Two musical orderings

Abstract

Orderings of one sort or another arise naturally whenever there is a notion of size or precedence among objects in a musical space. Paying attention to the order properties of the space can lead to new musical insights. In this paper, we study two musical orders that illustrate this point. In the first, we show that many of the most familiar chord and scale types in Western music appear as extremal elements in certain partial orders induced by set inclusion on pitch class sets of T_n-type. In the second, we propose a family of partial orders for making timbral comparisons between musical tones. The ordering principle used is ‘unanimous agreement among informed listeners’. We make this idea rigorous, and then study some of the basic properties of the partial orders that arise from it. Finally, we use these orders to compare the timbres of 10 orchestral wind instruments in terms of their ‘brightness’ and ‘flute-likeness’. Our results show that these partial orders enable rigorous and fine-grained comparisons of timbres that are musically meaningful.

Keywords:

• partial order
• quotient space
• extremal element
• pitch set class
• makam
• timbre
• brightness

MCS/CCS/AMS Classification/CR Category Numbers:

• 06A99
• 00A65

Acknowledgements

The author thanks two anonymous referees, whose comments and suggestions improved this work significantly.