Prime Harmonic Profiles

Status: Active development within Prime Period Theory (PPT). This document captures the mathematical methodology for calculating objective harmonic signatures, findings so far, and the core algorithmic properties of the profile model.

1. Goal

Develop a consistent structural methodology to evaluate chords and intervals by comparing their full recursive sets of partials — not just their fundamental frequencies. By measuring the combinatorial acoustic interactions between these partials, we extract the structural “Prime Harmonic Profile.”

The aim is not to produce a subjective consonance/dissonance score, but to mathematically expose which prime families (Du, Tri, Qui, Sep, Undec) are physically interacting to bind a chord together.

2. Core Algorithmic Properties

2.1 Prime Family Vectors

Any rational frequency ratio factors uniquely into exponents of the prime series (2, 3, 5, 7, 11). For example, 16/15 = 2⁴·3⁻¹·5⁻¹ translates to the vector {Du: 4, Tri: -1, Qui: -1}. A combination profile ignores the sign and exponent to simply identify which prime families are active in a relationship (e.g., Du Tri Qui).

2.2 Recursive Partial Generation & Depth Truncation

For a fundamental tone with ratio 1, partial n exists at ratio n with a physical amplitude weighting of 1/n. The algorithm models harmonic generation recursively:

• Depth 1 (Primary Partials): [n] at ratio n with weight 1/n.
• Depth 2 (Sub-partials): The partial itself acts as a virtual fundamental, generating sub-partials [n, m] at ratio n·m with compounded weight 1/(n·m).
• Depth > 2: Recursion continues theoretically infinitely, but is truncated for practical computation. Because physical amplitude decays quadratically through recursion, deeper partials contribute exponentially diminishing acoustic power. (Calculations in this document are capped at Depth 2).

2.3 Amplitude Aggregation (Deduplication)

Different recursive branches often arrive at the exact same physical frequency ratio (e.g., the 2nd sub-partial of the 3rd partial [3, 2] is ratio 6, as is [2, 3]). These paths are strictly collapsed into a single canonical point, and their weights are summed linearly to model physical constructive interference.

2.4 Power Weighting

When evaluating the interaction between two points in the pooled set, the physical interaction strength is proportional to acoustic power. The algorithm squares the aggregated amplitude weights of each point (weight²), multiplying them together to find the combination power. This aggressively suppresses weak, unreinforced partials while heavily rewarding reinforced structural nodes.

2.5 Just Noticeable Difference (JND) Snapping

When evaluating intervals from Equal Temperament (12-TET), irrational frequencies ensure that partials never perfectly align. The algorithm models neurological auditory grouping by defining a JND limit (e.g., 15 cents). If the raw interaction between two irrational partials falls within this tolerance limit to a pure rational fraction, it “snaps” to that rational lattice point. This neurologically models how out-of-tune systems like 12-TET leverage auditory tolerance to mimic true harmonic interference.

2.6 Tone Attribution

The total combination power generated by an interaction between two partials can be distributed back to their originating fundamental tones based on the ratio of their contributing weights. This allows the structural power of a complex chord to be attributed strictly to individual tones, mapping their relative structural gravity.

3. Structural Findings

3.1 Organic Root Bias Emergence

By treating summed weights as amplitudes and squaring them, a natural asymmetry emerges without any heuristic rules. Because the root note generates the simplest partials (ratio 1), it organically accumulates the highest constructive interference. This naturally produces a magnitude gap favouring the root-to-third and root-to-fifth relationships, explaining the primacy of root position chords mathematically.

3.2 Inversions: Shifting Gravity

The power distribution gap between Major and Minor triads is voicing-dependent. In root position, Major is significantly more powerful than Minor in the Du Tri combination because the root sits securely in the bass. In first inversion, this structural gravity shifts, and Minor edges out Major due to the geometric realignment of interacting partials.

4. Master Data Tables: Combination Profiles

Methodology Note: These tables reflect additive amplitude aggregation with Depth 2 recursive partial generation and power=2 weighting. The algorithm now strictly enforces prime factoring without naive octave reduction mapping—for example, a perfect twelfth (3/1) is classified strictly as Tri without inheriting a false Du relationship, ensuring the profiles represent true prime interference.

4.1 Just Intonation (JI) Triads: Combination Profiles

Calculated using pure 5-limit integer ratios (Root position).

4.2 Just Intonation (JI) 7th Chords: Standard Group

The standard 7th chords calculated purely within the 5-limit.

4.3 Tone Attribution Distribution

By attributing combination power back to its generative sources proportionally based on the relative weight of the interacting partials, we observe the gravitational pull of specific chord members.

Notice how in the Major triad, the Root structurally out-pulls the Third, and the Fifth acts as a secondary anchor. In the Diminished triad, power is distributed almost entirely symmetrically.