Sequence Sheets in Music Modeling

• Sequence sheets are a structured, event-based symbolic representation that encodes musical events with local sparsity, enabling clear abstraction from full scores.
• They employ deterministic and probabilistic methods—like the skyline algorithm and differentiable top-k selection—to optimize music generation and reconstruction.
• Sequence sheets support conditional generation and rigorous evaluation, impacting tasks from automatic arrangement to generative modeling in computational music research.

A sequence sheet is a structured, event-based symbolic representation used to encode and manipulate sequences of musical elements—almost exclusively notes and chord symbols—for both generative and analytical purposes in computational musicology and music information retrieval. Sequence sheets function as a compressed mid-level abstraction, distinct from both full multitrack scores and lower-dimensional representations such as monophonic melodies or simple chord charts. They serve as foundational data objects in generative modeling pipelines, automatic music arrangement, and evaluation benchmarks for sequence modeling techniques in music AI research (Novack et al., 2023, Roy et al., 2017, Makris et al., 2021).

1. Formal Definition and Mathematical Structure

A sequence sheet encodes a polyphonic symbolic music excerpt as a flat, temporally ordered sequence of tokens, where each token describes a musical event, such as a note onset or a chord symbol. The most common formalization is as a subsequence $X_{le} \subset X_{sc}$ of events drawn from a complete score $X_{sc} = \{x_i\}_{i=1}^{N_{sc}}$, where $x_i$ is a tuple including fields (type, beat, position-in-beat, pitch, duration, instrument); for chords, token fields include (type=chord, beat, position, root pitch class, quality).

A key operational constraint is local sparsity—typically, at every unique onset time $o$, the sheet contains at most $k$ events:

$$C_k(X_{sc}) = \{ X_{le} \subset X_{sc} : \forall o,\, | \{ x_i \in X_{le} : \mathrm{onset}(x_i) = o \} | \leq k \}$$

This supports tunable compression, facilitating the capture of the most salient structural elements while omitting orchestrational or textural detail (Novack et al., 2023).

2. Methodologies for Construction and Semantic Compression

The construction of sequence sheets may proceed deterministically, as with the “skyline” algorithm (e.g., select highest-pitch note per onset and all chords), or via probabilistic and differentiable modeling. Recent advances adopt unsupervised semantic compression: learning an encoder $P_{enc}(X_{le}\,|\,X_{sc})$ constrained to $C_k$, jointly with a decoder $P_{dec}(X_{sc}\,|\,X_{le})$, optimizing the reconstruction objective:

$$L(\theta) = -\mathbb{E}_{X_{le} \sim P_{enc}(\cdot \mid X_{sc})} \left[\, \log P_{dec}(X_{sc}\mid X_{le})\, \right]$$

This framework supports end-to-end learning of “informative” lead sheets as discrete subselections. Differentiable top-$k$ subset selection (using Gumbel-Softmax relaxation and straight-through estimators) enables gradient-based optimization under this highly discrete bottleneck (Novack et al., 2023).

For structured variation, graphical models synchronize melody and chords and sample variations at controlled edit distances using belief propagation with local fields, and the Mongeau–Sankoff distance quantifies sequence similarity to enforce proximity or global form constraints (Roy et al., 2017).

3. Sequence Sheet Representation and Tokenization

In practical implementations, sequence sheets use Multitrack Music Transformer (MMT)-style tokenization: each event (note or chord) is represented by a comprehensive tuple. Chord tokens (typically at the start of each beat) are derived using pre-trained chord recognizers, standardized to include only specific chord types as per experimental constraints (Novack et al., 2023). This representation supports flexible selection and efficient linearization for input into sequence models (Transformers, LSTMs, etc.).

Event sequences are organized such that each token is temporally placed, and local sparsity is controlled per onset group. At inference, deterministic selection (e.g., arg top-$k$) is used, and at training, fractional $k$ regimes allow adaptivity to local token density, yielding better musical coverage and more accurate reconstructions than rigid $k=1$ skyline selection (Novack et al., 2023).

4. Conditional Generation and Control

Advanced generative paradigms treat sequence sheet generation as a conditional sequence-to-sequence problem, where lead-sheet content is generated to match user-specified or inferred conditions, such as valence (emotional positivity/negativity label per bar), time signature, phrase grouping, and note density (Makris et al., 2021). Neural architectures include both multi-layer LSTM encoder–decoder and Transformer encoder–decoder designs, with conditions encoded as additional discrete tokens interleaved with musical content.

The conditional setup supports explicit control over higher-level musical attributes: affective valence is quantified and discretized via musicological mappings of chord-type “mood tags,” and included as input to the encoder. This design allows for precise steering of generated sequences' stylistic and emotional qualities and is empirically validated by both objective statistics and human listener agreement with desired attributes (Makris et al., 2021).

5. Evaluation, Comparison, and Significance

Sequence sheets enable rigorous benchmarking through both automatic metrics (e.g., MuTE F1 score, Jaccard similarity, Mongeau–Sankoff distance, pattern redundancy measures) and human studies (listening and score-reading for accuracy, fluency, and faithfulness) (Novack et al., 2023, Roy et al., 2017). Probabilistic, differentiable construction (as with Lead-AE) outperforms deterministic reductions (skyline) in both score-reconstruction tasks and in subjective fluency/accuracy, attested to by increased F1, Jaccard, and user preference scores at comparable or slightly increased event density. For example, fractional-$k$ selection yielded note density of 39% and chord coverage at 95%, outperforming skyline's fixed 37%/100% on reconstruction and human preference metrics (Novack et al., 2023).

Sequence sheet methodologies support imposition of large-scale musical form (A–A–B–A), enforcement of local thematic similarity, and enable structured generative workflows. Empirical data demonstrates that carefully tuned local fields, sparsity constraints, and sequence modeling architectures robustly enforce both stylistic variation and high-level control (Roy et al., 2017).

6. Applications and Impact in Computational Music Research

Sequence sheets are foundational for a variety of downstream tasks: generative music modeling, automatic arrangement, conditional music generation, lead-sheet harmonization, and as intermediates in multitrack music reconstruction. They facilitate structured sequence modeling that bridges symbolic content (melody, chords) with higher-dimensional representations (full scores), enabling both compression and expansion in neural pipelines.

In symbolic OMR workflows, sequence sheet-derived representations allow for model outputs in standard event-sequence formats (e.g., Humdrum **kern token sequences with polyphony markers), enhancing interoperability and downstream processing (Ríos-Vila et al., 12 Feb 2024).

Sequence sheet research underpins objective and subjective evaluation criteria for generative systems, provides mechanisms for explainability in event selection, and supports task-specific steering via semantic bottlenecks, advancing the capabilities and interpretability of music generation models.

7. Future Directions and Challenges

Emerging research is extending sequence sheet frameworks to accommodate more flexible sparsity—fractional and adaptive $k$, increased control conditions (including lyrics, dynamics, articulations), and richer evaluation paradigms harmonizing edit distance with pitch/rhythm correctness (Novack et al., 2023, Ríos-Vila et al., 12 Feb 2024). Modeling challenges include handling very high local polyphony, out-of-domain symbolic styles, robust generalization from synthetic to real-world data, and integrating direct musicological priors (key, meter, phrase structure).

A plausible implication is that sequence sheets will remain central to iterative workflows in both dataset curation and model conditioning, serving as a critical interface between human-understandable musical abstraction and high-capacity neural sequence modeling.