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

402 their vibrations meet a certain number of times, and being reunited they make a similar impression as if a third sound were formed, the vibrations of which were tuned with the meeting of the first two. The two sounds may make a fifth, or be expressed by the numbers 2 and 3, and while the first makes two vibrations the other may make 3: let us set a time in which the first makes 200 vibrations, and the other 300, and if the first two vibrations have met once, the same encounter must happen 100 times in the given time, and these represent a sound which makes 100 vibrations in the given time, which will correspond to the number 1. So two sounds expressed by the numbers 2 and 3 produce a third sound expressed by the unit; and in general two sounds expressed by two numbers whatsoever produce a sound expressed by 1, or by a common divisor of the first two. All the examples by Signor Tartini are congruent with this conclusion; and the sound produced according to this rule does not differ from that observed by the author but by one or a few octaves, which is not a substantial difference. So leaving aside the octaves, the major third 4 : 5 produces the sound 1 which is two octaves below the lower element 4; the minor third 5 : 6 produces a sound which is of two octaves with a major third below the lower 5, or simply of one major third below. The fourth 3 : 4 produces a third sound 1, a fifth below the lower element 3. The minor sixth 5: 8 produces a sound 1 of a few octaves below the high element 8, and the major sixth 3 : 5 a 1, which is a fifth below the lower 3: all the other examples proceed equally well. Therefore, Signor Tartini’s rule to find the bass, given that being two notes it comes back to this, that the bass must contain a sound expressed by a common divisor of the numbers which express the given sounds. It is not certain therefore that the practice of this excellent composer often moves away from this rule, as one can see in the examples reported at the end of his treatise; hence it appears that the judgement of the chords must be repeated by their exponents, as I have already established. The chords that musicians call consonances, are all comprised in the exponent 3 · 5 = 15 multiplied by any power of 2. All the other exponents not comprised in these give the chords that are called dissonances. So the exponent 36 = 2 · 2 · 3 · 3 giving the sounds expressed by the numbers 4, 6, 9 gives the chord ? www evaluated as dissonant, not for the two successive fifths as much as because of the exponent 36 not included in the above-mentioned form. Again, the chord B wwwww 6 9 12 15 20 which includes the sounds expressed by the numbers 6, 9, 12, 15, 20 and will have as exponent 180 = 2 · 2 · 3 · 3 · 5 is considered dissonant and similarly again that which has as exponent 400 = 2 · 2 ·2 · 2 · 5 · 5, which is of a different nature from those which