From musical imagination to an expressive instrument

GAMUT research agenda · 13 September 2026
Status: discussion document and proposed research program; not an implementation specification or a claim of demonstrated musical utility.
Repository baseline: bf76929 (full manuscript review).

This document develops the artistic motivation and practical directions discussed with the project’s founder after the first Lean release and the full manuscript review. It connects the current mathematics to instrument design, performance, learning, and musical analysis. The companion studies examine Coltrane at Antibes and two Bruckner symphonic movements. A separate study protocol defines how to test the proposals. The proposed Eschatology Report series scopes a public-facing exegesis of these ideas.

1. Artistic origin and the question the project should serve

In the founder’s account, this work grew from years of listening to and studying John Coltrane, especially the interaction of geometric and symmetric organization with musical expression. A recurring aspiration is to translate interior musical conception and feeling into exterior musical expression with fewer interruptions and less unwanted loss.

One formative dream involved playing an instrument through which music seemed to flow directly from mind to sound, accompanied by geometric, fractal, Platonic, and symplectic forms. The founder also identifies cross-sensory listening experiences as part of the project’s visual imagination. These are autobiographical sources of the artistic questions. They are not experimental evidence for a physical geometry of consciousness or a shared code linking particular shapes to emotions.

The dream is useful as a description of desired experience: continuity between intention and action, immediacy of response, a sense of authorship, and visible relationships that seem to belong to the sound. It does not oblige the first instrument to use neural sensing. Voice, breath, touch, movement, and familiar instrumental gestures can test important parts of the aspiration much sooner.

The working question is:

Can a musical instrument make relationships and transformations directly playable, so that a performer can preserve and develop an idea while it is still vivid?

That question accommodates both realizing an already imagined phrase and discovering an idea through playing. The instrument should support deliberate reproduction, controlled variation, and productive surprise without treating them as the same task.

2. What “direct” and “faithful” should mean

A practical engineering objective is less conscious translation and less unwanted information loss. A literal absence of representation is not an available specification: sensors, synthesis, and any neural decoder involve mediation. A relationship can nevertheless become familiar enough that managing it ceases to dominate attention.

Several kinds of fidelity must be evaluated separately:

Representational fidelity: the stored data retains the distinctions its contract promises.
Operational fidelity: a transformation does what its controls advertise.
Temporal fidelity: response timing supports the intended gesture and phrase.
Expressive fidelity: the performer recognizes the result as realizing a relevant intention.
Experiential continuity: making and adjusting the sound does not repeatedly interrupt the developing musical idea.

The current Lean library addresses selected mathematical contracts in the first two categories. It supplies no automatic guarantee of the other three. A transform can be invertible on pitch-class patterns while the representation has already omitted a decisive breath, timing deviation, register choice, or change of tone.

Some abstraction is useful. A recognizable interval relationship can become a reusable gesture across keys. The design objective is therefore to make reductions explicit, retain a route back to richer information where possible, and let the musician choose which distinctions matter for the present task. A transposition/inversion class useful for cataloguing cannot serve indiscriminately as a performance identity: concrete major and minor sonorities, registers, and tonal functions can matter even when a quotient identifies them.

3. What the project actually supplies today

The proof manifest, trust record, and paper review define the current boundaries. F01–F10 cover discrete reconstruction, fiber counts, finite complex Fourier identities, content invariance, full order-spectrum reconstruction, layered injectivity, the twelve-tone census, and exact examples of information loss.

The formal pattern domain is a rooted cyclic tuple of distinct pitch classes. It is not a complete model of an improvised line, an orchestral score, or a performed recording. General periodicity/support and the higher geometry remain paper proofs. The published visualizations illustrate selected representations and samples; their appearance does not certify perceptual distance, useful control mappings, or musical meaning.

The repository also has an event pipeline richer than the Lean seed. USAGE.md describes onsets, durations, tone/rest kinds, frequency ratios, amplitude, and categorical timbre. Its purpose is candidate-signal analysis and RMCP experiments. It is not a complete performance-capture format or an expressive instrument engine.

A source inspection matters here. The current corpus converter rounds pitch offsets to semitones and maps them to a small rational-ratio table; fixes amplitude and timbre; enforces a minimum duration; and either skips chords or uses their first pitch. Its traversal of notes does not preserve explicit rests or independent voice identities. These are concrete reasons not to use its output as the sole archival representation for Antibes or Bruckner. Exact fractions after conversion do not reverse losses introduced during conversion. No converter or runtime code is changed by this agenda.

