" minor third., but rather the ancient Semiditone: The Pythagorean minor third (32:27), which is slightly narrower and more dissonant than the pure 6:5 version.. Considering now the Fourth in the example shown, both for B-flat: b molle and B-natural: h duro, each composed of the interval I mentioned and the Major Tone, we will find that by adding together the numbers containing them, they will give a complete Fourth, as you will see from the following example.

Two musical staves showing melodic intervals. The first staff has a C-clef on the fourth line with a flat sign on the middle line; the second staff has a C-clef on the fourth line without a flat. Each staff contains two notes separated by a vertical bar.

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The Tritone, and why it is so called.

From these two intervals added together, we have obtained the Fourth in its true form; this did not happen before, because the Third of this last example—compared to the first one shown—is diminished by the amount that the Tone in that one was superfluous. When composed of such parts, the Fourth can still be considered in the ancient Diatonic: The standard musical scale of seven notes.. Regarding its consideration, I must make known to you by how much it exceeds the major and minor Thirds; the two examples given to you will do this sufficiently, where we considered it composed of such intervals and the Major and Minor Semitone.

I come now to tell you some details of the Tritone: An unstable, dissonant interval spanning three whole tones.. Modern practitioners say it is contained by the ratio "super 13-parting-32" original: "Super 13 partiente 32"; the ratio 45:32 between these numbers, 45 and 32, and that it contains within itself three tones; it perhaps took its name from this content. But in the Pythagorean Diatonic species original: "Diatona Ditoniea", these tones are equal. One can consider this dissonant interval as being composed of a Major Third and a Major Tone, whether in the lower or upper part. That the aforementioned ratio is its true proportion in the Syntonic: Ptolemy’s "Just" tuning system, which uses pure thirds (5:4). can be confirmed in several ways. Among these, I will now show you one which is the simplest and easiest: by adding together the numbers that contain the aforementioned intervals (its neighboring parts), which will yield the same numbers I said contained it.

Two musical staves showing the tritone interval. Both staves use a C-clef on the fourth line. Notes are placed on F and B, representing the tritone.

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Nature rarely gives birth to perfect things, and to monsters.

From the product of the Major Third and the Major Tone, the Tritone has been obtained in its true form and in its lowest terms. This interval, together with the one following it, has a form and extremes that are (so to speak) quite disproportionate, and consequently are understood by the intellect with difficulty. They cannot be considered, nor are they naturally found in the Octave: Ottaua, on any strings other than two. Nor is this to be wondered at; for nature rarely gives birth to disproportionate and monstrous things (just as she rarely produces perfect ones). However, we can clearly see by how much it exceeds the Fourth by subtracting the Fourth from it.