Dialogue by Galileo.

Two musical staves showing intervals and ratios. The top staff contains two notes labeled with numbers 9 and 8, followed by the text "Forma del Tuono maggiore" (Form of the Major Tone). The second staff contains two notes labeled 16 and 15, followed by the text "Forma del maggiore Semituono" (Form of the Major Semitone). Below these is a larger musical snippet showing notes D and F (natural) with numbers 114 and 120, bracketed with the number 24; this is labeled "Forma della Terza minore fuore de minori suoi termini" (Form of the Minor Third outside its lowest terms). Directly below these are the reduced numbers 6 and 5, labeled "Ne suoi minori termini" (In its lowest terms).

STROZZI. Is this Semiditone: An interval roughly equivalent to a minor third, but mathematically different depending on the tuning system. of ours the same as that of the ancients?
BARDI. It is not the same in any way; for our own is consonant, as you know, and is produced in the Superparticular: A type of ratio where the numerator is exactly one greater than the denominator (like 3:2 or 6:5). genus by the Sesquiquinta: The ratio 6:5. proportion; and that of the ancients, as all musicians affirm, is dissonant, contained in the Superpartient: A ratio where the numerator is larger than the denominator by more than one (like 32:27). genus between these numbers: 32 and 27.
What the Semiditone may be.

STROZZI. How are we to understand and distinctly recognize several different things by the same name?
BARDI. With great difficulty and confusion, certainly, for both the speaker and the listener. However, I will proceed with as much distinction and ease as I can so as to be clearly understood by you, even though my voice may be naturally hoarse. Since it is true, then, that the Minor Third: An interval spanning three scale degrees. consists of one major tone and one major semitone, and consequently that it is contained by the ratio of 6 to 5 (the original: "Sesquiquinta"), as Gioseffo Zarlino: An influential 16th-century theorist who argued that the first six integers—the senario—were the foundation of all musical harmony. particularly affirms in the 26th proposition of the second discussion of his Demonstrations; it follows necessarily (which is against the opinion of the practical musician) that the notes below are not actually minor thirds of the proportion and measure of the first two shown. This occurs for no other reason than that each of them contains nothing more than a minor tone and a major semitone: these two intervals joined together are not capable of giving us a minor third of the measure and proportion of those first ones; but rather a dissonant semiditone of the ancient Diatonic Ditonic: The ancient Pythagorean tuning system based on stacking perfect fifths (3:2 ratios). system. And although the senses (as has been said) accept it without resistance, we will nonetheless prove it to the intellect in this manner: Let us sum together (following the example that follows below) the numbers that contain one and the other interval, and see what product results. From having summed together the two aforementioned
The minor third is not found between D and F-faut.

10 . 9 . } intervals, we have the original: "superquintapartiente" 32 to 27, which is the true form of the
16 . 15 . } semiditone of the Diatonic Ditonic. This semiditone is dissonant for no other reason
160 . 135 . } than being smaller by one Comma: A very small pitch difference; here, the Syntonic Comma (81:80). (as has been said elsewhere) than the
32 . 27 . } minor third of the Syntonon of Ptolemy: A tuning system (just intonation) proposed by the ancient Greek scholar Claudius Ptolemy.; besides its form being found in the Superpartient genus, which according to Pythagoras was not suitable for the consonances of his time and falls outside the original: "numero Senario" number six.

STROZZI. From where do we believe that the musicians of today have drawn this so subtle consideration: that within the parts of the Senario: The set of numbers from 1 to 6. is contained every simple consonant interval and the parts of the compound ones?
BARDI. I hold for certain that considering the order in which the proportions are placed in the second genus of greater inequality called "Superparticular" gave them this opportunity; by having coupled the first ten intervals two by two in their natural order, and then reducing them to their lowest terms.
From whom the number six (senario) took its power.

STROZZI. How so, please?
BARDI. Here is a convenient example for you; which without more words will make known to you all that I feel about it.

Numbers arranged according to the nature of the Superparticular Genus; among which is found in act the form not only of any simple musical interval, but in potential each of the mixed and compound ones: and he who went further would also find those that contain the major and minor semitone. These numbers, if considered otherwise, would give the form of any other desired interval.

A complex diagram illustrating musical ratios and intervals. At the center is a horizontal line of numbers from 1 to 12. Above this line, ratios between consecutive integers are labeled: "Sesquialtera" (ratio 3:2), "sesquiterza" (4:3), "sesquiquarta" (5:4), "sesquiquinta" (6:5), "sesquisesta" (7:6), "sesquisettima" (8:7), "sesquiottaua" (9:8), "sesquinona" (10:9), "sesquidecima" (11:10), and "sesquiundecima" (12:11). Below the line, curved arcs connect integers to name specific intervals: "Diapason" (Octave, 1-2), "Diapente" (Fifth, 2-3), "Diatessaron" (Fourth, 3-4), "Terzamaggiore" (Major Third, 4-5), and "Terza minore" (Minor Third, 5-6). The diagram is contained within a rectangular frame.