4. Musical anchors and what they ask of a model

Coltrane: recognizable material in a developing performance

The equally spaced tonal centers of Giant Steps provide a clear example of a compact relationship that can guide musical movement. They do not exhaust the piece. Pellegrin’s original analysis relates collection, motive, and voice leading while attending to irregularities within the construction. The design inference is to make a structure and departures from it playable. Pellegrin (2020)

The requested live A Love Supreme is a different case. The Antibes dossier identifies the 26 July 1965 performance and distinguishes it from the studio suite, incomplete surviving film, and later Seattle performance. Its evidence suggests questions about how identity survives extension, altered delivery, interaction, and changed relationships to text. It should not be forced into the harmonic-cycle explanation used for Giant Steps.

For an instrument, the proposed lesson is to preserve a musical point of reference while giving the performer control over continuation, interruption, response, and return. Which features actually maintain recognition is an empirical question.

Bruckner: recognition across time and coordination across voices

The Bruckner dossier selects the Eighth Symphony’s Adagio and the Fifth Symphony’s finale as initial anchors. It identifies editions rather than treating every score or recording as interchangeable. The proposed questions concern extended trajectories, recurrence after intervening events, independent lines, orchestral identity, and the preparation of arrivals.

These cases suggest a system with memory. A state described only by the present pitch collection cannot say whether an event is a first statement, a deferred return, a culmination, or a recollection. Nor can a single cyclic ordering adequately represent simultaneous contrapuntal voices.

The comparison is deliberately limited. Coltrane’s quartet practice and Bruckner’s composed orchestral works involve different histories, spiritual contexts, kinds of authority, and performance traditions. Shared analytical questions do not establish a common compositional method, psychological mechanism, or geometry of transcendence.

5. Connections that could produce real utility

5.1 Embodied control and the development of technique

Wessel and Wright’s work on intimate musical control joins accessibility with potential for long-term skill, clear gesture relationships, and low, consistent latency. This is a better instrument-design precedent than assuming a large choice space alone supplies expressive freedom. Wessel and Wright (2001)

For GAMUT, the proposal is to give gestures stable musical consequences. A performer should be able to predict a local change, make a fine adjustment, repeat it, and recover from an unwanted result. Stable relationships may permit technique to develop; they do not remove the need for practice. Breath and touch can also carry continuous detail that a menu of named chords cannot.

5.2 Transformational thinking and voice leading

Musical intention often concerns a relationship: continue upward, retain a contour, return to a tone, open the voicing, displace the rhythm, or carry the phrase into another harmonic environment. These suggest controls over operations and constraints rather than a requirement to specify every output event independently.

Possible operations include transposition, interval inversion, rhythmic displacement, permutation of a selected pattern, register redistribution, and selection of voice-leading paths. Each needs its own domain and advertised invariants. Transposing a pitch-class pattern is not the same operation as reharmonizing a recorded phrase while preserving vocal timbre.

Graph adjacency and perceptual closeness must remain distinct. Tymoczko provides examples where paths in familiar music-theoretic graphs fail to reflect the voice-leading distance one might expect. A display can therefore show a named relation without presenting its drawing distance as a universal musical metric. Tymoczko (2010)

5.3 Audiation, feedback, and the discovery of intention

The project can investigate internal hearing without assuming that every intention arrives as a completely specified sound. Sometimes a musician can sing a contour; sometimes the clearest feature is a physical impulse, texture, or anticipated arrival. The preferred input modality remains an open question for the founder and may differ across tasks.

The proposed interaction is iterative: form an intention, act, listen, and refine. The output can help the performer discover what they meant as well as realize what they already knew. Evaluation should retain that distinction. An attractive surprise is evidence for exploratory usefulness; it is not evidence that a hidden prior intention was decoded accurately.

5.4 Personal mappings and interactive learning

Wekinator demonstrates an approach in which users supply gesture/output examples, train mappings, audition them, and revise the examples. It provides a precedent for a musician participating directly in the construction of an instrument’s control vocabulary. Fiebrink, Trueman, and Cook (2009)

A GAMUT experiment could let a musician teach the association between an expanding movement and a family of musical changes. Expansion might mean increased register span, a different voicing, a timbral trajectory, or a combination. This association would be personal and task-specific. “More tension” should not become an unexplained scalar pretending to encode a universal emotion.

