On Music.

Warning.

The Tritone exceeds the Fourth by the "Super 7-parting-128" The ratio 135:128; this proportion consists of a minor Semitone and—contrary to the opinion of the practical musician—one additional Comma: A tiny mathematical difference between two notes that are theoretically the same, often used to measure the "purity" of a tuning system., as has been demonstrated elsewhere above. It seems impossible to the practical musician that, having subtracted the Fourth from the Tritone, there should remain anything more than an ordinary minor Semitone. All the difficulty he has in understanding these novelties arises from not having that knowledge he ought to have regarding the intervals he has constantly manipulated. Such knowledge completely satisfies those who possess it perfectly. Nor does this happen without bringing him some blame; for it is impossible for those who do not understand the property and power of a thing (whatever it may be) to practice it well. And many say (indeed, all the judicious and learned) that this is one of the principal reasons, among many others, why the practical music of our times does not have the power to produce in the minds of listeners any of those marvelous and virtuous effects that ancient music once produced. I return to saying, therefore, that the Tritone cannot be obtained by adding together the form of the Fourth and that of the minor Semitone; for it is a Major Tone that must be divided to do so, which is capable of containing—besides the major and minor Semitone—a Comma, as has been proven another time, and as can be proven anew by adding them together. And because there might sometimes occur some proportion between large numbers that are so little different from one another that one might not recognize (so to speak) the difference between one interval and the other when trying to see it with diligence: for it is not impossible in this manner of exercising numbers (as in some others) to subtract a very large thing from a very small thing; but warned by what I am now about to tell you, and shown by example, you will be certain which of them contains the greater and which the lesser. Wherefore I first say that a large interval can be subtracted from a small one; as for example, to subtract a Major Tone from a Comma, even though the latter is by far smaller than the former. And that this can be done, we shall prove with the present example to the senses.

Warning.

At first glance, it seems as though we have subtracted a Minor Tone from a Comma, and consequently, that the Comma exceeds the Tone by such an amount; everyone knows how much this contradicts both the intellect and the senses. However, it must be noted that the smaller number of that interval which we sought to subtract from the Comma has come to the place of the larger, and conversely, the larger to the place of the smaller. For this reason, they do not have the form of the Sesquinona: The ratio 10:9, representing a Minor Tone. in the usual way, because, as I have told you another time, that is contained between 10 and 9, and not conversely between 9 and 10.

STROZZI: What interval, then, will be produced by such a proportion?

Semidiapente: why it is so called.

BARDI: That is a Subsesquinona; which in such a place reveals by how much the interval from which it was drawn is exceeded by the Sesquiottava: The ratio 9:8, representing a Major Tone., and not by how much the Comma exceeds it. Therefore, such proportions are deservedly called "Privative" and "Rational," while those others are "Positive" and "Real." Let this suffice for all other such cases that might occur to you. I come now to discuss the Semidiapente: A "half-fifth" or diminished fifth., which according to the opinion of our practical musicians is that interval consisting of a major tone, a minor tone, and two major semitones. Perhaps it took such a name because its extremes sound like a "scanty" Diapente: A Perfect Fifth (ratio 3:2)., divided however into four intervals by five terms and strings. It is contained in its lowest terms by these numbers: 64 to 45. We shall consider it primarily as being composed of a Fourth and a major Semitone, both in the low range and the high. One can grasp these truths in several ways, but the simplest and shortest is to add together the numbers containing the two aforementioned intervals and see that their product will give us the form we have stated.