Organ flue pipe scaling

In general, the larger the diameter of a given pipe at a given pitch, the fuller and more fundamental the sound becomes.

One of the first authors to publish data on the scaling of organ pipes was Dom Bédos de Celles.

[3] This constant compensates for the inappropriate narrowing of the highest pipes, and if chosen with care, can match modern scalings to within the difference of diameter that one would expect from pipes sounding notes about two semi-tones apart.

The system most commonly used to fully document and describe scaling was devised by Johann Gottlob Töpfer.

[4] Since varying the diameter of a pipe in direct proportion to its length (which means it varies by a factor of 1:2 per octave) caused the pipes to narrow too rapidly, and keeping the diameter constant (a factor of 1:1 per octave) was too little, the correct change in scale must be between these values.

Töpfer reasoned that the cross-sectional area of the pipe was the critical factor, and he chose to vary this by the geometric mean of the ratios 1:2 and 1:4 per octave.

Töpfer was able to confirm that if the diameter of the pipes in a rank halved on the 17th note, its volume and timbre remained adequately constant across the entire organ keyboard.

He established this as a standard scale, or in German, Normalmensur, with the additional stipulation that the internal diameter be 155.5 mm (6.12 in) at 8′ C (the lowest note of the modern organ compass) and the mouth width one-quarter of the circumference of such a pipe.

If a rank does not halve exactly at the 17th note, then its relationship to the Normalmensur will vary across the keyboard.

Normalmensur scaling table, 17th halving ratio: From Organ Supply Industries catalog