Probabilistic Multilabel Graphical Modelling of Motif Transformations in Symbolic Music

We review the notion of representing music using point sets and argue that such representations are better adapted than sequential representations for matching patterns in unvoiced, polyphonic music, such as keyboard music. Musical transformations such as transposition, inversion, diminution, augmentation and retrograde can be modelled by geometric transformations in pitch-time representations that combine translation with scaling parallel to and reflection in the time axis. We identify eight types of geometric pitch-time representation that use chromatic pitch, morphetic pitch, morph or chroma to represent pitch and either onset time or midtime to represent time. We illustrate how these types of representation allow us to characterise different types of musical transformation. For example, by using midtime instead of onset time, we can precisely characterise certain retrograde relationships; and by using morph and chroma pitch representations we can characterise transformations involving octave displacements and duplications. We present the concept of a transformation class and consider the three specific classes, $F_{\mathrm{2STR}}$, $F_{\mathrm{2STRMod7}}$ and $F_{\mathrm{2STRMod12}}$. We introduce the notion of an inter-pattern transformation graph for a set of patterns, $S$, and a transformation class, $F$. Each vertex in such a graph represents a pattern in $S$ and there is an edge in the graph from pattern $P_1$ to pattern $P_2$ if and only if $P_1$ can be mapped onto $P_2$ by a transformation in $F$. We show, with the aid of such graphs, that the musical relationships between the occurrences of the HAYDN theme in Ravel's Menuet sur le nom d'Haydn can be precisely described in terms of transformations in $F_{\mathrm{2STR}}$, $F_{\mathrm{2STRMod7}}$ and $F_{\mathrm{2STRMod12}}$ within the pitch-time representations considered.

Transformer models trained on symbolic music have become remarkably capable. However, it is yet unclear how they represent music structure, and whether understanding their internal representations can be used to unlock methods for model control. We examine this through a case study of harmonic analysis: when a music transformer navigates a cadence, what does it internally "know" about its harmonic context, and can that knowledge be used to affect the generation? Using Aria, a LLaMA-based symbolic music transformer evaluated on Bach chorales, we find that harmonic decisions are causally localized progressively from middle layers onward. Linear probes show that mode becomes decodable in the same layer range where patching identifies growing causal effect, while key identity emerges later. We then confirm these representations are actionable for music generation control: steering via latent directions corresponding to harmonic changes reliably produces a deceptive or minor resolution, while preserving structural coherence and generation quality. This demonstrates that, at inference time, the model's internal harmonic knowledge can be directly manipulated to control the harmonic output.

Deep learning has transformed symbolic music generation by borrowing the training paradigms of large language models, with systems such as NotaGen now producing complete, stylistically convincing classical scores from a short prompt. These systems could become powerful creative partners, helping musicians generate endless possibilities. However, current systems expose almost no control handles on the music itself. In principle, control handles could be built into a foundation model trained from scratch, but this is rarely practical without massive amounts of quality annotated data and compute resources. We therefore present MotiGen, a recipe for retrofitting pretrained symbolic music models to use new instruction prompts. MotiGen injects a musical motif as a structured prompt line, reinforces it with a scalar attention bias toward the motif tokens, and learns the association with a two-phase curriculum. First, it learns from focused excerpts cropped around motif occurrences in the training data, then full scores including the motifs. Our experiments show that our model composes with the prompted motif in over 92.3% of generated pieces. Generated pieces using a variety of motifs are included in our sample site: https://motigen-site.github.io/.