Learning should be inspectable and reversible. Explicit practice and calibration sessions can change mappings; a performance mode should retain a chosen mapping until the performer requests otherwise. Fiebrink and colleagues’ evaluation research supports taking user criteria beyond prediction accuracy seriously, including how users revise the problem itself through real-time interaction. Fiebrink, Cook, and Trueman (2011)

5.5 Multisensory representations as instruments and learning aids

A visible form should have an accountable relationship to sound. A rotating shape might implement a specified transposition; a path might identify a sequence of operations; separation might encode register span. If a shape closes, the interface should specify what has returned: content, ordering, voicing, or the entire captured phrase.

Visual and tactile feedback have at least three proposed roles: expose relationships during learning, provide limited landmarks during performance, and support retrospective exploration. These uses should be evaluated separately. A display that helps analysis may divide attention during live playing. A display-hidden transfer task can test whether learned relationships remain usable away from the visualization; an audiovisual performance can instead make the display part of the artistic medium.

A synesthetic association can guide a personal mapping without being generalized to all listeners. Consistency, learnability, and expressive usefulness are better initial questions than whether a color or polyhedron is the objectively correct image of a sound.

5.6 Musical time, expectation, and memory

Both case studies motivate relationships that depend on history. A repeated object can acquire a different role after intervening events. A future model could therefore retain phrase identity, returns, transformations, active voices, and the performer’s chosen goals alongside current acoustic or symbolic state.

This does not require a global numerical theory of expectation before any experiment begins. Start with interpretable controls: hold a destination, delay a return, retain a bass anchor, interrupt a repetition, or reintroduce a previous phrase in a changed context. Ask whether the performer hears and uses those distinctions.

5.7 Ensemble interaction and shared control

A further instrument could allocate different transformations to different players. One controls a motif’s development, another its harmonic environment, and another the timing of arrivals or interruptions. The musical research question is how individual agency and shared responsiveness coexist.

A system should make responsibility legible: who changed the material, whether a response was automated, and which relationships remain under each player’s control. A solo controller that generates several parts is not automatically an adequate model of a quartet or orchestra. The Antibes case particularly warns against reducing ensemble activity to a melody with background accompaniment.

5.8 Self-similarity, polyhedra, and symplectic dynamics

Self-similarity offers a concrete compositional proposal: related transformations act on a note group, phrase, and larger section, while the performer chooses exceptions and stopping points. A finite repeated hierarchy need not be described as a literal mathematical fractal.

Platonic forms could organize a finite interface or family of operations. The mapping must say what vertices, edges, and rotations do musically. The existence of a pleasing solid supplies no independent evidence that its chosen musical assignment is natural.

Symplectic geometry offers a language for structured evolution, commuting actions, and conserved quantities. The paper provides a chosen ambient framework in which to investigate such ideas. A proposed instrument still needs a realization of those paths as events or sound. Generic continuous trajectories leave the finite musical seed; returning by selection or quantization can introduce discontinuities or lose the properties of the ambient flow. Geometric smoothness, perceived continuity, and emotional expression require separate arguments or experiments.

6. Three candidate instruments

A. Phrase-transformation instrument — recommended first experiment

Capture a short phrase by singing, playing, or tapping. Preserve the original performance. Expose two or three operations with clear boundaries, such as transposition, rhythmic displacement, and a selectable destination. Keep an immediate route back to the original.

A later version could support controlled reharmonization and voice leading, but those should not be silently bundled into the first operations. A major-third cycle is a useful small test family; it is not a general recipe for Coltrane’s music.

Potential utility: developing an idea without re-entering it note by note; practicing transformations; controlled variation in improvisation.
Main uncertainty: whether the chosen controls align with how the performer conceives a phrase.
Current support: discrete transformations and reconstruction provide a foundation for restricted symbolic operations. Rich performance capture and expressive rendering remain new work.

B. Personal gesture-and-timbre instrument

The performer teaches relationships among gestures, sounds, and named musical operations. Continuous touch, breath, or vocal features control a small chosen sound space. Voice-timbre interfaces and qualitative evaluation already have research precedent, including studies with beatboxers. Stowell, Plumbley, and Bryan-Kinns (2008)

Potential utility: shaping sound through a vocabulary closer to the performer’s physical or visual imagination.
Main uncertainty: mapping stability, unwanted coupling, and the time required to learn reliable control.
Current support: the mathematical layer can constrain selected operations; the mapping and timbre model would need separate development and evaluation.

