Validation Results

The system has been validated through analysis of 192 temperaments from two datasets (PianoScope and CyberTuner). Results include:

Key Findings

• 52 very similar pairs (Euclidean distance < 10 cents) between PianoScope and CyberTuner datasets
• 184 similar pairs (Euclidean distance < 30 cents)
• 72 same comma fraction matches across datasets
• Pattern detection identifies comma fractions (1/4, 1/5, 1/6, etc.) with ±0.5 cent tolerance
• Label validation: 49% of PianoScope "meantone" labels match mathematical definition

See: Validation Report | Detailed Analysis

Available Tools

• Similarity Explorer — Discover similar temperaments through multiple similarity metrics. Find relationships based on pattern matching, transposition, and mathematical similarity.
• Educational Explorer — Visualize comma detection in any temperament. Explore fifth analysis and relationships through degrees of separation.
• Family Tree Explorer — Build family trees showing how temperaments relate through pattern inheritance using CyberTuner as validated base patterns.

Methodology

Comma Detection

The system detects comma fractions by analyzing perfect fifths (P5 = 701.955 cents). The method calculates the average fifth deviation from pure and determines the fraction using:

fraction = syntonic_comma / |avg_fifth_deviation|

Detection tolerance: ±0.5 cents. Validation requires standard deviation < 2.5 cents.

Similarity Metrics

Multiple metrics are computed on normalized temperament offsets:

• Euclidean Distance: sqrt(sum((norm1[i] - norm2[i])^2))
• Manhattan Distance: Sum of absolute differences
• Max/Mean Differences: Largest and average differences
• Circular EMD: Earth Mover's Distance accounting for octave circularity

Normalization

Temperaments are normalized to ensure transpositional invariance: minimum offset shifted to 0, wrapped to [0, 1200) range. This ensures temperaments in different keys are comparable.

Mathematical Constants

Comma Values (cents)

• Syntonic Comma: 21.5062896 cents (1200 × log₂(81/80))
• Pythagorean Comma: 23.4600104 cents (1200 × log₂(531441/524288))
• Schisma: 1.9537208 cents (1200 × log₂(32805/32768))

Expected Meantone Fifth Deviations

• 1/4 comma: -5.3766 cents per fifth
• 1/5 comma: -4.3013 cents per fifth
• 1/6 comma: -3.5844 cents per fifth
• 1/7 comma: -3.0723 cents per fifth
• 1/8 comma: -2.6883 cents per fifth
• 1/9 comma: -2.3896 cents per fifth
• 1/10 comma: -2.1506 cents per fifth

API

Endpoints:

• GET api_temperament_similarity.php?list=1 — List all 12-degree temperaments
• GET api_temperament_similarity.php?temperament_id=X&pattern=1 — Get pattern analysis
• GET api_temperament_similarity.php?temperament_id1=X&temperament_id2=Y — Compare two temperaments
• GET api_temperament_similarity.php?temperament_id=X&limit=N — Find similar temperaments
• GET api_temperament_similarity.php?pattern_fraction=X — Find temperaments by comma fraction

📖 View Complete API Documentation | Technical Documentation

See Also

• Similarity Explorer
• Educational Explorer
• Family Tree Explorer
• Documentation Index