Twelve Books on Harmony

The same principle of magnitude is proven by the examples I will bring forward in the book on Bells. There are many thicker objects that sound higher than thinner ones. This is true not only for different materials, but even for the same material. For instance, among bells of the same size, the one that is thicker sounds higher.

Now let us consider the examples he relates when he says that things strike "more collectively" original Greek: athroōteron when they are "thinner" original Greek: leptotera, because they can pass through or be moved more quickly, as well as being "more dense" original Greek: pyknoteros mallon. He compares bronze to wood, and a gut string to flax. Certainly, it cannot be denied that bronze is denser than a string, and yet a string of the same size (that is, of equal length and thickness) and even under the same tension, sounds far higher, as will be demonstrated in the following books.

It may be doubted whether hollow arteries and reeds sound higher when they are denser. Therefore, we must see whether, for example, a bronze pipe sounds higher than a lead or tin pipe of exactly the same size. I conducted an experiment with the excellent flute-maker original Greek: aulopoion Vacherius. A pipe of the same cavity and material does not change its pitch, no matter how thick or thin it is. It seemed it should resist the sound more when it is thicker, just as it does when it is harder.

Furthermore, he compares the hardness of bronze with the density of lead. He says a higher sound is produced by bronze because bronze exceeds lead in its hardness more than lead exceeds bronze in its density. By this reasoning, the objection I raised above concerning lead and bronze can be explained. This also applies to what I brought up regarding thicker bells from the points that follow. A larger and thicker bronze object sounds higher than a smaller, thinner one when the ratio of the magnitudes exceeds the ratio of the thicknesses. That is, as I believe, when the length or width of the thinner object exceeds the thickness of the larger one by more than the thickness of the larger exceeds the thickness of the smaller. But these are mentioned here as a sketch and in passing, as we shall say more in the following book. From these points, he concludes that low and high pitch pertain to quantity, a view I freely support.

We must not omit his discourse which completes the chapter. He says: As the greater distance from the starting point is to the lesser, so is the sound which proceeds from the lesser distance to that sound which proceeds from the greater. This is like weights. For as the greater distance of the weight or fulcrum original Greek: hypomochlion is to the lesser, so is the weight in the smaller part or distance to the weight in the larger: original Greek: hōs hē meizōn apochē tou hypomochliou pros tēn elattona, outōs hē apo tēs elattonos rhopē pros tēn apo tēs meizonos. He explains this by the example of a string and a flute. Because a string is shorter, and because the holes of the flute (or the mouth-pieces) are closer, the sound is higher. As the sharpness of the sound in the shorter string relates to the longer string, so the lower sound of the longer string relates to the shorter. As sound is to sound, so is string to string, as will be discussed more fully in the next book.

Furthermore, what he adds about the artery and its comparison with the flute must be discussed in the book on the Voice. In Chapter 4 of the same book, he teaches that sounds from the lowest toward the highest are "potentially" original Greek: dynamei infinite, but "actually" original Greek: energeia finite. One limit of these belongs to the sounds themselves, and the other to hearing. It is certain that the latter is greater than the former. For there is no one who can reach such a high pitch that the ear cannot perceive a higher one. Perhaps the same should be said of low pitch. He says some sounds are "equal-toned" original Greek: isotoni and others are "unequal-toned" original Greek: anisotoni, which is to say, unison and non-unison. In this context, "tone" is taken for a certain degree of the system, and it is common to both low and high pitch, much like "place" original Greek: topos, acting as a boundary for the end and the beginning. Again, some unequal-toned sounds are "continuous" original Greek: synecheis and others are "discrete" original Greek: diestēkotes, about which we spoke in proposition 23.

I add only the comparison of the rainbow which he uses. Continuous sounds that serve speech do not have certain and manifest transitions from low to high and high to low, just as the colors of the rainbow do not. The first color passes through nearly innumerable others to the second, and the second to the third. An example of sounds that pass gradually and "imperceptibly" original Greek: aphanōs from high to low is taken from the lowing of cattle, which he calls mooings original Greek: emykēthmus. Others end in a high pitch from a low one.

Finally, he teaches that melodic sounds are those which, being "tuned to one another" original Greek: emmeloumenoi pros allēlous, attain a "clear expression" original Greek: emphasin to the hearing. Discordant sounds are those which behave in the opposite way. Among these, he adds "consonant" original Greek: symphōnous sounds, which he says are vocal and the most beautiful of all. "Dissonant" original Greek: diaphōnous sounds are "those which do not behave in this way." I did not want to omit these points, lest anyone think I overlooked what the ancients taught about sounds, and so that everyone might behold both the old and the new in a single view. I would add the things Aristoxenus holds concerning the tension, relaxation, low pitch, high pitch, and standing of sounds, if I were not going to explain them in another place.

PROPOSITION XXV.

To explain the generation of both low and high sounds.

In addition to what we have brought from Aristotle and Ptolemy, it is pleasing to relate what is read in the Mirror of Music by Johannes de Muris, Canon and Dean of the Church of Paris. This work is kept in the Royal Library. In it, after the seven books of the Mirror, a treatise is added without the author's name. In the first chapter, after saying that three things are necessary for sound according to Aristotle's opinion cited in proposition 24 (and sound is defined as the breaking of air from the impact of the striker upon the struck), the following is held.

A low sound is created from slower movements, while a high sound is created from faster and thicker movements. (For it should be said "faster" and not "sharper," as is read in that same place).

Although there are many distinct movements in both, which make many sounds in number (as is clear in the vibrations of strings, of which we shall speak at length in the following book), they are nevertheless heard without interruption like a single sound because of the speed of the vibrations. This happens in nearly the same way that a circular line is seen on a moving spinning top original Latin: trochus, even though the color exists only as a point on its very edge. A high sound is made from a low one by the addition of movements, just as a low sound is made from a high one by the subtraction of movements. Therefore, there are more sounds in high pitches and fewer in lower ones. Every small number is related to a plurality under a certain number, and thus every high sound is related to a low one. Whether a low sound is of greater importance in music than a high one, and whether it is the basis of harmony, will be discussed in the book on the Composition of Melodies. For it is enough that we notice Aristotle, in Book 5 of The Generation of Animals, chapter 7, thinks that a low sound is to be preferred to a high one because it is more noble and greater: "wherefore a low voice seems to be of a better nature, and in melodies the low is better than the sharper."