C. Form-and-memory instrument

The performer captures several ideas, retains their relationships, and controls returns, interruptions, combinations, and long-range arrivals. Independent voices or orchestral roles remain separately adjustable. A visual history can show what has happened without prescribing a single future.

Potential utility: long-form improvisation, composition, rehearsal, and ensemble experimentation.
Main uncertainty: whether formal controls remain playable without demanding continuous planning or visual monitoring.
Current support: the content/order distinction is useful but insufficient; phrase hierarchy, temporal state, voice identity, and orchestration are new modeling requirements.

These are research directions, not three approved implementation projects. The initial experiment should be small enough that the musician can explain exactly what each control contributed.

7. A proposed information architecture

The proposed layers are:

Performance capture: available audio, event timing, pitch trajectories, gestures, and source identity.
Derived views: selected voices, pitch classes, gaps, spectra, phrase annotations, and display projections, each with explicit losses.
Musical operations: transformations with declared domains, preserved properties, and recoverable history where possible.
Control mapping: gestures and examples mapped to those operations or directly to sound parameters.
Realization: synthesis, playback, or instrumental output with measured response behavior.
Feedback and memory: sound, optional visual/tactile landmarks, annotations, and the performer’s evaluation.

The raw performance is not a fully observable record of inner intention. It is the richest available external evidence and should remain distinct from analyses derived from it. An operation may be reversible at the event level without being reversible on recorded audio; this distinction belongs in its contract.

8. What further verification could usefully prove

Future formal work should follow a chosen musical operation. Candidate contracts include composition and inverse laws for transposition, retention of onset relationships under a pitch-only operation, preservation of voice assignments, and explicit conditions for reconstructing a selected symbolic view. An undo operation needs a defined state and history; it cannot promise recovery of information never stored.

Repeated notes, noncyclic phrases, polyphony, register, and timing should be modeled deliberately when the instrument needs them. The existing proof domain should not be stretched by informal language. Extending the Lean library and implementing a performer-facing system are separate tasks whose correspondence would itself require review.

General periodicity/support is a reasonable next mathematical target. Higher symplectic formalization may strengthen the theory. Neither is a prerequisite for discovering whether a two-control musical experiment is useful, and neither certifies that usefulness.

9. Neural interfaces as a separate research horizon

Bellier and colleagues reconstructed aspects of a heard musical stimulus from intracranial recordings using decoding models. That study concerns responses to presented music; it does not demonstrate unrestricted realization of voluntary musical imagination. Bellier et al. (2023)

A neural interface could eventually become another input modality, with its own measurement, training, latency, and agency questions. It would still mediate the relation between person and sound. The first useful GAMUT instrument can investigate the desired experience without making neural decoding part of its critical path.

10. Research sequence and decisions for discussion

Begin with source-grounded musical study and a small performer study, then choose an input modality and operation family. Only after that choice should an implementation specification be written. The protocol separates reconstruction, expressive control, exploratory discovery, and transfer so that one kind of success cannot stand in for another.

The unresolved decisions are concrete:

Which part of an idea usually arrives first for the founder: melody, rhythm, texture, movement, or visible form?
Which familiar instrument or workflow is the appropriate comparison?
What must remain recognizable when a phrase changes?
Should the first control affect pitch relationships, temporal placement, or timbre?
When should the system surprise the player, and how should that permission be expressed?
Which Bruckner performances best match the founder’s listening experience?
Is the visualization primarily for playing, learning, retrospective analysis, or audiovisual performance?

The initial criterion is whether the system helps the performer retain and develop a musical idea through controllable action. Musical usefulness, the quality of the resulting work, and broader adoption will require sustained playing and evaluation beyond this document.

11. Review and provenance of this write-up

The originating discussion and research were recorded on 13 September 2026. Separate AI research passes examined the Antibes recording sources and the Bruckner texts; a further AI review checked all four documents for internal consistency, proof-domain correspondence, source attribution, and separation of completed research from proposed experiments. Two minor clarity corrections were applied. This is not independent human peer review.

Local documentation links were checked, the Antibes metadata arithmetic was recomputed, and git diff --check passed. No Lean, runtime, or site implementation was changed, so this documentation pass does not claim a fresh proof build or software test run. The README was linked to these documents and several older overview phrases were aligned with the completed manuscript review.