atonal 0.0.4

Tools for atonal theory -- pitch class sets, forte numbers etc.

atonal

Tools for mathematical analysis of 12‑tone musical structures.

1) Context and goals

This package focuses on pitch‑class set theory (Forte-style set classes and related relationships) as a practical, data-oriented toolkit.

The core abstraction is a pitch class: pitches modulo 12, typically labeled 0..11 (0=C, 1=C♯, …, 11=B). A pitch‑class set (pc‑set) is an unordered collection of pitch classes, ignoring octave and enharmonic spelling.

This is the same abstraction pipeline described in A mathematics of musical harmony:

• move from sound → pitch,
• discretize to 12‑TET,
• quotient by octave/enharmonic equivalence,
• study subsets of the 12 pitch classes,
• and then factor by musically meaningful symmetries (especially transposition $T_n$ and inversion $I$).

Under $T/I$ equivalence, many different pc‑sets collapse into a single set class represented by a canonical prime form. That enables combinatorial analysis of harmonic “shapes” independent of key.

atonal aims to make these objects and relations easy to compute, and (crucially) easy to analyze as data.

2) Data in misc/tables

The folder misc/tables/ contains precomputed tables (Parquet format) that represent:

• Nodes: every subset of the 12 pitch classes ($2^{12} = 4096$ rows)
• Links: several useful relations between those subsets (edge tables)

These tables are generated in misc/making_pitch_class_sets_data.ipynb using functions from atonal/base.py.

2.1 Nodes table: pitch_class_sets/twelve_tone_sets.parquet

This file contains 4096 rows, one for each pitch‑class subset.

You can load it with pandas:

import pandas as pd

nodes = pd.read_parquet('misc/tables/pitch_class_sets/twelve_tone_sets.parquet')

Field guide:

• id_ (int, 0..4095)

  • A 12‑bit bitmap encoding the set: bit i is 1 iff pitch class i is present.
  • This is a compact single‑identifier for joins across all link tables.

• pcset (sequence of ints)

  • The sorted pitch classes present in the set.
  • Example: (0, 4, 7) is the C major triad pc‑set.

• cardinality (int)

  • The number of pitch classes in the set (size of pcset).

• contains_zero (bool)

  • Convenience feature: whether pitch class 0 is present.
  • Useful because many canonical representatives are transposed to include 0.

• complement_id (int)

  • Bitmap id of the complement set with respect to the 12‑tone universe.
  • Computed as 4095 ^ id_.

• prime_form (sequence of ints)

  • Canonical representative of the set class under $T/I$ equivalence.
  • Computed by comparing best normal order of the set and its inversion, transposed to start at 0, then choosing the lexicographically smallest.
  • Example: major/minor triads share prime form [0, 3, 7].

• forte_name (str or null)

  • Forte set‑class label for cardinalities 3..9 when defined (e.g. "3-11", "6-Z29").
  • For 7..9 the name is derived via complement labeling (same suffix).
  • Null outside 3..9.

• is_forte_set (bool)

  • True when the row’s pcset is itself a canonical prime‑form representative for a Forte-labeled class (i.e. pcset == prime_form and that prime form is in the module’s Forte lookup table).
  • Practically: “one chosen representative per set class” (where available).

• interval_vector (length‑6 sequence of ints)

  • The interval‑class vector $(ic_1, ic_2, ic_3, ic_4, ic_5, ic_6)$.
  • Each entry counts unordered pitch‑class pairs at that interval class, where interval class is the shortest distance mod 12.
  • This is a key invariant for characterizing harmonic color.

• is_t_symmetric (bool)

  • True if the set is invariant under some non‑zero transposition $T_n$.
  • Symmetric sets have fewer than 12 distinct transpositions.

• z_correspondent_prime_form (sequence of ints or null)

  • Prime form of the Z‑correspondent, if any.
  • Z‑related sets share the same interval_vector but are not $T/I$ equivalent.
  • Reported only when a correspondent exists in the module’s Forte catalog.

• z_correspondent_forte_name (str or null)

  • Forte label for the Z‑correspondent (when available).

• n_T (int)

  • Number of distinct transpositions in the set’s $T$ orbit.
  • For symmetric sets this is less than 12.

• n_I (int)

  • Number of distinct inversions in the set’s $I T$ orbit.

• kh_size (int)

  • Size of the set’s Forte $K_h$ subcomplex under the operational definition used in atonal.base.kh_complex_size().
  • Intuition: counts how many sets are linked to the nexus set by reciprocal inclusion relations involving both the set and its complement.

