Music theorists have devised three measures of musical distance, each giving rise to different geometry. In my talk I will describe these geometries and discuss some of ways in which they are related. My conclusion is that while there are important musical circumstances in which the three geometries converge, this is largely a coincidence—an artifact of the special cases that music theorists habitually consider. Understood more robustly, the three geometries are fundamentally different, and suggest very different ways of understanding music. Unclarity about this fact has led to some interesting twists and turns in recent music theory.