The Scratch Engine: A Dihedral Permutation Sequencer for Structural Control in Musical Systems

Simon Kilshaw 2025

1. Conceptual Overview

Traditional musical sequencers operate by ordering discrete musical events over time: pitches, rhythms, dynamics, or timbral changes. In contrast, the system presented here operates on a different ontological level: it sequences transformations of control structure.

At any moment, the instrument is defined by a fixed vector of six numerical values. These values are not intrinsically musical; rather, they acquire meaning through their assignment to six parameter roles. The core musical operation is therefore not variation of values, but permutation of their functional interpretation, identified here as containerisation.

2. Parameter Containers and Structural Roles

The instrument defines six persistent parameter containers, represented geometrically as the vertices of a regular hexagon as well as Each container corresponds to a stable control role within a synthesis or processing architecture, for example:

• Pitch
• Note length or rhythmic interval
• Harmonicity or spectral density
• Pitch multiplier or register
• Resonant filtering
• Distortion or nonlinear processing

These roles remain spatially and conceptually fixed. What changes is which numerical value occupies each role at a given moment.

3. State Vectors as Instrument Configurations

A single interaction generates a six-element numerical vector, for example:

[54 4 32 6 40 121]

This vector constitutes a complete instrument configuration. Importantly, the system treats this vector as immutable material. No stochastic processes act upon the values once generated. All subsequent change arises exclusively through permutation.

This design choice foregrounds structural transformation over parametric noise, aligning the system with compositional approaches that privilege invariance, constraint, and formal coherence.

4. The Dihedral Group D₆ as a Control Grammar

The set of allowable transformations is restricted to the twelve elements of the dihedral group D₆, comprising six rotations and six reflections. Each group element defines a bijective mapping of the six containers onto themselves.

Rotations correspond to cyclic reassignment of parameter roles, producing gradual and perceptually continuous structural shifts. Reflections introduce inversional symmetries, often yielding more abrupt or contrastive reconfigurations.

Because D₆ is finite and closed under composition, all possible transformations are known, repeatable, and structurally related. This imbues the system with a sense of formal unity even under rapid interaction.

5. Sequencing as Structural Traversal

In performance, the instrument behaves as a live permutation sequencer. Each button press advances the system to a new structural state, without altering the underlying material.

The cumulative log of permutations functions as a structural score: a record of transformations applied to a single material source. Such a score can be replayed, analysed, or mapped onto other domains (e.g. spatialisation, orchestration, or form).

In "performance mode" The Scratch Engine locks into a user set tempo through Ableton Link tempo networked synchronisation. At bar 1, a set of 6 random integers populate a 3 x 2 matrix. That string of number stays in the system for 16 bars, before the next random set is generated. At every beat for the duration of the 16 bar cycle, the vector permutes its position in the matrix symmetrically, through 12 dihedral symmetries, ramping between states. A fluid and contiguous coherence in phrasing, rhythm and pitch emerges. Crucially, the listener perceives continuity because energy and material are conserved, even as functional relationships are reconfigured.

6. Relation to Serialist and Algorithmic Practice

The system resonates with serialist techniques, particularly total serialism, in its concern with the permutation of musical parameters. However, it diverges by permuting not parameter values themselves, but the assignment of values to roles.

This distinction shifts the focus from ordered rows to ordered structures, placing the work closer to contemporary algorithmic and post-spectral practices concerned with morphogenesis and transformation rather than variation.

7. Computational and Design Implications

From a systems perspective, the instrument exemplifies a separation of concerns: material generation, structural transformation, and presentation are cleanly decoupled. This architecture ensures robustness under rapid interaction and supports extensibility to other group actions or parameter cardinalities (the number of elements in the given set).

The use of explicit group theory provides both compositional rigour and computational predictability, offering a viable framework for interactive systems that balance freedom and constraint.

8. Conclusion

By embedding dihedral symmetry into the control layer of a live instrument, this work proposes an alternative model of sequencing grounded in structural permutation rather than event ordering. The resulting system enables performers and composers to navigate a finite yet expressive space of transformations, fostering coherence, contrast, and formal articulation in real time.

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