Dialogue on Ancient and Modern Music

On Music.

The Tritone is surpassed by the Semidiapente by the Super 23-part-of-2025: A very small mathematical ratio (2048:2025) used to define the difference between certain musical intervals., which consists of approximately half a Comma.

STR. Are the Semidiapente and the Tritone of the ancients of the same measure and proportion as those of our modern musicians?

BAR. No, sir; because the interval that was considered a Tritone among the ancients fell under these numbers 729 to 512, which exceeds ours by a Comma: A tiny interval, the difference between two sounds that are nearly identical but mathematically distinct in different tuning systems.; and their Semidiapente fell under these others, 1024 to 729, which conversely is smaller than the Super 19-part-of-45: A ratio (64:45) defining the modern diminished fifth., called by the moderns the Semidiapente, by such an interval; as can be clearly seen by subtracting one from the other.

Correct names of the intervals, No. 116.

STR. So, in such a way, are the majority of the names of musical intervals today (not to say all of them) corrupted and ruined?

BAR. Do not doubt it for a moment.

STR. Would it not be good to provide correct names for them before the matter grows old and spreads further into the minds of men?

Glareanus, in chapter 12 of book 3, calls the Sesquialtera, Sesquipla.

BAR. Truly yes, but this is a task for a man of authority and great value; however, regarding the names of intervals, we will proceed by following common usage (even if it is bad), if for no other reason than to be understood. But let us now come to the examination of the Diapente: The ancient Greek term for the interval of a Perfect Fifth.; which they say is contained in its true proportion of the Sesquialtera: A ratio of 3:2, where the larger string length or frequency is one and a half times the smaller., also called Sesquipla, between these numbers in their lowest terms, 3 to 2. It contains primarily two major tones, one minor tone, and one major semitone. It can be considered in many ways, but its proper and principal way is as a composite of the major and minor Third, as the product we obtain from the numbers containing it will show (according to the example that follows); and it will be in its true harmonic essence whenever the major Third holds the lower (grave) position, and the minor holds the higher (acute) position.

Proximate and remote parts of intervals, what they are.

I have said the Fifth is composed of such intervals before any others because those are its most proximate parts: The primary components that sum directly to the ratio of a larger interval.. And so that you may well understand the reason for this too, I say: according to the opinion of theorists, the most proximate parts of intervals are those which, when the numbers containing them are multiplied together, give a product that is not only the same proportion as the whole, but very close to—and sometimes (as you have seen with the Tritone and the Semidiapente) in—its very lowest terms. Conversely, those are the "remote" parts which, compared to the lowest terms of their whole, give a more distant product. This is brought to us by their more imperfect and unequal parts, and the former by the more perfect and equal parts. And coming to consider the other parts of the Fifth and composing it from them, I say: it will still be a Fifth if it consists of a Fourth and a major Tone; because when the numbers of their proportions are multiplied together, the product they give (though a little further from its lowest terms) will yield its shown form.

Zarlino, at proposition 30 of the second discussion of his Dimostrationi.

The Diapente is not found between these strings.

Since the Fifth arises from those two shown intervals as was said, one can argue along with Gioseffo Zarlino (1517–1590), the most famous music theorist of the Renaissance and Galilei’s teacher, whom Galilei often challenges in this work. Zarlino that the intervals in the following example are not truly Fifths at all, but of a different proportion and genus. This goes against the opinion of the practical musician, and even more so when it is proved to him that they are dissonant. Their being of a different proportion from the first ones arises because they contain a Fourth and a minor Tone; these two intervals added together cannot give us a Fifth of the same proportion as the first two shown, but another that will be a Comma smaller, and therefore dissonant. To show how much it is diminished, here is the example that will confirm it.