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

263 LETTERS 1) The order of the harmonic progression is found created by nature. 2) The order of the number proves and demonstrates the necessity of the same progression. 3) One must consider the number as geometric at least in its root, which is the duple, because in fact nature simply doubles or multiplies the duple, dividing the said duple harmonically, as it is divisible. 4) Therefore, the first duple is indivisible, since between 1 and 2 there falls no other number. The 2nd duple between 2 and 4 is divisible, because in the middle there is 3, which forms a fifth with the 2, a fourth with the 4, that is to say the first, sesquialter, the second, sesquithird. The 3rd duple between 4 and 8 is more divisible as the numbers 5, 6, 7, fall in between; 4 and 5 sesquifourth or major 3rd; 5 and 6 sesquififth or minor third; 6 and 7 sesquisixth, 7 and 8 sesquiseventh, both true and legitimate consonanc- es unknown to both Greeks and Latins, and even less known to those Italians who referred back to the Greek institution. In Nature nothing is superfluous or missing. If the above-mentioned last two proportions are present in all the three instruments mentioned above, that is to say made by nature, or where nature at least acts alone, then they are not superfluous but necessary; and if everything which perfectly starts, must still perfectly end, nature starting its progression in the duple which is the most perfect of all proportions, then it will have to end all its proportions in the same duple. Therefore not in the senary number, as Zarlino says, 7 but in the octonary it will complete its division, and so it will be true that between 4 and 8 there will be nothing missing; which was indeed not true in the 6, because between 4 and 6 there is no duple but sesquialtera. More divisible still is the 4thduple between 18 and 16, as the numbers 9, 10, 11, 12, 13, 14, 15 fall between them. From this last division one realises evidently that our scale is lacking one element, because whereas our scale is composed of eight elements, and seven steps or distances, the one made by nature is of nine elements and eight distances or steps. There is furthermore much diversity in the proportions themselves, because in the third and fourth element of our scale there is a semitone of sesquififteenth proportion; in that made by nature there is a sesquitenth proportion. In the 4th and 5th element of ours there is the sesquioctave tone, in the other there is the sesquitenth proportion. Between the fifth and sixth element of ours there is the sesquininth tone, in the other there is the sesquitwelfth proportion. Between the sixth and seventh element of ours there is the sesquioctave tone, and in the other there are two proportions: the first of a sesquithirteenth, the second, which is that lacking in our scale, of a sesquifourteenth. However, one of the most important observations that must be made in the scale cre- ated by nature, is that between its 4th element and the first, being the supertripartient octave proportion, this forbids the fourth of the tone from being able to be modulated, 7 See Claude V. Palisca, “Zarlino, Gioseffo”, in Ng, vol. 27.