Whither Tone Ring Ring?

TL;DR

This paper presents a simple model to understand how a banjo's tone ring interacts with the wood rim, relating their resonances and explaining the perceived sound characteristics.

Contribution

It introduces a straightforward model that captures the interaction of the tone ring and rim, matching measurements within 5% for low frequencies.

Findings

Crude measurements align with the model for low-frequency modes.

The system's sub-components become independent at higher frequencies.

The model relates individual part sounds to the combined system's sound.

Abstract

An extremely simple model captures the essence of the interaction of a banjo tone ring with the wood rim. The large scale, low frequency resonances of the assembled system are related to the weights and resonant frequencies of the tone ring and rim separately. Very crude measurements satisfy the derived relations within about 5% for the lowest frequency modes and give qualitative agreement for the next ones on a particular, heavy-tone-ring resonator banjo. The two combined sub-systems become increasingly independent for higher frequency, shorter-lived modes. Nevertheless, the ringing sounds of the struck individual parts, which dominate the perception of their pitch and sustain, are related by the simple model to the sound when the parts are struck when combined into one.