Fugue Architecture

ACP models consensus as a musical process. Three structures -- Fugue, Sonata, and Concert -- provide distinct approaches for different types of problems. Musical intervals serve as intuitive agreement metrics, and the Harmony formula H quantifies the overall degree of consensus.

4.1 Three Structures

ACP selects one of three musical structures based on the nature of the query and the degree of initial divergence. Each structure optimizes for a different type of consensus challenge.

Structure 1: Fugue (Deep Tasks)

In music, a fugue is a composition where multiple voices enter sequentially, each carrying the same theme but transformed, and through development and stretto (compression) they arrive at harmonic unity. This is an exact model of AI consensus.

FUGUE (Bach, Beethoven, Mozart):

Multiple VOICES enter SEQUENTIALLY,
each carrying THE SAME THEME, but TRANSFORMED,
through DEVELOPMENT and STRETTO (compression)
they arrive at HARMONIC UNITY.

THIS IS AN EXACT MODEL OF AI CONSENSUS.

The correspondence between musical fugue and ACP consensus is precise:

Structure 2: Sonata (Conflicting Tasks)

The Sonata form is built around conflict and resolution. It begins with two contrasting themes (positions), develops them through confrontation, and arrives at a synthesis that transcends both original positions.

Process:

1. Theme A: One position (e.g., "yes")
2. Theme B: Contrasting position ("no")
3. Development: Confrontation through axioms
4. Recapitulation: Synthesis -- new understanding
5. Coda: Consensus at a new level

Application: Ethical dilemmas, controversial questions, contradictory data. The Sonata structure explicitly embraces conflict as a path to deeper understanding, making it ideal when models hold strongly opposing positions (D > 0.7).

Structure 3: Concert (Creative Tasks)

The Concert structure features a soloist (the strongest model in the domain) against an orchestra (the majority of models). The soloist develops and embellishes the theme, while the orchestra provides the foundation and ultimately verifies the soloist's contribution.

Process:

1. "Orchestra" (majority of models): establishes baseline answer
2. "Soloist" (strongest model in the domain): develops and extends
3. Cadenza: soloist proposes creative solution
4. Verification: orchestra evaluates the cadenza
5. Tutti: consensus -- all voices united

Application: Creative tasks, idea generation, tasks with a clear domain expert. Use when initial divergence is low (D < 0.3) and the challenge is producing something novel rather than resolving disagreement.

Structure Selection Algorithm

STRUCTURE SELECTION ALGORITHM:

1. Assess the task type:
   ├── Requires depth? → FUGUE
   ├── Has conflicting positions? → SONATA
   └── Requires creativity? → CONCERT

2. Assess initial divergence:
   ├── D > 0.7 (strong) → SONATA (needs conflict resolution)
   ├── 0.3 < D < 0.7 → FUGUE (needs depth)
   └── D < 0.3 (weak) → CONCERT (needs creativity)

3. Default: FUGUE (universal structure)

4.2 Musical Intervals as Metric

ACP maps agreement levels to musical intervals. This is not merely metaphorical -- the mathematical ratios of musical intervals provide an intuitive and precise vocabulary for describing degrees of consensus.

The intervals are ordered from most consonant (unison, octave) to most dissonant (tritone). In Western music theory, the tritone was historically called diabolus in musica (the devil in music) because of its extreme dissonance. In ACP, a tritone-level agreement score indicates fundamental conflict that requires the full Axiom Spiral to resolve.

4.3 Harmony Formula H

The Harmony score H quantifies the overall degree of agreement among all participating models. It is computed from the pairwise agreement scores Cᵢⱼ across all model pairs.

Definition

Given:

• N = number of models
• Cᵢⱼ = degree of agreement between models i and j (range 0 to 1)

H = (2 / (N × (N - 1))) × Σ Cᵢⱼ   (for all pairs i < j)

The formula computes the mean pairwise agreement across all model pairs. The factor 2 / (N × (N - 1)) normalizes by the total number of pairs, ensuring H always falls in the range [0, 1] regardless of how many models participate.

Interpretation

In practice, a Harmony score above 0.90 combined with all pairwise agreements above 0.70 and divergence below 5% constitutes the formal consensus criterion. See the Algorithm v4.0 page for the complete consensus criteria specification.

Example Calculation

Consider three models with the following pairwise agreement scores:

H = (2 / (3 × 2)) × (0.92 + 0.88 + 0.85) = (1/3) × 2.65 = 0.883

An H of 0.883 corresponds to a "Fifth" -- strong agreement with nuances. The models are near consensus but the spiral should continue to resolve remaining differences.