A recently emerging view in music cognition holds that music is not only social and participatory in its production, but also in its perception, i.e. that music is in fact perceived as the sonic trace of social relations between a group of real or virtual agents. While this view appears compatible with a number of intriguing music cognitive phenomena, such as the links between beat entrainment and prosocial behaviour or between strong musical emotions and empathy, direct evidence is lacking that listeners are at all able to use the acoustic features of a musical interaction to infer the affiliatory or controlling nature of an underlying social intention. We created a novel experimental situation in which we asked expert music improvisers to communicate 5 types of non-musical social intentions, such as being domineering, disdainful or conciliatory, to one another solely using musical interaction. Using a combination of decoding studies, computational and psychoacoustical analyses, we show that both musically-trained and non musically-trained listeners can recognize relational intentions encoded in music, and that this social cognitive ability relies, to a sizeable extent, on the information processing of acoustic cues of temporal and harmonic coordination that are not present in any one of the musicians' channels, but emerge from the dynamics of their interaction. By manipulating these cues in two-channel audio recordings and testing their impact on the social judgements of non-musician observers, we finally establish a causal relationship between the affiliation dimension of social behaviour and musical harmonic coordination on the one hand, and between the control dimension and musical temporal coordination on the other hand. These results provide novel mechanistic insights not only into the social cognition of musical interactions, but also into that of non-verbal interactions as a whole.