Residue-pitch calculation from spare partial data recovers a bell’s perceived pitch

Evidence

The Hosanna Bell in Freiburg, Germany, was commissioned in 1258. It is of a design now considered antiquated; its partials are not well spaced by modern criteria, owing ultimately to its shape. The bell is “long waisted” and “shaped like a large flowerpot with a heavy rim” according to William A. Hibbert, whose excellent 2008 PhD thesis (The Open University, Milton Keynes, United Kingdom) on bells includes a study of the Freiburg Hosanna. The Hosanna has a very definite pitch, which most people agree is near 308 Hz, rather high for a bell of this size. The partials are at frequencies 135.4, 267.4, 346.4, 365.8, 615.8, 912, 1231.6, 1582, 1962, 2356 Hz, with relative amplitudes 0.28, 0.5, 0.82, 0.1, 1.1, 0.85, 0.75, 0.2, 0.3, 0.25. Figure 23.21 shows the power spectrum and the autocorrelation using just this data. The residue pitch (formula 23.1) using Nn = (0, 1, 1, 2, 2, 3, 4, 5, 6, 8)—arrived at by counting the nearest whole number of periods of each partial present—is 308.6 Hz. — Eric J. Heller, Why You Hear What You Hear- An…, 2130

Categorization

• DEFINE: Concretely illustrates ‘residue pitch’: a pitch derived from sparse partial spacing, computing nearest whole-number periods per partial.
• FICTION: A 13th-century bell with a secretly awkward shape whose perceived pitch only modern mathematics can explain is a nice anchor for a story about craft vs. physics.