gesture-hypergestures — Инструмент для генерации топологических жестов

--- name: gesture-hypergestures description: Gesture Hypergestures Skill version: 1.0.0 --- # Gesture Hypergestures Skill > *"A gesture is a continuous curve in a topological category."* > — Guerino Mazzola, Topos of Music III: Gestures **Trit**: +1 (PLUS - generative) **Color**: #C42990 (from seed 137508, index 23) **Foundation**: Mazzola's Diamond Conjecture ## Overview **Gestures** are the missing link between structure (forms/denotators) and performance (physical action). This skill implements Mazzola's gesture theory from *Topos of Music III*. ``` Form (static) → Gesture (dynamic) → Performance (physical) ↓ ↓ ↓ Denotator Hypergesture Sound wave ``` ## Core Concepts ### Gesture Definition A gesture is a continuous curve `γ: [0,1] → X` in a topological category: ```julia struct Gesture{T} domain::Interval # [0, 1] or [a, b] target::T # Topological space curve::Function # t → target end # Example: pitch gesture (glissando) glissando = Gesture( (0.0, 1.0), PitchSpace, t -> 60 + 12 * t # C4 to C5 ) ``` ### Hypergestures A **hypergesture** is a gesture of gestures - a higher-order curve: ```julia struct Hypergesture{T} base_gestures::Vector{Gesture{T}} interpolation::Function # Gesture × Gesture → Gesture end # Hypergesture: morphing between two melodic contours melody_morph = Hypergesture( [melody_a, melody_b], (g1, g2, t) -> interpolate_gesture(g1, g2, t) ) ``` ### Diamond Conjecture The fundamental theorem relating local to global: ``` H^n(Gesture) ≅ H^n(Skeleton) ⊗ H^n(Body) Local gesture fragments glue iff cohomology obstructions vanish. ``` ## Integration with Loaded Skills ### Gestures ↔ topos-of-music Gestures extend the Form/Denotator framework: ```julia # Form → Gesture NoteGestureForm = GestureForm(NoteForm) # Denotator → Gestured Denotator performed_note = GesturedDenotator( note, timing_gesture, # Micro-timing dynamics_gesture # Expression curve ) ``` ### Gestures ↔ catsharp-sonification Sonify gesture trajectories: ```julia function sonify_gesture(g::Gesture, seed::Int) Gay.gay_seed!(seed) for t in 0:0.1:1 point = g.curve(t) color = Gay.next_color() freq = pitch_to_freq(point) trit = hue_to_trit(Gay.hue(color)) play_tone(freq, waveform_for_trit(trit)) end end ``` ### Gestures ↔ persistent-homology Track gesture stability across filtrations: ```julia # Gesture persistence: which contours survive simplification? filtration = [ε for ε in 0.1:0.1:1.0] persistence = compute_gesture_persistence(gesture_space, filtration) # Stable gestures = robust performance features stable_gestures = filter(g -> g.persistence > 0.5, all_gestures) ``` ### Gestures ↔ sheaf-cohomology Verify gesture gluing conditions: ```julia # Local gesture patches left_hand = Gesture(...) right_hand = Gesture(...) # Check if they glue into coherent performance H1 = compute_gesture_cohomology([left_hand, right_hand]) if H1 == 0 println("Gestures glue correctly!") else println("Obstruction detected: hands don't coordinate") end ``` ### Gestures ↔ interaction-nets Gesture composition as net reduction: ``` ┌──────┐ ┌──────┐ │ G1 ├─────┤ G2 │ └──┬───┘ └───┬──┘ │ │ └──────┬──────┘ │ ┌────┴────┐ │ G1;G2 │ (composed gesture) └─────────┘ ``` ### Gestures ↔ stellogen Gestures as continuous star trajectories: ```stellogen ' Gesture as constellation with time parameter (def gesture_star [(+time T) (+pitch (interp P0 P1 T)) (-note N)]) ' Hypergesture: nest gestures (def hyper [(+meta_time S) (+gesture (gesture_star S)) (-performance P)]) ``` ## Mathematical Structure ### Gesture Category ``` Objects: Topological spaces (pitch, dynamics, timing, ...) Morphisms: Continuous curves γ: I → X Composition: Path concatenation (up to homotopy) ``` ### Hypergesture Homology ``` H_0(G) = Connected components of gesture space H_1(G) = Loops in gesture space (repeating patterns) H_n(G) = Higher-dimensional voids (complex structures) ``` ### Diamond Diagram ``` Skeleton ↑ Body ←─── G ───→ Gesture ↓ Performance G = Gesture object relating all three domains ``` ## Commands ```bash # Create gesture from MIDI just gesture-from-midi performance.mid # Interpolate between gestures just gesture-interpolate g1.json g2.json --steps 10 # Verify hypergesture cohomology just gesture-h1 hypergesture.json # Sonify gesture trajectory just gesture-sonify contour.json --seed 137508 # Visualize gesture space just gesture-viz --output gesture.svg ``` ## GF(3) Triads ``` gesture-hypergestures (+1) ⊗ topos-of-music (0) ⊗ rubato-composer (-1) = 0 ✓ gesture-hypergestures (+1) ⊗ catsharp-sonification (0) ⊗ persistent-homology (-1) = 0 ✓ gesture-hypergestures (+1) ⊗ interaction-nets (0) ⊗ sheaf-cohomology (-1) = 0 ✓ ``` ## Example: Rubato Performance ```julia # Rubato = tempo gesture rubato = Gesture( (0.0, 1.0), TempoSpace, t -> 120 * (1 + 0.1 * sin(2π * t * 4)) # Oscillating around 120 BPM ) # Apply to note sequence for note in score performed_onset = apply_gesture(rubato, note.onset) play(note.pitch, performed_onset, note.duration) end ``` ## Files - **Core implementation**: `lib/gesture.jl` - **Hypergesture homology**: `lib/hypergesture_homology.jl` - **Sonification bridge**: `lib/gesture_sonify.jl` ## Related Skills | Skill | Trit | Relationship | |-------|------|--------------| | topos-of-music | 0 | Forms → Gestures | | catsharp-sonification | 0 | Sonify trajectories | | rubato-composer | -1 | Execute performances | | persistent-homology | -1 | Gesture stability | | sheaf-cohomology | -1 | Gluing verification | ## References - Mazzola, G. *The Topos of Music III: Gestures* (2017) - Mazzola, G. *Musical Performance* (2011) - Mazzola et al. "The Diamond Conjecture for Hypergestures" --- **Skill Name**: gesture-hypergestures **Type**: Musical Performance Theory **Trit**: +1 (PLUS - GENERATOR) **GF(3)**: Generates continuous performance curves **Sonification**: C#4 sine (hue 55°, warm)