Spring tuning models adaptive retuning with vertical springs (just ratios), horizontal springs (pitch stability), and grounding springs (tonal center)

DeLaubenfels’ spring tuning algorithm models adaptive intonation using a physical analogy: each pair of simultaneously sounding notes is connected by a ‘vertical spring’ that pulls the interval toward the nearest just ratio; ‘horizontal springs’ resist melodic pitch changes over time, preventing excessive wandering; ‘grounding springs’ pull the overall tuning toward a fixed reference (e.g., 12-tet), preventing global pitch drift. The spring constants can be adjusted: stronger vertical springs produce purer intervals at the cost of more melodic instability; stronger horizontal springs smooth melody but allow more harmonic roughness. The model can also be used in ‘non-real-time’ mode to compute a Calculated Optimum Fixed Tuning (COFT) — the best fixed tuning for a given piece.

Examples

A performance of a Bach chorale through the spring tuning model: sustained chords breathe into just intonation while melodic lines maintain pitch identity. With all horizontal springs rigid, the algorithm finds an optimal fixed tuning for the piece — analogous to historical well-temperament optimization.

Assessment

A piece modulates frequently and the spring tuning system is producing excessive pitch fluctuation in the melody. Which spring type should be strengthened to reduce this? What is the trade-off of doing so? Explain how the ‘grounding spring’ prevents the phenomenon of global pitch drift.