Plotting each stop’s scale against the Töpfer Normal Scale, with the sounding pitch (pipe length) on the horizontal axis, makes the relationships within a principal chorus immediately visible. The foundation stop appears as a roughly level line; the upperwork falls below it, each stop progressively narrower as the pitch rises. The spacing between lines shows how much each stop’s power is reduced relative to the foundation — approximately 0.5 dB for every half-tone of scaling.
In a well-balanced chorus in a small, dead acoustic, the Octave 4′ is typically about 2 HT narrower than the Principal 8′, the Octave 2′ a further 2 HT narrower, and the Mixture 1′ another 2 HT narrower still. This mirrors the harmonic series, where each ascending overtone naturally carries less energy. The degree to which these steps are compressed or expanded depends on the acoustics and the desired tonal result.
High-frequency sounds are absorbed by air more strongly than low-frequency sounds, and the effect increases with distance. In larger rooms, the highest-pitched pipes need to be scaled wider to compensate — sometimes dramatically. This is why the curves in the graphs typically rise above 1′ pitch, and why the compensation is greater in a large, dead room than in a small, live one.
McNeil’s key insight is that graphing the normalized mouth widths is often more revealing than graphing the diameters. The diameter sets the upper limit of harmonic development, but the mouth width — together with the flueway — determines the actual power of the pipe. Different builders use different mouth width ratios within the chorus; these differences become clearly visible when plotted against the Normal Scale of Mouth Widths (circumference / 4).
Sources
Michael McNeil, The Sound of Pipe Organs, Mead, CO: CC&A, LLC, 2012. ISBN 978-0-9720386-5-2.
Johann Gottlob Töpfer, Lehrbuch der Orgelbaukunst, Weimar, 1855.