These fragments attributed to Gerland deal with the measurement of organ pipes and the proportional ratios between length, diameter and musical consonances. The text describes the mathematical criteria used to obtain intervals such as the diapason, diapente and diatessaron, together with the construction procedures for bass, treble and super-treble pipes according to medieval music theory.
Gerland again on pipes
If the pipes have the same diameter and the larger contains the smaller twice in its length, and moreover the diameter of the bore is added, they will produce between them a consonance of diapason (octave).
If instead the larger pipe contains four times the length of the smaller, and moreover includes the measure of the diameter, it will produce a double consonance of diapason.
If the larger pipe contains the smaller entirely, and moreover a third of its length and also a third of the diameter, it will produce a consonance of diatessaron (fourth).
If the larger contains the smaller entirely, and moreover half of its length together with half the diameter, it will produce a consonance of diapente (fifth).
If the larger pipe contains the smaller entirely, and moreover an eighth of its length, a tone will result.
If instead the larger pipe contains the smaller entirely, and moreover a sixteenth of its length, a semitone will result.
Again on the “nolae” (notes/weights)
If you wish to build the notes, take any weight and call it A.
Then take the whole of A and add its eighth part, and you obtain B: you will thus have a tone.
Take B and add its eighth part, and you obtain C; take the whole of A and add a third of its length and obtain D, and you have a diatessaron consisting of two tones and a semitone. Take the whole of D and its eighth part, and the whole of A and its half, and you obtain E. Take the whole of E and its eighth part, and you obtain F. Take the whole of F and its third part, and you obtain G. Take the whole of G and its eighth part and you obtain a, and thus you have the complete diapason, and the diatessaron and diapente connected to each other. If instead you wish to add the synemmenon, take the whole of G and its eighth part, divide it in half and add this half to G: you will obtain the synemmenon, which must be placed between G and a.
On pipes
Add an eighth of the diameter to the first pipe and divide the remaining length into nine parts, of which you will assign eight to the second. In the same way add an eighth of the diameter to the second pipe and divide the remaining length into nine parts, of which you will assign eight to the third.
Then, returning to the first, add a third of the diameter and divide the remaining length into four parts. In the same way as you obtained the fourth from the first, so from the second you will obtain the fifth, from the third the sixth and from the fourth the seventh, always adding a third of the diameter and dividing by four.
Then, returning to the third, add half the diameter and divide the remaining length into three equal parts, of which you will assign two to the eighth pipe.
The treble pipes are derived from the bass pipes as follows: to each bass pipe assign the entire measure of the diameter, and divide the remaining length into two parts, assigning one of them to the pipe. In the same way, starting from the treble, you will find the super-treble.
