We present aligned hierarchies, a low-dimensional representation for sequential data streams. The aligned hierarchies encode all hierarchical decompositions of repeated elements from a high-dimensional and noisy sequential data stream in one object. These aligned hierarchies can be embedded into a classification space with a natural notion of distance. We motivate our discussion through the lens of Music Information Retrieval (MIR), constructing aligned hierarchies by finding, encoding, and synthesizing all repeated structure present in a song. For a data set of digitized scores, we conducted experiments addressing the fingerprint task, a song comparison task in MIR, that achieved perfect precision-recall values and provide a proof of concept for the aligned hierarchies. We also introduce aligned sub-hierarchies and aligned sub-decompositions. Both derived from the aligned hierarchies, these structure based representations for songs can be embedded into classification spaces and can address additional MIR tasks. We will compare properties of the aligned hierarchies, aligned sub-hierarchies, and the aligned sub-decompositions. Katherine M. Kinnaird researches the dimension reduction problem, representing high-dimensional and noisy sequential data as a low-dimensional object that encodes relevant information. She applies her work to tasks from the interdisciplinary field of Music Information Retrieval (MIR), such as locating the chorus of a given musical song or finding all copies of a particular recording of a song. Katherine earned her Ph.D. at Dartmouth College from the Department of Mathematics in 2014. Currently, Katherine is a Visiting Assistant Professor in the Department of Mathematics, Statistics, and Computer Science at Macalester College, where she is the founder and Principal Investigator for the Data Science TRAIn Lab. She serves as the President of the Executive Board of the Women In Machine Learning Workshop, and in the Fall of 2016, she will be a Data Sciences Postdoctoral Fellow at Brown University, affiliated with the Division of Applied Mathematics.