Scaling is the ratio of an organ pipe's diameter to its length. The scaling of a pipe is a major influence on its timbre. Reed pipes are scaled according to different formulas than for flue pipes. In general, the larger the diameter of a given pipe at a given pitch, the fuller and more fundamental the sound becomes. The effect of the scale of a pipe on its timbre The sound of an organ pipe is made up of a set of harmonics formed by acoustic resonance, with wavelengths that are fractions of the length of the pipe. There are nodes of stationary air, and antinodes of moving air, two of which will be the two ends of an open-ended organ-pipe (the mouth, and the open end at the top). The actual position of the antinodes is not exactly at the end of the pipe; rather it is slightly outside the end. The difference is called an end correction. The difference is larger for wider pipes. For example, at low frequencies, the additional effective length at the open pipe is about , where is the radius of the pipe. However, the end correction is also smaller at higher frequencies. This shorter effective length raises the pitch of the resonance, so the higher resonant frequencies of the pipe are 'too high', sharp of where they should be, as natural harmonics of the fundamental note. This effect suppresses the higher harmonics. The wider the pipe, the greater the suppression. Thus, other factors being equal, wide pipes are poor in harmonics, and narrow pipes are rich in harmonics. The scale of a pipe refers to its width compared to its length, and an organ builder will refer to a flute as a wide-scaled stop, and a string-toned gamba as a narrow-scaled stop. Dom Bédos de Celles and the problem of scaling across a rank of pipes The lowest pipes in a rank are long, and the highest are short. The progression of the length of pipes is dictated by physics alone, and the length must halve for each octave. Since there are twelve semitones in an octave, each pipe differs from its neighbours by a factor of . If the diameters of the pipes are scaled in the same way, so each pipe has exactly the same proportions, it is found that the perceived timbre and volume vary greatly between the low notes