Playable BooksJean-Philippe Rameau · section 14 of 109
Chapter eleven
Read and hear this section in the playable edition →How to Relate Ratios Given for Divisions to Vibrations and Multiplications of Lengths
Take separately, from the proposed string, the lengths resulting from each division, extending rightward from the number to the end of the string. Arrange them so that their vibrations can be distinguished, assuming the strings differ only in length. We shall find that the vibration ratios agree with the division ratios. We may then divide a string into as many parts as are needed to obtain the ratios of dissonances, and shall always find the same agreement. See above, Chapter III, Article VI, p. 15.
To obtain the ratios of lengths, take separately the two lengths arising from two different divisions and give each a common measure with a compass. Each length will contain that common measure as many times as the numbers marking the divisions contain units, except that the comparison will be reversed. To explain: compare the lengths resulting from the divisions marked 2 and 3. String 2 contains the measure three times, and string 3 only twice. Thus on one side we compare 2 to 3, and on the other 3 to 2. This would amount to the same thing if the first number of each ratio did not represent the lowest sound. We therefore need only reverse in this way any ratio conceived in the divisions to relate it to lengths. But what seems very easy in comparing two sounds becomes more troublesome as their number increases. In harmony, as in a continuous quantity, the middle sounds or terms must be related to each extreme. Thus we do not find in 4, 3, 2 what we find in 2, 3, 4: the interval generated by comparing 2 to 3, which is first here, is last on the other side. This obliges us to invert our arithmetic proportion, as we have said elsewhere, by multiplying both extremes by the mean and then one extreme by the other, restoring the intervals to their natural order through the numbers 12, 8, 6. In short, if the most perfect harmony that the union of consonances can produce is represented in divisions by 1, 2, 3, 4, 5, 6, 8, it can be represented in the multiplication of lengths only by 120, 60, 30, 24, 20, 15. This therefore demands more attention than the rest.
We could use this common measure to find at once the ratios of intervals contained in the lengths taken to the left of the proposed string, or proceed through the inversion just observed. We need only say: if string 3, compared with string 1, contains 1 in one of its parts, it will contain 2 in its other two parts; string 1, compared with 3, must consequently contain 3. Thus these left-hand lengths give the fifth, whose ratio is 2 to 3. If I next compare string 3 with string 2, since 3 contains 2 in one of its parts, it will contain 4 in the other two; and since 2 contains 3 in one part, it will likewise contain 3 in the other. The left-hand lengths therefore give the fourth's ratio between 3 and 4. Similarly, if string 3 compared with string 4 contains 4 in one of its parts, it will contain 8 in its other two parts; and if string 4 compared with string 3 contains 3 in one of its parts, it will contain 9 in its other three. I thus see that the ratio of the whole tone arising from the difference between the fifth and fourth is contained in 8, 9 according to divisions, and 9, 8 according to multiplications. This presents no difficulty and can be verified with every kind of interval.