Of Music.

STROZZI. How many of our commas will the major and minor tone, and the semitone, come to contain?
BARDI. I will tell you this briefly for now. The Sesquioctogesima: The ratio 81:80, known as the Syntonic Comma. consists of three commas, and a bit more than a fourth of one, but less than half of one. The Sesquiquindecima: The ratio 16:15, or the major semitone. consists of five commas and a little more than an eighth of one. The Sesquinona: The ratio 10:9, or the minor tone. exceeds eight commas by a little less than a half. And the Sesquioctava: The ratio 9:8, or the major tone. exceeds nine commas by the same amount that the minor tone exceeds eight. How many of the ancient commas the Tone and their major and minor semitones contained, Boethius declares to you very well. From the example given to you above—in proving to you that the minor tone contained nothing more than the major and minor semitone—one can also draw true knowledge of how much the tone exceeds each of those separately. And having seen and understood that the Comma added to the minor tone produces the major tone, one comes to know by how much the former is exceeded by the latter. Thus, of the intervals we have treated until now, it only remains to make known to you how much the major tone exceeds the minor and major semitones; this knowledge will be given to you by the two examples I place here at the bottom.

The minor semitone is exceeded by the major tone by the Superbipartient 25; this consists of a major semitone and a comma. This is so clear that no other proof is required. However, we will use the following example to subtract the major semitone from the major tone, to see what remains of it.

The major semitone is exceeded by the major tone by the Super-sept-partient 128. This interval (as can be understood from what we have said above) contains within itself a minor semitone and one additional comma. I am proving these truths to you in several ways to further confirm you in the true opinion one must have regarding the value and content of intervals, and by how much one exceeds the other. From what has been declared so far, we can very well see and understand the distance that exists from the flat b molle marked on A-la-mi-re or E-la-mi, to the sharp diesis X marked on G-sol-re-ut or D-la-sol-re, as seen noted in this example. And so that a complete and correct judgment may be made, we will consider—besides what has been said above—the quality of the interval in which such accidental signs occur, and furthermore the power they have to operate in such a place. Wherefore I say first, that both intervals where these effects are produced are, in their essence, a minor tone. Each of these is diminished at both extremes by the sharp (X) and by the flat, by one minor semitone. It is now necessary to see what remains of a minor tone after two minor semitones are subtracted from it. And because part of this occurred above when we examined the quality of the semitones, this single observation will suffice as necessary: that is, by taking two minor semitones from a minor tone, there will remain the same interval that remained when the minor was subtracted from the major semitone; and by this same amount is the flat of A-la-mi-re and E-la-mi higher in pitch than the sharp (X) of G-sol-re-ut and D-la-sol-re.

A musical staff with five horizontal lines containing four notes with accidentals. The first note is a B-flat on the second space (A), followed by a G-sharp on the second line (G). The third note is a B-flat on the first space (E), and the fourth is a D-sharp on the fourth line (D). The staff is segmented by vertical bars.

STROZZI. This has indeed been a new and pleasing speculation for me, which I have many times desired to understand—not so much for the sake of counterpoint, as for the keyboard instrument.
BARDI. I wish furthermore to warn you that when such a case arises in major tones (though it happens rarely), the remaining interval would be a comma smaller than that one—provided, however, that one considers its minor semitones in the form and measure that we shall say befits them in the proper place. The minor third, which has also been called the Semiditone and Sesquitone by moderns, is said to be that interval contained in its radical terms by the Sesquiquinta: The ratio 6:5. proportion, between these numbers 6 and 5. In the Syntonic Diatonic strings of Ptolemy: A tuning system favoring "pure" thirds based on simple ratios.—where we shall presently and principally consider all the intervals we need to measure to complete what we have proposed—this contains a semitone and a tone, both of which are major. Thus, by summing together the numbers that enclose the said intervals, one will have their true form from their product, as appears in the example that follows.

What the Sesquitone is. Fabro in chapter 24 of the third book of The Elements of Music. Refers to Jacques Lefèvre d'Étaples (Jacobus Faber Stapulensis).