Dialogue on Ancient and Modern Music

On Music.

STROZZI. This will be exceptionally helpful to me, as well as new and welcome.

BARDI. I am coming to treat every smallest detail that one could desire regarding the form of the Minor Sixth, also understood by modern practitioners as the Minor Hexachord A scale or interval of six notes.. They say this interval is that imperfect consonance contained by the Supertripartientequinta: A ratio where the larger number contains the smaller number plus three-fifths of it (8:5). in its radical terms. It consists of two major tones, one minor tone, and two major semitones. It can be considered primarily as being composed of a Fourth and a Minor Third, as we shall see from the product yielded by adding together the smallest numbers of their proportions.

In this regard, there are some who criticize Franchino Franchinus Gaffurius (1451–1522), a highly influential Italian music theorist. for having said that the Minor Sixth is composed of the Diatessaron: The Greek term for a perfect Fourth. and the "greater chromatic interval." By this, he meant the Trihemitone, using it in place of the minor third, or more accurately, the Semiditone. These intervals, when added together, cannot in any way produce the minor sixth in the form shown in the following example; nor can they do so according to the Diatonic Ditonic The Pythagorean tuning system, which uses only major tones (9:8). system in which Franchino considers it. This perhaps happened because he did not hold this interval in such high esteem—being an imperfect consonance—as many others do.

Musical staff showing two notes (A and F) with mathematical ratios to the right.

Warning.

Regarding this imperfect consonance, another consideration must not be kept silent: the common people held for certain that every musical interval has one fewer "species" A specific arrangement of tones and semitones within an interval. than the number of its strings Notes.. However, this is not true, because the species of the Minor and Major Sixth are no more than three; furthermore, this rule does not hold true for several other intervals which we will omit for brevity.

Returning to the Minor Sixth, I say it is also that which consists of the Semidiapente A diminished fifth. and the Major Tone. By adding together the numbers by which they are contained, their product will give its true form, as shown in the example that follows.

Musical staff showing two notes (B and G) with mathematical ratios to the right.

Minor Sixths are not found between these strings.

This truth being established, it follows (contrary to the opinion of the practical musician) that some of these intervals—or any other similar ones—are not Minor Sixths of the same proportion as the first ones shown. This arises because, between the notes shown, the Tone and the Semidiapente together cannot yield the Minor Sixth as they did above. This is because in these cases the Tone is minor; whereas, conversely, in the previous ones it was major. Thus, these necessarily become smaller than the others by a Comma. We shall now see which interval is produced from them by the product we get from adding their lowest terms together.

Musical staff showing two notes with mathematical ratios to the right.

The interval obtained from these is the Super 47-part-of-81, which contains the true Ancient Minor Hexachord of the Diatonic Diatonic system in its lowest terms. This is one Comma smaller than the Supertripartientequinta (8:5) known today as the form of the Minor Sixth. This interval is also called the Minor Hexachord by some and is consonant, unlike that ancient one which is dissonant because it is (as has been said) a Comma smaller. Anyone who takes the care to subtract its form from the other will be certain of this. One can also obtain this imperfect consonance from the Fifth and the Major Semitone, as we shall see from their product by adding together...