lamm-mit/scales-12tet-defects

Musical Scales in the 12-Tone Equal Temperament System: Defects Dataset

All 12-TET pitch-class sets that include the root (pc 0), with size ≥1 and no max-gap filter. Includes per-scale metrics and binary vectors.

Contents

• mask: bitmask of present pitch classes (LSB = pc 0).
• n_pcs: total pitch classes (12).
• pcs: list of pitch classes present.
• steps: circular step vector (sorted pcs differences, wrapping at 12).
• k: number of notes in the scale.
• x_0 ... x_11: binary vector indicating presence of each pitch class.
• pcs_json, steps_json: JSON-encoded pcs/steps for convenience.

Defect and entropy metrics

• zeitler_defect: count of degrees whose perfect fifth (p+7 mod 12) is missing.
• evenness_defect: normalized spacing unevenness. Compute std of step sizes vs ideal (12/k); divide by the maximum std observed for this k across all scales to map to [0,1].
• evenness_defect_count: integer unit moves to nearest even partition. Build the ideal integer step pattern (floor/ceil of 12/k summing to 12), sort both actual and ideal steps, take L1 distance, divide by 2 (each +1/–1 pair counts as one move).
• arrangement_defect: palindromic symmetry defect in [0,1]; 0 if step vector matches some rotation of its reverse (max cosine similarity), 1 for least symmetric.
• unique_intervals_defect: in [0,1]; 0 if only one distinct step size, 1 if as many distinct step sizes as possible (capped at min(k,6)).
• composite_defect: weighted mix 0.5*evenness_defect + 0.3*arrangement_defect + 0.2*unique_intervals_defect, in [0,1].
• entropy_bits: Shannon entropy (bits) of the step-size distribution.
• entropy_norm: entropy_bits normalized by log2 of the number of distinct step sizes (0–1).
• lz_norm: normalized LZ76-style complexity of the step sequence (c/L, heuristic).

Generation settings

• Enumeration: all masks with pc 0 present, k ≥ 1, no max-gap filter (--min-k 1 --no-gap-filter --max-gap 0).
• Pitch-class system: 12-TET.

How we generated this dataset

• Scale enumeration: Iterate over all (2^{12}) masks; keep only those with root (pc 0) set and size (k \ge 1). No circular-gap filter is applied in this build.
• Intervals and steps: For each retained mask, convert to pcs, sort, and build the circular step vector (including wrap-around at 12).
• Zeitler defect: Count notes whose perfect fifth (p + 7 mod 12) is absent.
• Evenness metrics:
  • Normalized evenness defect: std of step sizes vs ideal (12/k), divided by the maximum std observed for that (k) across all scales (yields [0,1] within each (k)).
  • Evenness defect count: minimal semitone “unit moves” to reach the nearest integer even partition of 12; sort actual and ideal step patterns, take (\ell_1) distance, divide by 2.
• Symmetry/interval diversity:
  • Arrangement defect: 1 minus the best cosine similarity between the step vector and any rotation of its reverse (0 = palindromic under some rotation).
  • Unique intervals defect: 0 if one distinct step size; 1 if as many distinct step sizes as possible (capped at (\min(k,6))).
• Composite defect: Weighted mix (0.5 \cdot \text{evenness} + 0.3 \cdot \text{arrangement} + 0.2 \cdot \text{unique_intervals}).
• Entropy/complexity:
  • entropy_bits: Shannon entropy (bits) of the step-size distribution.
  • entropy_norm: entropy_bits divided by (\log_2) of the number of distinct step sizes.
  • lz_norm: heuristic LZ76-style normalized complexity of the step sequence (c/L).
• Representations: Store pcs/steps, mask, k, and binary indicator columns x_0...x_11. Also keep JSON-encoded pcs/steps for CSV friendliness.

Sample results

Reference

@misc{buehler2025selectiveimperfectiongenerativeframework,
      title={Selective Imperfection as a Generative Framework for Analysis, Creativity and Discovery}, 
      author={Markus J. Buehler},
      year={2025},
      eprint={2601.00863},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2601.00863}, 
}