PROPORTION AND SYMMETRY AS MUTUAL ANTAGONISTS IN TUNING: SOME QUARTER-TONE RESOURCES

Abstract

Quarter-tones have the dubious honour of being the microtonal default in Western art music, yet they have been of little recent interest to those most involved with extended intonation. Other microtonal equal divisions have appealed as pragmatic approximations of consonant just-intonation intervals, something that quarter-tones do not offer. This article proposes that quarter-tones can be valued in a different way, for their ability to generate symmetrical harmonic resources that divide the fourth and fifth as the tritone does the octave. These resources are offered as examples of a broader aesthetic of symmetry, which is contrasted with an aesthetic of proportion. These antagonistic principles are explored through the case of the ever problematic tritone, illustrating how proportion and symmetry are best understood using the symbolic resources of just intonation and equal temperament respectively. Drawing on the work of Robert Hasegawa, Georg Friedrich Haas and Ivan Wyschnegradsky, the article argues for a hybrid approach that embraces both just intonation and equal temperament.