Dialogue on Ancient and Modern Music

Of Music.

Such a consideration could also have been drawn from the eighth chapter of the third book of Ptolemy's Harmonics, or from the fourteenth chapter of the first book of his Tetrabiblos original: "Quadripartito"; Ptolemy's famous work on astrology.; Ptolemy compares the aspects of the planets to musical intervals. where he ingeniously compares the aspects In astrology, "aspects" are the angles between planets. of the planets to the forms of the musical intervals of his time, saying thus: The Tetragon: The square aspect (90 degrees) in astrology. and the Square, compared to the Trine 120 degrees., make a Sesquiterza: The ratio 4:3, which forms a perfect fourth.; compared to the Hexagon (or Sextile, if we wish to call it so) 60 degrees., it makes a Sesquialtera: The ratio 3:2, which forms a perfect fifth.; compared to the opposition 180 degrees., it makes a Dupla: The ratio 2:1, which forms an octave.; and with the whole circle of the Zodiac, it makes a Diapason-diapente: An octave plus a fifth (a twelfth).. All of this, compared again to the Square, makes a Disdiapason: A double octave (ratio 4:1).; and finally, comparing three squares to two trines, they have the same relationship between them as 9 has to 8 This ratio, 9:8, defines the "Major Tone" in Pythagorean and Just intonation..

STROZZI. Are the forms of the Imperfect Consonances: Intervals like thirds and sixths, which were considered "imperfect" compared to octaves and fifths. not also found among these aspects?

BARDI. No, sir, because the heavens do not permit nor tolerate imperfection. One could also derive, following the same order from each of the two cited passages—for anyone who wished to go more subtly into sophisticated details—the forms of all the other musical intervals of our times; but let enough be said of that.

STROZZI. I believed that this faculty of the Senary: The number six. The theorist Gioseffo Zarlino argued that all musical consonances are derived from the first six integers. was an entirely new invention, and I see it is not so; which makes me doubt whether there are other things (regarding invention) that are very ancient, yet are preached to us as new by this or that person.

BARDI. Do not doubt it for a moment; for simple people, when reading a book on any subject, often believe (due to little experience) that those things are not found elsewhere but in that book; whereas most of the time they were written in many others, thousands of years before.

STROZZI. That division of the square into three equal parallelograms—the middle one of which is separated into two equal parts, then intersected by a line starting from one of the angles of the square and resting on the opposite side such that it divides it into two equal parts—the various portions of which lines, compared to one another, have the power to give us the forms of a greater number of musical intervals than the Senary and its parts: is this an ancient or a new invention?

BARDI. That very thing is taken wholesale from the second chapter and book of the Harmonics of the same Ptolemy; of which, as a joke, he tells as much as was necessary for his purpose to denote the musical intervals of those times. From what I am now about to tell you, you will easily be able to recognize when you have an interval outside of the true and lowest numbers, whether it is "superfluous" Too large. or "diminished" Too small from its true being. However, it should be noted that if its lesser term is diminished from what suits its natural effect, it is a manifest sign of being superfluous; and being greater than usual, such an interval will be diminished. By how much more or less, you will be certain by subtracting from it or adding to it this or that part which you judge most appropriate; and the opposite effect would occur in its greater term.

STROZZI. I have not entirely understood this last concept of yours.

An Advisory. BARDI. Let me explain myself better. In place of the Sesquiquinta: The ratio 6:5, the "pure" minor third. (assigned to us by the masters of this modern practice of counterpoint as the form of the minor third of Ptolemy’s Syntonon: The "Syntonic Diatonic" scale, which uses different ratios than the Pythagorean system.), we have the Superquintapartient 27: The ratio 32:27, the Pythagorean minor third.. Wanting now to see if it exceeds or is exceeded by the Sesquiquinta, the excess with which 32 (its greater term) exceeds the lesser (which is 27) can sufficiently teach us: this excess is five. Now consider how many times 5 enters into the lesser term of that proportion (27); it enters five times and has two left over. Whereas in the Sesquiquinta (6:5), the excess by which the greater number exceeds the lesser is one; and it enters into the lesser number exactly five times. From this, it clearly appears that what I have said is true; namely, that because the lesser term of the Superquintapartient 27 is "superfluous" compared to that of the Sesquiquinta, it is consequently smaller; by how much, has been said in its place.

STROZZI. I have understood it all very well, but tell me another detail: since the Semiditone: A minor third in the Pythagorean system (32:27). is not the same thing as the minor third, with which of these does the Triemitone: Literally "three half-tones," used here for the ratio 19:16. agree? For I have found it named as the same thing by some, using each of the three different names.

Triemitone, what it may be. BARDI. The Triemitone is, as you know, the highest interval of each tetrachord of the ancient Chromatic [genus], and it does not agree with the Semiditone, nor with the minor third of the Syntonon; for it falls under the proportion Supertripartient 16: The ratio 19:16.. Thus it is smaller than the minor third, and larger than the semiditone; an opinion shared by Zarlino in chapter 34 of the second part of his Harmonic Institutions. It remains for us to see, in this regard, by how much the minor third exceeds the Major and Minor Tone, which will be made known to us by the two following examples.

This calculation compares the minor third (6/5) to the major tone (9/8). By cross-multiplying (6x8=48 and 5x9=45), we find the difference between them is the ratio 48:45. When divided by 3, this reduces to 16:15, which is the mathematical definition of a Major Semitone.

The minor Third exceeds the Major Tone by a major Semitone; the proof of which...