Giuseppe Tartini - Lettere e documenti / Pisma in dokumenti / Letters and Documents - Volume / Knjiga / Volume II

261 LETTERS but it does me little good, because objections cannot be raised of such a type that they can make me study or change my opinion. The monochord is divided with the rule of the common arithmetic number 1, 2, 3, 4, 5, 6, 7, 8, etc. without ever altering the order of the number; because just as it not allowed to pass from 6 to 8 without numbering 7, which is in the middle of 6 and 8, in the same way in the division of the monochord it is never allowed to neglect any number and pass from 5 to 7, from 9 to 11, but from 5 one must go to 6, then to 7, and likewise from 9 to 10 and then to 11; and one proceeds with this order, if one wants, until the infinite in the same way, in which one counts up to the infinite, if one wants. The reason for the necessity of this rule is clear, and actually it is not a reason, but a demonstration, assuming the principle to be true (as in fact it is) that in a harmonic progression one must pass from the higher to the lower proportions by successive degrees, and so it is true that it must be done in this way as it is never possible to pass in another way. For example, the greatest and first of all the proportions is the duple. There shall never be a passage from the duple to the sesquithird, if one does not pass first by the sesquialter, the mean between the duple and the sesquithird. But the duple is like 1 to 2, and the sesquithird like 2 to 3, the series of the number 1-2-3-4 [inserire segni grafici] is necessarily born thereof; that is to say, duple, sequialtera and sesquithird. The axiom is true, and infallible in any series of proportions. Having supposed all of this, the division of the monochord is easy to understand. The string of the instrument is one; it is marked with the number 1, and it is intended as the unison, which is not and cannot be a consonance; as, if the consonance is the proportion of two unequal sounds, neither two equal sounds nor a single sound shall ever be a consonance. From 1 one necessarily passes to 2, and this means that the string 1 is divided into two parts, each of which is in duple proportion with the entire string, that is to say in octave. These two parts, because they are equal (given that the string is divided into two halves), are in unison, not in consonance. The duple is therefore the first of all consonances for the ratio of 2, which is the prime number after 1. It is still the biggest in proportion to all those following, as there is more distance from 1 to 2 than from 2 to 3; that is to say, half of the string is more than a third of the same string. Because even though the difference between 1 and 2 is one, and between 2 and 3 is likewise one, this does not make these two differences equal, as the first 1 indicates half, and the second 1 indicates a third, as we shall see immediately. From the division of the string in two equal parts one proceeds to the division of the same in three equal parts by the ratio of 3, which succeeds 2. Each of these parts is in sesquialtera proportion, that is of the fifth, with the octave of the whole string; and they are in unison, because they are equal. The intonation of those two proportions reduced to practice is this.