Galilei's Dialogue.

it is not necessary, as it has just been seen in composing it from both of these; however, let us see by how much the minor Third exceeds the minor Tone.

2 } 54 : 50 Ratio of the Superbipartient 25: A ratio where the larger number is the smaller plus two units (27:25). outside of its lowest terms.
} 27 : 25 In its lowest terms.

The minor Tone is exceeded by the minor Third by the Superbipartient 25, which consists of a major Semitone and a Comma: A very small interval, specifically the Syntonic comma (81:80), representing the difference between a major and minor tone., as was proved above. They say the Major Third is that imperfect consonance arising from the Sesquiquarta: The ratio 5:4, which forms a pure or "just" major third. ratio; and that it is contained in its lowest terms by these numbers 5 and 4; and furthermore that it consists of the major and minor tone: which we shall see (according to this example) by adding them together and seeing the product they yield.

Two musical staves with notes, followed by numerical ratios and their descriptions on the right.

9 : 8 Ratio of the major Tone.
10 : 9 Ratio of the minor Tone.

18 } 90 : 72 Ratio of the Major Third, outside of its lowest terms.
} 5 : 4 In its lowest terms.

The Major Third arises from having added together both the Major and Minor Tone; from which knowledge, along with subtracting the minor Third from it, we can also know by how much the latter is exceeded by the former.

Thus the minor Third is exceeded by the major by a minor Semitone; the proof of which would be to add this same minor Semitone to it. The simple practitioner, on various strings accidentally by means of these signs # and ♭ The text uses "X" for a sharp and "b" for a flat, standard notation of the era., has other types of major Thirds, which he considers to be of the same size as the first ones shown: which (in my opinion) is one of the greatest abuses he has among many, many others that I am about to demonstrate to you, beyond those already mentioned.

STROZZI. I marvel greatly that among so many excellent men who have written about this faculty of Music so subtly and learnedly, there have not yet been any (that I know of) who have revealed and uncovered these obvious errors for the common benefit.

BARDI. There is no reason at all to wonder at this; for those who believed they were singing the Ditonic Diatonic: The Pythagorean tuning system based on pure fifths (3:2), resulting in sharp, dissonant thirds. could not conceive of such a consideration, as neither the cause nor the effect of it could be found there in any way. As for those who have said that the species sung today is Ptolemy's Syntonic: A tuning system (Just Intonation) that uses simpler ratios like 5:4 for thirds to make them sound sweet and consonant., I will easily believe, given the many correspondences, that they noticed the error all too well; but because revealing it did not suit their purpose, they have (as I said another time) kept silent. Driven by an ambitious and vain thought, they valued the irrelevant novelty of their own whims more than the benefit they could have brought to the public by explaining the truth; they have even corrupted and spoiled many names of the most important things belonging to this faculty. Therefore, the Thirds I mentioned are contained between the strings of the present example; and
The major third is not found between these strings.

because each consists of two major Tones, they consequently fall under this ratio of 81 : 64 The "Ditone" or Pythagorean major third, which is wider and more dissonant than the just 5:4 ratio., which, if well considered, will be found to contain those particular qualities found in the Superquintapartient 27: A complex ratio (32:27) representing the Pythagorean minor third.. It cannot in any way be the natural Major Third of the Syntonic Diatonic; but rather, as Zarlino affirms (in the fourth discourse of his Demonstrations at the Third Definition), the dissonant Ditone of the Ditonic Diatonic; which from high to low in each of its Tetrachords: A scale of four notes spanning a perfect fourth. proceeds by Tone, Tone, and Lemma: Also called a Leimma; the Pythagorean minor semitone (256:243).. It is true that those placed below are contained by their Sesquiquarta ratio, and consequently...

Two musical staves illustrating different musical intervals with accidentals. Below the chords are the ratios 81 : 64, 5 : 4, and 5 : 4 respectively.