Dialogue on Ancient and Modern Music

Galilei’s Dialogue.

The meaning of the word Lemma. What the Diaschisma is.

...called by them Semitones: which, in order to distinguish one from the other, they accompany—as you know—with one of these words: Major and Minor. They say the minor Semitone is that which is contained between these numbers: 25:24. This interval has been given the name sesquiquinquattresima: A ratio of 25:24, where the larger number contains the smaller once plus 1/24th of its value. by its authors, just as they call this one, 16:15, sesquiquindicesima: A ratio of 16:15, where the larger number contains the smaller once plus 1/15th of its value., which constitutes the major Semitone. On the contrary, in the Diatonon: The ancient Pythagorean tuning system based on perfect fifths., the minor semitone—which is also called the Lemma—is contained between these terms: 256:243. Half of this they then called the Diaschisma. The major semitone, also called the Apotome, is found between these other terms: 2187:2048.

STROZZI. For what reason did those first researchers establish the major and minor Semitone within such numbers in the Diatonic Distribution?

Why the major and minor Semitones are within those numbers.

BARDI. For this reason: the ancient Musicians called the minor Semitone that original: "avanzo" remainder of the Diatessaron: The interval of a perfect fourth. after two whole tones are subtracted from it. Because when you subtract two sesquioctaves: The ratio 9:8, which defines a standard whole tone. from the sesquitertia: The ratio 4:3, which defines a perfect fourth., the remaining proportion is 256:243, and in this they established such a Semitone. They called it "minor" because two of them added together do not fill the space of a whole Tone; whereas, on the contrary, two major ones exceed it. That remainder of the Tone, after the minor Semitone has been taken away, they called the major Semitone.

STROZZI. What does that word, Lemma, mean?

What the Apotome is.

BARDI. Lemma Greek: λείμμα means the same thing as "residue" or "remainder." This was highly appropriate because the Lemma is nothing other (as I said) than the leftover part of the Fourth after two whole tones are removed. The Greeks also call Lemma that part of a thing which, when taken twice, does not reach the whole. Furthermore, they called the major Semitone the Apotome, a word which in that language signifies "severance" or "detachment"; as when (for example), if you take an Apotome from a Ditono: A major third consisting of two whole tones., a Semiditono: A minor third. remains. There have been others who understood the major Semitone as the superquintapartiente 76: A complex ratio where the larger number is 81 and the smaller is 76., which is the form of the second interval of each Tetrachord: A scale of four notes spanning a perfect fourth. of the ancient Chromatic scale; which, in its lowest terms, falls within these numbers: 81:76. From this knowledge, by subtracting the sesquiquinquattresima: 25:24 ratio (minor semitone). from the sesquiquindicesima: 16:15 ratio (major semitone).—which are the forms of the Semitones in the Syntonic scale—we can know by how much the minor is exceeded by the major. According to Boethius: An influential 6th-century philosopher whose works on music theory were standard textbooks for centuries., the method of subtracting one musical interval from another is as follows: first, arrange the lower terms of their proportions in this manner;

(Form of the major Semitone.)
X

(Form of the minor Semitone.)

Method of subtracting one interval from another.

Place those numbers containing the smaller interval below, and those of the larger interval above. Arranged thus, they appear at first glance to be the opposite of the method an Arithmetician uses to subtract one number from another; nonetheless, the effect is the same. This is because the Musical Theorist does not simply consider the value of the numbers like the Arithmetician, but rather the quantity of sound they enclose between them. And because most often the smaller numbers contain the larger intervals, it happens that the truth appears to the senses to be the opposite of what the intellect understands.

STROZZI. Is it not always true, then, that the smaller terms contain the larger intervals?

Method for finding the Divisor from proportions.

BARDI. No, sir; not in superparticulars: Ratios where the numerator is exactly one greater than the denominator (like 3:2 or 16:15). and thereafter. These exceptions (besides other intervals shown to us in superpartients: Ratios where the numerator is more than one greater than the denominator (like 5:3).) are the Lemma and the Apotome; and among multiples: Ratios where the larger number is a multiple of the smaller (like 2:1 or 3:1)., the double and the triple. Once the numbers are arranged in the manner shown, we will multiply the 16 (the major term of the 16:15 ratio) by the 24 (the minor term of the 25:24 ratio), and the 25 (the major term of the latter) by the 15 (the minor term of the former). From these multiplications, we will have these products: 384 and 375. By finding the common Divisor, we will see them reduced to their lowest numbers, so that one can more easily comprehend the quantity of the interval and sound contained within their extreme terms. To do this, I first measure the 384 (the larger term) by the smaller, which was 375, and I have 9 left over. Since 9 is not a common measure of them, it cannot be the sought-after Divisor. Therefore, I measure the smaller term (375) by the 9, and I have 6 left over. Considering this, I find that it too is not a common measure of each, but only of the larger (384), into which it goes 64 times. Therefore, I return to measure the first and largest excess (which was 9) by the smaller (6), and I have 3 left over. This 3, being a common measure of each term, is necessarily the sought-after Divisor. Dividing the two original large numbers by this 3, the results are 128:125. Since these cannot be reduced to smaller numbers in any way, they are in their minimal and radical terms. The interval contained by them is a little more than a Comma: A very small musical interval, the difference between two nearly identical notes in different tuning systems. and a half. We can further verify that the major Semitone exceeds the minor by this amount by adding it together with the proportion and numbers that contain the minor Semitone; for the product they yield will have the same form as that which the major contains. To do this, one shall follow this order: first, arrange the numbers of their forms in this manner.