Mathematical Symmetry Research Articles

XVIII.–The Musical Scale

I. The Tonal Order of the Chromatic Scale.The exact ratios of the chromatic scale were first made known by Claudius Ptolemy of Alexandria about A.D. 150. We give them here as he found them, unaltered and unalterable.It is a most remarkable fact that while the ratios of the chromatic scale have thus been known for nearly two thousand years, its mathematical symmetry and origin have remained unrecognised down to the present day.For the concrete musical faculty is compounded of two distinct elements, physical and mathematical.In its physical aspect (with which we are familiar), the scale is a series of notes arranged in order of pitch upwards from the tonic (or keynote).In its mathematical aspect, the scale is a system of intervals measured symmetrically upwards and downwards from tonic and dominant as central interval (or tonal centre).The physical order of the scale is that which we most easily appreciate, but the mathematical order is that in which it originates. The chromatic scale therefore is, primarily, not a series of notes, but a system of intervals (or tonal system).Diagram I shows the scale in its familiar order of pitch, and it is remarkable how effectually this physical order conceals the mathematical from view.