Loudness-Factor Alignment at the Grand Piano Bass/Treble Break: A Decision-Support Framework for String Replacement and Scale Design

The bass/treble break (BTB) of a grand piano is among the most acoustically demanding regions of the string scale, yet no published workshop tool exists to diagnose and adjust the BTB Z shift from routine measurements. This paper presents a decision-support framework for BTB loudness-factor alignment based on Roberts' loudness/sustaining factor Z. It first shows that wire diameter cancels identically from the Z formula for both plain and wound strings once tension is derived from measured m, L, and d data, reducing cross-instrument comparison to a unison-adjusted T/L relation. The anchor adopted is c_Bt = 0 ⇔ T_Bt = 183.7 lbf at N_Bt = 2, derived from Roberts' Steinway D normalization and cross-checked by five pianos fitted with wound bass strings manufactured by J.D. Grandt, whose bass-top tensions cluster around an lp-corrected geometric mean of 184.2 lbf. The framework combines a position diagnostic and a scale-geometry design diagnosis. The position diagnostic derives the theoretical lowest plain-wire position m̂_lp from speaking-length geometry; the displacement m̂_lp − m_lp(actual) distinguishes structural Z-constraint from Z-adjustability reserve. The design diagnosis evaluates the bass-top tension reference, lp-zone tension floor, I₄ diameter ceiling, and breaking-load percentage as coupled constraints on a technician-selected c_Bt. Across the 16-instrument dataset, the single-step Z discontinuity ΔZ(actual) spans from below 1 cent (S&S D HM) to 576 cents (S&S L), and the composite ratio r_t·r_b predicts the appropriate c_Bt setting with R² = 0.95. Two speaking-length measurements, L_lp and L_Bt, thus initialize the BTB position diagnostic, the Z-jump target, and the tension-design starting point simultaneously.