Pseudo-Chord Measures in Music Analysis

• Pseudo-chord measures are a family of analytical techniques that compare surrogate chord properties such as pitch-class sets, rational ratios, or symmetry invariants.
• They encompass diverse frameworks: continuous pitch-content accuracy in chord estimation, invariant complexity metrics in just intonation, and combinatorial counts in 12-TET pseudo-palindromic chords.
• Applications range from refining chord recognition evaluation via pseudo annotations to investigating harmonic similarity, though challenges include normalization and domain-specific limitations.

Pseudo-chord measures are not a single standardized construct. In current arXiv usage, the term spans several technically distinct families of objects: a pitch-content accuracy for chord estimation that compares the pitch classes implied by chord labels rather than treating labels as independent classes (Devaney, 2022); invariant measures for pseudo-chords in Just Intonation, where a pseudo-chord is any finite multiset of rational ratios or integers after normalization (Ryan, 2016); counts and symmetry classes of pseudo-palindromic $n$-chords in 12-note equal temperament (Caimmi et al., 2017); and pseudo accuracies derived from pseudo annotations in large-scale chord-recognition evaluation (Ni et al., 2011). By contrast, recent work in convex and integral geometry introduces chord measures $F_q(K,\cdot)$ and $G_q(K,\cdot)$ but explicitly does not define “pseudo-chord measures” (Lutwak et al., 12 Feb 2025).

1. Terminological scope and conceptual distinctions

In music-information-retrieval usage, “pseudo-chord measures” refers to comparing the pitch-class content implied by chord labels rather than the labels themselves. A chord label is treated as shorthand for a pitch-class set, and the measure operates on the resulting sets. This directly addresses the limitation of label-only metrics, which treat chord labels as independent classes and therefore fail to represent musical proximity through shared pitch content (Devaney, 2022).

In Just Intonation harmony analysis, the relevant object is a pseudo-chord understood as any finite multiset of rational ratios, or of integers after common-denominator scaling and normalization by the greatest common divisor. The associated measures are invariant functions under transposition, including Complexity, Otonality, Utonality, pairwise-ratio coefficients, prime projections, and weighted variants (Ryan, 2016).

In 12-TET combinatorics, the term appears in the narrower form “pseudo-palindrome $n$-chords.” These are $n$-tuples of positive integers summing to $12$ that are invariant under reflection followed by a suitable rotation, that is, there exists $k$ with $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$. Their “measures” are enumerative and geometric: counts of repeating chords, $T_n$ set classes, and $T_n/T_nI$ set classes inside the regular inclined $F_q(K,\cdot)$0-hedron $F_q(K,\cdot)$1 (Caimmi et al., 2017).

A different usage arises in meta-song evaluation for chord recognition. There, pseudo annotations are time-aligned chord labels generated from untimed online chord charts, and pseudo accuracy is the frame-wise accuracy of a system’s output against those pseudo annotations. Statistical models then map pseudo accuracy to estimated ground-truth accuracy (Ni et al., 2011).

This distribution of meanings suggests that “pseudo-chord measure” functions as a family resemblance term rather than a unique formalism. The shared pattern is the replacement of direct, fully specified chord identity by a surrogate structure: pitch-class sets, normalized ratio multisets, symmetry classes, or pseudo annotations.

2. Pitch-content pseudo-chord measures in chord estimation

The formulation introduced in chord estimation begins from a 12-tone equal-temperament pitch-class model with octave invariance and enharmonic equivalence. A parser $F_q(K,\cdot)$2 maps a chord label $F_q(K,\cdot)$3 to a pitch-class set $F_q(K,\cdot)$4. The framework explicitly specifies common cases: inversions and slash chords preserve the underlying pitch-class set, suspensions replace the third by the second or fourth, added-tone chords include the added pitch classes, power chords contain only root and fifth, seventh chords add the relevant seventh, diminished and augmented chords receive their standard triadic pitch classes, and no-chord is represented by the empty set $F_q(K,\cdot)$5 (Devaney, 2022).

