Notes On Tuning


The intonation does not conform to 12-tone equal temperament. Follow the intonation of the guide track as closely as possible. Some deviation is, naturally, inevitable. Although a comprehensive understanding of the tuning system employed in this piece is not essential to performance, the system is described below.


The present work takes a type of extended mean-tone temperament as its point of departure. Musicians familiar with the music of the 16th and 17th centuries may recognize 1/4-comma meantone as one of the tuning systems commonly employed in the performance of the music of the Renaissance and early Baroque eras. Quarter-comma meantone tuning possess two semitone intervals of differing sizes, a small semitone of approximately 76 cents, and a large semitone of about 117 cents. Dividing the small semitone in two results in an interval of about 38 cents and multiplying this number by 3 results in an approximation of the large semitone. Dividing 1200 (the number of cents in an octave) by 38 approximately yields 31, so by rounding and approximation, we can arrive at a 31-tone equal temperament that can very closely approximate 1/4-comma meantone, with no interval deviating from 1/4-comman meantone by greater than two cents. Thirty-one-tone equal temperament then extends 1/4-meantone tuning, filling in the missing semitones and offering some new harmonic possibilities. The present piece does not employ all 31 potential pitch classes, the "double-flat" and "double-sharp" series being notably absent.

More information on 31-tone equal-temperament can be found here:
Wikipedia Article
Overview of 31-tone Equal Temperament>
Paul Rapoport's Notes on 31-Tone Equal Temperament