Sauveur's ideas took their development from his having instituted at all points more exact quantitative investigations than his predecessors. He is first desirous of determining as the foundation of musical tuning a fixed note of one hundred vibrations which can be reproduced at any time; the fixing of the notes of musical instruments by the common tuning pipes then in use with rates of vibration unknown, appearing to him inadequate. According to Mersenne (_Harmonie Universelle_, 1636), a given cord seventeen feet long and weighted with eight pounds executes eight visible vibrations in a second. By diminishing its length then in a given proportion we obtain a proportionately augmented rate of vibration. But this procedure appears too uncertain to Sauveur, and he employs for his purpose the beats (_battemens_), which were known to the organ-makers of his day, and which he correctly explains as due to the alternate coincidence and non-coincidence of the same vibrational phases of differently pitched notes.[137] At every coincidence there is a swelling of the sound, and hence the number of beats per second will be equal to the difference of the rates of vibration. If we tune two of three organ-pipes to the remaining one in the ratio of the minor and major third, the mutual ratio of the rates of vibration of the first two will be as 24: 25, that is to say, for every 24 vibrations to the lower note there will be 25 to the higher, and one beat. If the two pipes give together four beats in a second, then the higher has the fixed tone of 100 vibrations. The open pipe in question will consequently be five feet in length. We also determine by this procedure the absolute rates of vibration of all the other notes. It follows at once that a pipe eight times as long or 40 feet in length will yield a vibrational rate of 12½, which Sauveur ascribes to the lowest audible tone, and further also that a pipe 64 times as small will execute 6,400 vibrations, which Sauveur took for the highest audible limit. The author's delight at his successful enumeration of the "imperceptible vibrations" is unmistakably asserted here, and it is justified when we reflect that to-day even Sauveur's principle, slightly modified, constitutes the simplest and most delicate means we have for exactly determining rates of vibration. Far more important still, however, is a second observation which Sauveur made while studying beats, and to which we shall revert later. Strings whose lengths can be altered by movable bridges are much easier to handle than pipes in such investigations, and it was natural that Sauveur should soon resort to their use.