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 cBt = 0 ⇔ TBt = 183. 7 lbf at NBt = 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 m̂ₗp from speaking-length geometry; the displacement m̂ₗp − mₗp (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 cBt. 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 rₜ·rb predicts the appropriate cBt setting with R² = 0. 95. Two speaking-length measurements, Lₗp and LBt, thus initialize the BTB position diagnostic, the Z-jump target, and the tension-design starting point simultaneously.
Morito Kawamura (Wed,) studied this question.