Optimal Tuning of Two-Dimensional Keyboards
This work addresses a specific tuning problem in music theory, offering incremental improvements by extending solutions to new keyboard dimensions.
The paper tackles the problem of optimally tuning two-dimensional keyboards for harmonic approximation in music theory, formulating it as a linear programming problem and providing exact solutions for many new keyboard dimensions, with results showing that optimal tuning is achievable for any keyboard width given enough octave rows.
We give a new analysis of a tuning problem in music theory, pertaining specifically to the approximation of harmonics on a two-dimensional keyboard. We formulate the question as a linear programming problem on families of constraints and provide exact solutions for many new keyboard dimensions. We also show that an optimal tuning for harmonic approximation can be obtained for any keyboard of given width, provided sufficiently many rows of octaves.