Editing Organ flue pipe scaling (section)

==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).<ref name="Mainstone">Mainstone, John (1998). "The physics of the organ". In Nicholas Thistlethwaite & Geoffrey Webber (Eds.), ''The Cambridge Companion to the Organ'', p9. 37-41. Cambridge: Cambridge University Press.</ref> 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 <math>e = 0.6r</math>, where <math>r</math> 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. <ref name="Mainstone" /><ref>{{cite web |last1=Hedberg |first1=Dennis |title=Physics of End Correction in Organ Pipes |url=https://www.atos.org/sites/default/files/pdf/programs-awards/EndCorrection_Complete_1.pdf |access-date=19 Jun 2021}}</ref>

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.