Research
Technical articles on the theory and algorithms behind PitchGrid.
The Two-Dimensional Structure of Musical Scales
Western music theory implicitly treats scales as one-dimensional objects. Yet the structure that gives rise to keys, modes, accidentals, and the circle of fifths is inherently two-dimensional. This article formalizes this observation using rank-2 regular temperaments and Moments of Symmetry (MOS) scales, shows that the diatonic scale is one instance of a vast family of two-dimensional tonal structures, and develops the mathematical tools needed to navigate this space.
The PitchGrid Spiky Consonance Measure
We present a novel, timbre-aware measure of musical consonance derived from Plomp-Levelt sensory dissonance curves. By extracting a hull curve and computing the residual "spiky" consonance signal, the measure quantifies consonance on a perceptually meaningful 0–1 scale. We validate it across 12-TET, Pythagorean, quarter-comma meantone, and Bohlen-Pierce tuning systems, showing it correctly captures historical tuning preferences and the dependence of consonance on both interval structure and spectral content.