Dialogue on Ancient and Modern Music

Dialogue of Galilei

A woodcut of an empty five-line musical staff appears at the top left, followed by a mathematical calculation of musical intervals with ratios grouped by a brace on the left.

What the goal of the practical musician is.

This interval, just like the one preceding it, cannot be considered or composed among the natural Diatonic strings in any Octave from any parts other than the two already mentioned. And if the practical musician Galilei uses 'practical musician' to refer to performers or composers who rely on their ears or traditional rules rather than the mathematical 'science' of music. should tell me that this can be composed of two minor Thirds of the same form and equal proportion, I would answer and prove to him that he errs in this as in many other things. This is because his goal is only to satisfy—not the intellect or the sense of hearing—but very often that of sight. The sense of sight, like all others (or more than the others), is easily deceived; in distinguishing sounds, it plays as small a part as the sense of hearing does in discerning the differences between colors. The senses are particularly deceived by the smallest differences between common and specific objects, which even a healthy intellect can hardly grasp. That a Semidiapente of the form shown cannot be composed of two minor Thirds of the same measure is revealed to us first by the number of its larger term The number 64 in the 64:45 ratio.. Although this number is a square—which it must be if it is to be capable of Geometric mediation: A mathematical way of dividing an interval into two equal parts using the square root.—it nonetheless lacks another necessary condition: it must also be capable of providing the larger term for one of those intervals into which one wishes to divide it equally. This does not happen with the larger number and term of the Semidiapente, as it is not divisible by six, which is the larger term of the minor Third The ratio 6:5.. But let us come to the experience of the fact with a sensible example, which will be to sum together the terms of such intervals; from their product, one will not obtain the form of the Semidiapente, but rather that of the original: "Super 11 partiente 25"; a ratio where the larger number is the smaller plus 11, i.e., 36:25.

A mathematical calculation with ratios grouped by a brace on the left.

Warning.

It is clear that the interval we have obtained from summing two minor Thirds together is a Semidiapente increased by a Comma: A very small interval representing the difference between two nearly identical notes in different tuning systems.. That this is true can be seen by subtracting from it the "Super 19-parting-45" The ratio 64:45., which is the form of the Semidiapente, and we will be left with one Comma; this is the reason why. When the theorist considers the Semidiapente as composed of two minor Thirds, since it is impossible to rationally divide any interval of the first three simple types of proportions into equal parts (except for the Quadruple and its replicas, always speaking according to the faculty of Arithmetic), he necessarily finds one part larger and different from the other. I have clearly shown you the truth of this above regarding the minor Third. It only remains to see if, by summing such different minor Thirds together, their product will give the true form of the Semidiapente. From this example, it will be clearly recognized that the ancient Ptolemaic musicians: Followers of Claudius Ptolemy, who proposed a system of "Just Intonation" based on simpler mathematical ratios than the earlier Pythagorean system. also held them to be of this kind. For if they had held them to be of the same measure, they would not have assigned the shown proportion to the Semidiapente capable of containing both, but rather the one obtained above from the product of two minor Thirds, each contained by the Sesquiquinta: The ratio 6:5.. That the Thirds are therefore different in form will be made clear by the following example.

A mathematical calculation of musical intervals with braces on the left. The first brace groups the Semiditono and Terza minore. The second brace groups the resulting ratios with a '3' indicating simplification.

Behold the Semidiapente composed of the two aforementioned intervals in its true form. Nothing else remains to be seen regarding it except by how much it exceeds the Tritone; this will be made known to us by the following example, in which the Tritone will be subtracted from it.

A mathematical calculation showing the difference between two musical intervals. A large 'X' symbol is placed between the ratios 64:45 and 45:32 to indicate cross-multiplication.