and the high, and the result is not musically satisfactory. This effect has been known since antiquity, and part of the organ builder's art is to scale pipes such that the timbre and volume of a rank vary little, or only according to the wishes of the builder. One of the first authors to publish data on the scaling of organ pipes was Dom Bédos de Celles. The basis of his scale was unknown until Mahrenholz discovered that the scale was based on one in which the width halved for each octave, but with addition of a constant. 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. Töpfer's Normalmensur The system most commonly used to fully document and describe scaling was devised by Johann Gottlob Töpfer. 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. This meant that the cross-sectional area varied as . In consequence, the diameter of the pipe halved after 16 semitone intervals, i.e. on the 17th note (musicians count the starting-note as the first, so if C is the first note, C# is the second, differing by one semitone). 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 at 8′ C (the lowest note of the modern organ compass) and the mouth width one-quarter of the circumference of such a pipe. Töpfer's system provides a reference scale, from which the scale of other pipe ranks can be described by means of half-tone deviations larger or smaller (indicated by the abbreviation ht). A rank that also halves in