Let the ground-truth chord be $F_q(K,\cdot)$6 and the estimate be $F_q(K,\cdot)$7. The base quantities are

$F_q(K,\cdot)$8

The pitch-content accuracy is then

$F_q(K,\cdot)$9

For $G_q(K,\cdot)$0, the score lies in $G_q(K,\cdot)$1. Exact matches yield $G_q(K,\cdot)$2. Missing notes reduce $G_q(K,\cdot)$3 but do not subtract directly, whereas inserted notes reduce the score through $G_q(K,\cdot)$4. The stated consequence is a metric that is more forgiving of omissions than of insertions, which is useful when an algorithm predicts simpler chords in a complex vocabulary (Devaney, 2022).

The paper positions this quantity against exact label accuracy, WCSR, and root/third/quality metrics. Exact label accuracy and WCSR are binary and weight all errors equally; root/third/quality metrics remain categorical and evaluate selected components of the label. By contrast, the pseudo-chord measure is a continuous score in $G_q(K,\cdot)$5 that depends on overlap and insertions in the pitch-class domain. It is not a stand-alone evaluation protocol but a weighting that can be integrated into existing approaches (Devaney, 2022).

Temporal aggregation is defined by replacing binary matches with $G_q(K,\cdot)$6: $G_q(K,\cdot)$7 This supports direct substitution in exact-label or WCSR-style scoring, or modulation of root/third/quality metrics via $G_q(K,\cdot)$8. The computational cost is $G_q(K,\cdot)$9 per frame or segment because the relevant set operations range over at most $n$0 pitch classes, and $n$1 over $n$2 frames (Devaney, 2022).

The framework also gives a weighted extension. If $n$3 is assigned to each $n$4 and $n$5 weights false positives, then with

$n$6

the weighted measure is

$n$7

Setting $n$8 and $n$9 recovers $n$0. The intended use is to prioritize root, third, or seventh, or to impose penalties tied to note salience (Devaney, 2022).

Several worked examples clarify the asymmetry between omissions and insertions. For $n$1 and $n$2, the score is $n$3; for $n$4 and a single-note estimate $n$5, the score is $n$6; and for $n$7 versus $n$8, the same value $n$9 is obtained because the estimate shares two pitch classes and inserts one. The paper does not report formal correlation tests with human judgments, but motivates the metric on the grounds that pitch-content comparison should align better with perceived harmonic similarity than binary label matches, especially as vocabularies expand or annotation styles vary (Devaney, 2022).

3. Pseudo-chords and invariant measures in Just Intonation

In the Just Intonation framework, a chord is represented by rational ratios $12$0 with $12$1. For any local section, the ratios can be scaled so that all become whole positive integers $12$2. The chord is then written as an integer vector $12$3 in ascending order, and subsequent invariants are computed after reduction by the greatest common divisor. Invariance means invariance under multiplication of the entire chord by a constant factor, which models transposition or a change of base frequency (Ryan, 2016).

The central global measure is Complexity: $12$4 It is a positive integer, unchanged by transposition, and for $12$5 reduces to Benedetti Height. The paper interprets it as representing “the ratio of the largest to the smallest structures in the waveform of the chord,” with smaller $12$6 indicating simpler, more consonant structure and larger $12$7 indicating more complex, more dissonant structure (Ryan, 2016).

The logarithmic formulation uses

$12$8

and defines the LogMidpoint

$12$9

From this, Otonality and Utonality are

$k$0

with $k$1 for $k$2 and $k$3 for $k$4. These coefficients quantify whether a chord is lower-divisor-heavy or upper-divisor-heavy within ComplexitySpace, the invariant set of divisors of $k$5 (Ryan, 2016).

Local structure is captured by pairwise-ratio invariants: $k$6 The summary measures include

$k$7

together with

$k$8

Prime projections retain only selected prime factors, leading to projected complexity

$k$9

Important special cases are $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$0, which discards all powers of $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$1 and therefore represents pitch classes up to octave equivalence, $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$2, and Bohlen–Pierce Complexity $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$3, which discards primes $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$4 and $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$5 (Ryan, 2016).

