Generalizing Transduction Grammars to Model Continuous Valued Musical Events

We describe a generalization of stochastic transduction grammars to be able to model continuous values, the first models to natively handle continuous-valued musical events such as microtones while still gaining the advantages of STGs for describing complex structural, hierarchically compositional inter-part relationships. Music transduction modeling based on linguistic or grammatical models have commonly approximated continuous valued features like pitch by quantizing them into discrete symbols, which represent ‘clean’ notes on a scale. The sacrifice is worthwhile for modeling the learning of musical improvisation and accompaniment where musical sequences interact hierarchically at many overlapping levels of granularity; previous work has shown in flamenco and hip hop how discrete STGs allow each part to influence decisions made by other parts while also satisfying contextual preferences across multiple dimensions. We extend the modeling machinery toward the many musical genres where contextual relationships between continuous values influence improvisational and accompaniment decisions. Illustrating using the ‘bent notes’ prevalent in blues, we show how continuous STGs can be generalized from conventional discrete STGs which have until now only been able to handle symbolic events, thereby allowing musical signals to remain finely represented as continuous values without crude quantization into discrete symbols, while still retaining the ability to model probabilistic structural relations between multiple musical languages. We exemplify this new approach in learning blues notes biases via a new polynomial time algorithm for expectation-maximization training of continuous SITGs (stochastic inversion transduction grammars), a specific subclass of STGs that has already proven useful in numerous applications in both music and language. 1. STOCHASTIC TRANSDUCTION GRAMMARS To introduce continuous STGs (see Wu [4, 5] for details of conventional STGs) we will generalize a step at a time from context-free grammars, illustrating with the well-known c ⃝ Dekai Wu. Licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Attribution: Dekai Wu. “Generalizing Transduction Grammars to Model Continuous Valued Musical Events”, Extended abstracts for the Late-Breaking Demo Session of the 17th International Society for Music Information Retrieval Conference, 2016. twelve-bar blues form in which verses consist of three lines: a first four bars, a second four, and a third four called a turnaround. A stochastic context-free grammar could use these syntactic rules to describe twelve-bar blues with the first four in both basic [AA AA] and ‘quick to four’ [AD AA] variants, biased by rule probabilities: