A reading guide. The order is suggested; each article stands on its own.
A saw sings when its blade is bent. Two different chords have the same interval inventory. A composer asks a player piano to maintain a tempo ratio that no pair of integers can express exactly. These articles begin with such specific problems and follow the mathematics far enough to explain what is happening.
Beauty is the reason for collecting them, not a quantity they all claim to measure. Some concern a physical mechanism, some a mathematical construction, and some an experiment on a listener. Keeping those questions distinct leaves room for the connections between them.
Why a Saw Needs an S to Sing
Curvature, a localised vibration and the conditions under which topology helps explain a familiar musical instrument.Two Chords, One Interval Fingerprint
What an interval inventory preserves, what it loses, and how the same reconstruction problem appears in diffraction.When Consonance Changes with the Instrument
The interaction between partials, tuning and listener judgements, with roughness kept distinct from musical preference.A Melody That Contains Its Own Mirror
Two recursive rules, their embedded subsequences and the connection to Thue–Morse order.The Bell That Needed a Bulge
Choosing a shape to change a partial, while the rest of the bell’s spectrum imposes constraints.How Close Can Two Beats Come Without Meeting?
Nancarrow’s tempo canons, rational approximation and the difference between exact coincidence and a small timing error.