Mersenne's laws

The equation was first proposed by French mathematician and music theorist Marin Mersenne in his 1636 work Harmonie universelle.

[note 1] Higher-pitched strings typically are thinner, have higher tension, and may be shorter.

[4] Though his theories are correct, his measurements are not very exact, and his calculations were greatly improved by Joseph Sauveur (1653–1716) through the use of acoustic beats and metronomes.

[5] The natural frequency is: Thus, for example, all other properties of the string being equal, to make the note one octave higher (2/1) one would need either to decrease its length by half (1/2), to increase the tension to the square (4), or to decrease its mass per length by the inverse square (1/4).

Similar laws were not developed for pipes and wind instruments at the same time since Mersenne's laws predate the conception of wind instrument pitch being dependent on longitudinal waves rather than "percussion".

A string half the length (1/2), four times the tension (4), or one-quarter the mass per length (1/4) is an octave higher (2/1).
If the tension on a string is ten lbs., it must be increased to 40 lbs. for a pitch an octave higher. [ 1 ]
A string, tied at A , is kept in tension by W , a suspended weight, and two bridges, B and the movable bridge C , while D is a freely moving wheel; all allowing one to demonstrate Mersenne's laws regarding tension and length [ 1 ]