The framework explicitly extends to weighted pseudo-chords, where multiplicities or loudnesses become weights $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$6. Then

$T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$7

The psychoacoustic rationale given is that louder notes contribute more to the perceptual center of gravity on the tone lattice. The paper recommends collapsing duplicate notes into distinct pitch ratios with multiplicities encoded in $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$8, although duplicates may also be retained if ratio-based measures are handled carefully (Ryan, 2016).

A pseudo-chord is defined here as any finite multiset of rational ratios or normalized integers, possibly with duplicates, missing fundamentals, partial selections from an overtone or undertone set, or octave-equivalent collapses. This broad definition allows standard and non-standard sets to be analyzed within the same invariant apparatus. The stated applications include classifying consonance by $T_k I(\ell_1,\dots,\ell_n)=(\ell_1,\dots,\ell_n)$9 and $T_n$0, distinguishing otonal and utonal bias through $T_n$1, visualizing chords in the $T_n$2-limit tone lattice, and searching for scales or chords meeting explicit constraints on Complexity, prime content, and ratio spacing (Ryan, 2016).

4. Pseudo-palindromic $T_n$3-chords in 12-TET

The 12-TET combinatorial framework represents an $T_n$4-chord as an ordered $T_n$5-tuple of positive integers

$T_n$6

encoding successive semitone intervals between adjacent pitch classes around the octave, subject to

$T_n$7

Geometrically, these are positive integer points on the regular inclined $T_n$8-hedron $T_n$9 defined by $T_n/T_nI$0 (Caimmi et al., 2017).

The cyclic permutation operator $T_n/T_nI$1 rotates coordinates, and the reflection operator $T_n/T_nI$2 reverses them. Repeating $T_n/T_nI$3-chords are those whose $T_n/T_nI$4 orbit has size less than $T_n/T_nI$5, equivalently those fixed by some nontrivial rotation. Palindromic $T_n/T_nI$6-chords are invariant under reflection, while pseudo-palindromic $T_n/T_nI$7-chords are invariant under reflection followed by a suitable rotation: $T_n/T_nI$8 These pseudo-palindromes are precisely the repeating $T_n/T_nI$9-chords that become symmetric under the larger $F_q(K,\cdot)$00 relation (Caimmi et al., 2017).

The counting formulas distinguish positive-integer points, repeating points, $F_q(K,\cdot)$01 classes, and $F_q(K,\cdot)$02 classes. The total number of distinct $F_q(K,\cdot)$03-chords is

$F_q(K,\cdot)$04

If $F_q(K,\cdot)$05 is the divisibility indicator, the total number of repeating $F_q(K,\cdot)$06-chords is

$F_q(K,\cdot)$07

Then

$F_q(K,\cdot)$08

and if $F_q(K,\cdot)$09 denotes the number of $F_q(K,\cdot)$10 set classes containing palindromic or pseudo-palindromic $F_q(K,\cdot)$11-chords, then

$F_q(K,\cdot)$12

The paper states that for $F_q(K,\cdot)$13, every $F_q(K,\cdot)$14-chord is palindromic or pseudo-palindromic, so $F_q(K,\cdot)$15 (Caimmi et al., 2017).

The small-$F_q(K,\cdot)$16 cases reported in the paper are:

$F_q(K,\cdot)$17	$F_q(K,\cdot)$18	$F_q(K,\cdot)$19	$F_q(K,\cdot)$20	$F_q(K,\cdot)$21	$F_q(K,\cdot)$22
2	11	1	6	6	6
3	55	2	19	5	12
4	165	7	43	15	29

For $F_q(K,\cdot)$23, palindromes solve $F_q(K,\cdot)$24, producing $F_q(K,\cdot)$25, hence $F_q(K,\cdot)$26. For $F_q(K,\cdot)$27, the paper decomposes the pseudo-palindromic count into three Diophantine types and obtains $F_q(K,\cdot)$28 (Caimmi et al., 2017).

