A relationship is something that is necessarily built up over time, however, Human-Robot Interaction (HRI) trials are rarely extended beyond a single ses- sion. These studies are insufficient for examining multi-interaction scenarios, which will become commonplace if the robot is situated in a workplace or adopts a role that is part of a human's routine. Long term studies that have been exe- cuted often demonstrate a declining novelty effect. Music, however, provides an opportunity for affective engagement, shared creativity, and social activity. This being said, it is unlikely that a robot best equipped to build sustainable and meaningful relationships with humans will be one that can solely play music. In their day-to-day lives, most humans encounter machines and computer programs capable of executing impressively complex tasks to a high standard that may provide them with hours of engagement. In order to have anything that that could be classed as a social relationship, the human must have the sense that their interactions are taking place with another, a phenomenon known as social presence. In this thesis, we examine whether the addition of simulated social behaviours will improve a sense of believability or social presence, which, along with an engaging musical interaction, will allow us to move towards something that could be called a human-robot relationship. First, we conducted a large online survey to gain insight into relationships based in regular music activ- ity. Using these results, we designed, constructed and programmed Mortimer, a robotic system capable of playing the drums and a responsive composition algorithm to best meet these aims. This robot was then used in a series of studies, one single session and two long-term, testing various simulated social behaviours to compliment the musical improvisation. These experiments and their results address the paucity of long-term studies both speci cally in Social Robotics and in the broader HRI eld, and provide a promising insight into a possible solution to generally poor outcomes in this area. This conclusion is based upon the model of a positive human-robot relationship and the method- ological approach of automated behavioural metrics to evaluate robotic systems in this regard developed and detailed within the thesis.