• hexachord_combinatorial (str or null)

  • Combinatoriality label for hexachords (cardinality 6):
    • "a": all‑combinatorial (both T and I combinatorial)
    • "p": prime combinatorial (T combinatorial only)
    • "i": inversion combinatorial (I combinatorial only)
  • Null for non‑hexachords.

• max_T_invariance (int) and max_T_invariance_n (int)

  • Over all non‑trivial transpositions $T_n$ (n=1..11), max_T_invariance is the maximum number of pitch classes left invariant under the best such $T_n$; _n stores the argmax.

• max_I_invariance (int) and max_I_invariance_n (int)

  • Analogous invariance count for inversion‑transpositions $I T_n$ (n=0..11).

• best_normal_order (sequence of ints)

  • Forte “packed” normal order: a most‑compact cyclic ordering of the set.
  • This is the canonical pre‑prime template used before transposing to 0.

• label (str)

  • A human-friendly label for plotting/inspection.
  • Typically includes forte_name when present and the concrete pcset.

2.2 Links tables: misc/tables/pitch_class_sets/links/*.parquet

These Parquet files are edge tables that relate nodes by id_.

Load example:

import pandas as pd

nodes = pd.read_parquet('misc/tables/pitch_class_sets/twelve_tone_sets.parquet')
links = pd.read_parquet('misc/tables/pitch_class_sets/links/immediate_subset_links.parquet')

edges = links.merge(nodes.add_prefix('src_'), left_on='source', right_on='src_id_') \
						.merge(nodes.add_prefix('tgt_'), left_on='target', right_on='tgt_id_')

Link tables and meanings:

complement_links.parquet

• Columns: source, target
• Meaning: pairs each set with its 12‑tone complement.
• Construction: target = 4095 ^ source.
• Notes:
  • Stored once per undirected pair (2048 edges).

immediate_subset_links.parquet

• Columns: source, target
• Meaning: Hasse diagram edges of the subset lattice.
  • target is obtained by adding exactly one pitch class to source.
  • cardinality(target) = cardinality(source) + 1.
• Musical intuition: moving by one pitch class is the smallest possible change in “pitch material” at the pc‑set level.

ti_equiv_links.parquet

• Columns: source, target
• Meaning: connects pc‑sets that belong to the same $T/I$ set class (i.e. share prime_form).
• Notes:
  • This is an undirected, all‑pairs‑within‑class expansion.
  • Use it to move between different transpositions/inversions of the “same” harmonic shape.

z_relation_links.parquet

• Columns: source, target, interval_vector, prime_form_a, prime_form_b
• Meaning: connects sets that share the same interval_vector but have different prime_form (i.e. are not $T/I$ equivalent).
• Notes:
  • If you want the classical Forte Z‑relations, you will typically filter this table to cardinalities 3..9 and/or to rows where forte_name is non‑null.

rp_triads.parquet and rp_tetrads.parquet

• Columns: source, target, max_common
• Meaning: Forte $R_p$ similarity relation, computed within a fixed cardinality.
  • Two sets of size $n$ are $R_p$‑similar when, under some $T_n$ or $I T_n$, they share exactly $n-1$ pitch classes.
  • For rp_triads, max_common is 2.
  • For rp_tetrads, max_common is 3.
• Musical intuition: “closest neighbors” within a fixed chord size, allowing transposition/inversion.

3) Main functions used to build the data

The tables above are built in misc/making_pitch_class_sets_data.ipynb using these functions from atonal/base.py:

• build_pcset_nodes_df()

  • Generates the 4096‑row nodes table, computing prime forms, interval vectors, invariances, $K_h$ sizes, and labels.

• build_immediate_subset_links_df()

  • Generates the subset lattice edges (add one pitch class).

• build_complement_links_df()

  • Generates complement pairs.

• build_ti_equivalence_links_df(nodes_df)

  • Connects sets with the same prime_form (same $T/I$ class).

• build_z_relation_links_df(nodes_df)

  • Connects sets with the same interval_vector but different prime_form.

• build_rp_similarity_links_df(nodes_df, cardinality=3|4)

  • Generates $R_p$ similarity graphs for triads/tetrads.

Core primitives used in these builders (also useful directly):

• int_to_pcset() / pcset_to_int() for conversion between bitmap ids and explicit pc‑sets.
• prime_form(), best_normal_order(), interval_vector().
• forte_name() and z_correspondent_prime_form().

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Development note

The precomputed tables in misc/tables/ are meant to be usable as-is for analysis and graph workflows; the builders are provided so you can regenerate or extend the data with your own fields/relations.