Optimal tuning of multi-rank keyboards. Issue 3 (2nd September 2019) - Bannerman, Aricca; Emington, James; Venkatesh, Anil
Abstract : The Helmholtzian theory of sensory consonance by spectral matching has been used by many scholars to evaluate equal temperaments. This approach highlights certain temperaments, such as the 12-, 19-, and 31-note tunings, for their fit of popular just intervals. We study the spectral matching of linear temperaments, which are generated by the octave plus a second independent interval. Our project further develops the work of Chalmers on linear temperaments, with particular focus on the Miracle temperament discovered by Secor. Our results fall into three categories. We present a list of linear temperaments with notable spectral matching properties, and a computationally efficient algorithm for evaluating the spectral matching of linear temperaments for a variety of error functions. We also give a theoretical proof that bounds the number of keyboard ranks required to obtain optimal spectral matching.