The geometric interpretation is central. Repeating $F_q(K,\cdot)$29-chords lie on $F_q(K,\cdot)$30-faces shared by multiple $F_q(K,\cdot)$31; palindromes lie on symmetry lines determined by opposite faces; pseudo-palindromes are images of such symmetric loci under cyclic permutation. In this usage, pseudo-chord measures are therefore discrete, enumerative, and geometric rather than similarity-based.

5. Pseudo annotations and pseudo accuracies in chord-recognition evaluation

Meta-song evaluation addresses the scarcity of fully annotated songs by using online chord charts to generate “pseudo annotations” and then evaluating systems against those annotations. The source is a large online database, E-chords, from which untimed chord sequences are scraped. A reference system called Jump Alignment aligns the untimed chord sequence to the audio and produces high-accuracy time-stamped chord annotations, which are treated as pseudo annotations (Ni et al., 2011).

If a system outputs a time-indexed chord sequence $F_q(K,\cdot)$32 and the pseudo annotation is $F_q(K,\cdot)$33, pseudo accuracy is defined as the accuracy of the system’s prediction compared to pseudo annotation. The paper uses frame-wise accuracies, which can be written as

$F_q(K,\cdot)$34

or, in discrete form,

$F_q(K,\cdot)$35

The paper does not report root-only, quality-only, seventh-aware, or WCSR-like pseudo metrics; only a single pseudo accuracy aligned with frame-wise accuracy is used (Ni et al., 2011).

The contribution of the work is the statistical mapping from pseudo accuracy $F_q(K,\cdot)$36 to ground-truth accuracy $F_q(K,\cdot)$37. Three models are introduced. The Single Gaussian model assumes

$F_q(K,\cdot)$38

The Individual Gaussian model uses system-specific bias and variance,

$F_q(K,\cdot)$39

The Linear regression model uses a system-specific slope and intercept,

$F_q(K,\cdot)$40

All three models produce confidence intervals for song-level and mean accuracies on test sets without ground-truth annotations (Ni et al., 2011).

The validation set contains $F_q(K,\cdot)$41 Beatles songs with both ground-truth and pseudo annotations; the test set contains $F_q(K,\cdot)$42 songs from a variety of genres with pseudo annotations only. On the Beatles validation set, the reported estimated mean ground-truth accuracies and actual ground-truth accuracies are: HP, $F_q(K,\cdot)$43 under the Single Gaussian model and $F_q(K,\cdot)$44 actual ground truth; labROSA, $F_q(K,\cdot)$45 under Single Gaussian and $F_q(K,\cdot)$46 actual ground truth; Chordino, $F_q(K,\cdot)$47 under Single Gaussian and $F_q(K,\cdot)$48 actual ground truth; and the majority-vote consensus, $F_q(K,\cdot)$49 under Single Gaussian and $F_q(K,\cdot)$50 actual ground truth. All true ground-truth values lie within the reported $F_q(K,\cdot)$51 intervals. The paper attributes bias in the Single Gaussian model to shared chromagram features between HP and Jump Alignment and states that this bias is reduced or eliminated by the Individual Gaussian and Linear models (Ni et al., 2011).

Within this line of work, pseudo-chord measures are not measures of harmonic content but measures of evaluation surrogate quality. Their role is methodological: scalable evaluation on large, diverse song collections without full annotation, together with calibrated estimates of real performance. The limitations are equally explicit: human-entered charts can be simplified, approximate, incomplete, or cover only part of a song; the absence of exact timings can degrade alignment; and coupling between the pseudo-annotation generator and a test system can inflate pseudo accuracies (Ni et al., 2011).

