MUSICAL PHYSICS · PHASE · RECURRENCE

The Hirajoshi Wave

A polyrhythmic pendulum laboratory in which audible pitches retain their exact frequencies while visible motion preserves the same ratios through octave-folded time scaling.

Exact pitch ratios Center-crossing audio Comparative wave mathematics
Open Visualizer Read Method

PRIMARY LABORATORY

Polyrhythmic Pendulum Field

Release the system from alignment, listen at each center crossing, and watch phase relationships separate and partially reassemble.

Ready for interaction
C Hirajoshi · 25 spheres Temple Bell · A4 = 440 Hz · aligned release
Three.js Rapier Audio locked
Stopped 00:00.000

Twenty-five pendulums are aligned and waiting to begin.

Elapsed00:00.000Modeled system time
Near recurrenceCalculatingCircular coherence ≥ 95%
Recent crossings0Events in the last 5 s
Phase coherence100%Five note families
Motion divisor1024Common 210 fold
Render rate—Adaptive WebGL loop

AUDIBLE AND VISIBLE LAYERS

Wave Mathematics

Every visible frequency is derived from the audible pitch by the same divisor. The transformation changes timescale, not interval structure.

Pitchf(n) = A4 × 2(n−69)/12
Visible motionfv = fa / 1024
Pendulum lengthL = g / (2πfv)²
Angular positionθ(t) = A cos(2πft + φ)

ACTIVE SCALE

Frequency and Period Table

NoteIntervalAudio HzVisible HzPeriodModeled length

LIVE PHASE FIELD

Five-Family Phase Plot

0–2π

Phase is sampled from the five unique scale degrees; repeated spheres do not alter the family-level calculation.

Methodological distinction:

A macroscopic pendulum cannot literally oscillate hundreds of times per second. This model therefore keeps the exact note ratios while translating them into a visible octave-folded timescale. Modeled pendulum lengths refer to that visible layer only.

PITCH-SET COMPARISON

Scale Laboratory

Compare interval structures without changing the physical mapping method.

ACTIVE PRESET

C Hirajoshi

0 · 2 · 3 · 7 · 8

Comparative use

Preset names are treated as practical analytical labels. Japanese modal and tuning practices vary by repertory, instrument, historical period, and scholarly taxonomy; the methodological note documents the interval sets used by this implementation.

RELATED WAVE MATHEMATICS

Comparative Models

Each model reuses the active pitch set where mathematically appropriate and states when it is illustrative rather than physically complete.

REPRODUCIBLE STATE

Calculated Outputs

Inspect, copy, or download the current configuration and derived values.


            

        

    


    

        

            

                

                    
STUDENT EVIDENCE COLLECTION

                    
Evidence Notebook

                    
Capture the visual state, numerical configuration, research question, observation, and interpretation as a reproducible lab record.

                

                
0captured records

            

            

                

                    

                        

                        

                    

                    
                    
                    
                    
                    
                    
                    
                    
                    

                        
                        
                    

                    
Each evidence record includes a timestamp, active scale and instrument, sphere count, tuning, audible and visible frequencies, phase coherence, recurrence estimate, recent crossing count, comparative-model state, and a high-resolution frame capture.

                

                
            

        

    


    

        

            

                
METHODS, ASSUMPTIONS, AND LIMITS

                
Methodological Note

                
The full note documents the frequency transformation, synthesis engine, crossing logic, recurrence criterion, comparative models, and interpretive limits.

            

            

                

                    
Core methodological commitments

                    
    
                        
  1. Audible pitch and visible motion are explicitly separated.
    2. 
                        
  2. All visible frequencies share one power-of-two divisor.
    3. 
                        
  3. Crossing events are detected from sign changes, not animation-frame assumptions.
    4. 
                        
  4. Repeated note families are gain-normalized to avoid count-dependent loudness inflation.
    5. 
                        
  5. Recurrence is reported as a numerical near-recurrence with a stated coherence threshold.
    6. 
                        
  6. Cymatic and coupled-oscillator modules are labeled as approximations.
    7. 
                    

                    

                        
DocumentHirajoshi Wave Methodological Note

                        
Version1.1 · August 2026

                        
PublisherStella Nova Education

                    

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LOCAL SCHOLARLY RECORD

                
Scholarly Questions and Additions

                
Use the local annotation log for mathematical corrections, scale taxonomy, acoustical observations, implementation notes, or proposed comparative models.

            

            

                

                    

                        

                        

                    

                    
                    
                    

                        
Entries are stored in a local JSON file on this installation.

                        
                    

                

                

                    
LOCAL ANNOTATION LOG
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