Dialogue on Ancient and Modern Music

Galilei’s Dialogue.

Method for adding ratios together.

128:125. The form of the supertripartiente 125: A ratio where the larger number contains the smaller once plus three-125ths of its value (128/125).
25:24. The form of the minor Semitone.
Then multiply the 128 (the larger term of the Supertripartiente 125) by the 25 (the larger number of the Sesquivētiquattresima: The ratio 25:24, where the larger contains the smaller plus 1/24th.); and the 125 (the smaller term of the former) by the 24 (the smaller number of the latter). From these multiplications, one will obtain these products: 3200 and 3000. These have a relationship of a Sesquiquindicesima The ratio 16:15 proportion to one another. Within these numbers is contained the major Semitone, though not in its lowest terms. To reduce them to those terms, one observes the rule mentioned above.

STROZZI. Since the opportunity has arisen, please do not find it troublesome to repeat that rule for me.

Relatively prime numbers original: "contraesprimi" are those that are measured by no other number but unity; composite numbers can be measured by other numbers.

BARDI. I subtract the smaller term, which was 3000, from the larger term, 3200, and I have 200 remaining. Because this remainder is a common measure of each term, it is likewise their Divisor. Thus, there is no need to look further. When 3200 is divided by 200, the result is 16; and 15 results from dividing 3000 by that same 200. These lowest terms—which are 16 and 15, as I said—virtually contain that same major Semitone, but expressed in relatively prime numbers Numbers that share no common factor other than 1. rather than in the composite and communicating numbers of the larger set. Regarding the reduction of multiple: Ratios like 2:1 or 3:1. and superparticular: Ratios like 3:2 or 4:3 where the numerator is one greater than the denominator. intervals to their lowest terms, I could also give you the rule of dividing the former by their smaller term and the latter by their difference.

Ascending by degrees toward the high pitches (according to the promised order), the Sesquinona 10:9 proportion follows the Sesquiquindicesima. Its content is called the minor Tone by modern practitioners and theorists today. Immediately following this is the Sesquiottaua 9:8, or the Tone, called major to distinguish it from the minor. They say the minor Tone is contained in its true form by the Sesquinona proportion between the numbers 10:9, and the major by the Sesquiottaua between 9:8. With this bit of light, we can clearly see which of these consists exactly of a major and minor Semitone without any part missing or exceeding. It will be the one whose form is entirely similar to the radical terms of the product born from adding them together and reducing them to their lowest terms. Because I have already spoken sufficiently above about the method of adding, subtracting, and finding the divisor of proportions, it will be enough in the future (whenever these needs arise) that I format the examples for you in a way that you can easily understand everything. And wishing now to show you (in the manner I described) which of the two tones has extremes incapable of containing an interval larger than that produced by adding the two Semitones together, we shall see it from the example placed below.

Which tone is composed of the major and minor semitone.

{ 400 : 360. Form of the minor Tone, outside its radical terms.
40 {
{ 10 : 9. In its radical terms.

Zarlino, in Proposition 19 of the first discourse, and 33 of the third.

From two Semitones added together, we have obtained the minor Tone; this fact agrees with what Zarlino: Gioseffo Zarlino (1517–1590), the most influential music theorist of the Renaissance. proves in his Demonstrations.

STROZZI. This fact does not at all align with the opinion of the pure practitioner, nor is it without surprise for them, as they expect the major Tone from such a combination. However, please continue.

BARDI. By now subtracting the minor Tone from the major Tone, we will be able to see tangibly by how much the major exceeds the minor, which we will understand from the following example.

By how much the major tone exceeds the minor.

From the subtraction of the minor tone from the major, the Sesquiottantesima 81:80 is born; its content is called the Comma by modern practitioners.

STROZZI. Is this the same as the ancient one?
BARDI. No, sir.
STROZZI. What is the difference between them?

What the difference is between the ancient Comma and that of today.

What the Schisma is.

BARDI. Ancient musicians understood the Comma as the excess of the Apotome after the Lemma was subtracted from it; or we might say, the remainder of a Tone after two of their minor Semitones were subtracted. Today, the Comma is understood (as you have heard) as the remainder by which the Tone exceeds the Sesquinona. The moderns did not wish to derive it from the difference of the Semitones in the manner of the ancients for the reason that will be stated shortly. The ancient Comma is therefore enclosed within these numbers: 531441:524288. Half of this they then called the Schisma.

STROZZI. Is our comma larger, or that of the ancients?
BARDI. The modern one exceeds that of the ancients by an interval of 32805:32768.