6. Adjacent geometric usages and explicit non-usage of the term

Two further arXiv lines of work are relevant primarily because they clarify what pseudo-chord measures are not. In “Hausdorff Measures in Bertrand’s Random Chord Problem,” the objects are geometric chords of a circle viewed as compact subsets of $F_q(K,\cdot)$52, endowed with the Hausdorff metric

$F_q(K,\cdot)$53

The paper develops open and closed metric balls as “tubes,” computes their Hausdorff dimension, and defines a normalized $F_q(K,\cdot)$54 probability measure on the space of chords. It states that the chord space has diameter $F_q(K,\cdot)$55, that non-degenerate tubes have Hausdorff dimension $F_q(K,\cdot)$56, that $F_q(K,\cdot)$57, and that the Bertrand probability $F_q(K,\cdot)$58 equals $F_q(K,\cdot)$59. However, the paper does not explicitly use the term “pseudo-chord” (Zorine, 2023).

An even sharper terminological distinction appears in “Chord Measures in Integral Geometry and Their Minkowski Problems.” That paper explicitly states that it does not use or define the term “pseudo-chord measures.” Instead, it introduces chord measures $F_q(K,\cdot)$60, cone-chord measures $F_q(K,\cdot)$61, and their $F_q(K,\cdot)$62 generalizations $F_q(K,\cdot)$63. For $F_q(K,\cdot)$64,

$F_q(K,\cdot)$65

$F_q(K,\cdot)$66

The paper further notes that $F_q(K,\cdot)$67, $F_q(K,\cdot)$68, and formulates Minkowski and log-Minkowski problems for these measures (Lutwak et al., 12 Feb 2025).

The same source then proposes plausible “pseudo-chord” generalizations only by analogy: weighted, $F_q(K,\cdot)$69, Orlicz, and dual variants preserving homogeneity, orthogonal covariance, translation invariance, weak continuity, and variational origin. This is explicitly presented as a proposal rather than a definition. A plausible implication is that, in geometry, “chord measure” has already become a precise technical term, whereas “pseudo-chord measure” remains nonstandard and context-dependent (Lutwak et al., 12 Feb 2025).

7. Comparative perspective, limitations, and research implications

Across these literatures, pseudo-chord measures serve markedly different analytical purposes. In chord estimation, the measure quantifies musical proximity by shared pitch content and inserted content; in Just Intonation, it quantifies invariant structure under transposition and normalization; in 12-TET combinatorics, it counts symmetry classes under rotation and reflection; and in meta-song evaluation, it operationalizes a surrogate accuracy against pseudo annotations. The commonality is structural substitution rather than object identity.

The limitations are equally domain-specific. The pitch-content measure ignores voice-leading, register, and bass motion, does not model salience in the base formulation, and is out of scope for microtonality and non-12-TET (Devaney, 2022). The Just Intonation invariant framework requires rational inputs or rational approximation, can become numerically difficult when integers grow large, and warns that weighted otonality depends strongly on $F_q(K,\cdot)$70, so comparisons of $F_q(K,\cdot)$71 across chords should keep $F_q(K,\cdot)$72 fixed (Ryan, 2016). The 12-TET pseudo-palindrome framework is tied to positive integer compositions of $F_q(K,\cdot)$73 and to the symmetry actions $F_q(K,\cdot)$74 and $F_q(K,\cdot)$75, so its notion of pseudo-chord is specifically combinatorial (Caimmi et al., 2017). Meta-song pseudo accuracies depend on the availability and quality of online chord charts, on alignment quality, and on the absence of system–reference coupling (Ni et al., 2011).

For arXiv readers, the main methodological consequence is the need for disambiguation. A pitch-content pseudo-chord measure $F_q(K,\cdot)$76, a Just Intonation invariant such as $F_q(K,\cdot)$77 or $F_q(K,\cdot)$78, a pseudo-palindrome count $F_q(K,\cdot)$79, and a pseudo accuracy $F_q(K,\cdot)$80 are all legitimate “pseudo-chord measures” in their own literatures, but they are not interchangeable. This suggests that any use of the term in technical writing should specify the underlying object space—pitch-class sets, integer-normalized ratios, interval compositions, or pseudo annotations—and the symmetry or evaluation principle that turns those objects into measurable quantities.