The 31-Tone Tuning System of Nicola Vicentino and the Toroidal Tonnetz: An Annotated Bibliography

Nicola Vicentino’s treatise L’antica musica ridotta alla moderna prattica (1555), from here on L’antica musica, argues that contrapuntal practices based on modes derived from the diatonic tetrachord are insufficient to express the variety of emotions possible in vocal text settings, and that composers should be inspired by the other Ancient Greek genera as described by Boethius, the chromatic and enharmonic tetrachords. To employ these alternative genera, Vicentino devised an ingenious system that extended quarter-comma mean tone temperament to a thirty-one-tone system that can be neatly approximated by a division of the octave into thirty-one equal parts