Chord-Frame Parameterization

• Chord-frame parameterization is a representation framework that distinguishes the basic chord event from the structural context—such as durations, geometric sagitta, or matrix indices—to enhance higher-level analysis.
• In automatic chord recognition, the method decouples frame-level labels into a compressed chord sequence and an explicit duration process, leading to more accurate modeling of harmonic progression.
• Across domains like digital geometry and the framed Kontsevich integral, the approach enables precise estimation and reconstruction by using chord-based frames to encode spatial or topological relationships.

Searching arXiv for the cited papers to ground the article in the relevant literature. Chord-frame parameterization denotes a family of representations in which a primitive “chord” object is not treated in isolation but is linked to an explicit structural scaffold that mediates between local observations and higher-level organization. In automatic chord recognition, it is the decomposition of frame-level chord labels into a chord-sequence representation and an explicit duration process (Korzeniowski et al., 2018). In digital geometry, it is the representation, detection, and parameterization of a circular arc using a chord as a geometric frame and a sagitta as a scale variable (Bera et al., 2014). In the framed Kontsevich integral, it is a matrix-based encoding in which a chord is parameterized by component and strip indices, and framing is encoded through vertical slicing, hooks, and congruence operations on “pages” and “books” (Gauthier, 2010). Across these usages, the common idea is to separate the combinatorial identity of a chord-like event from the structure that determines how it is situated in time, geometry, or topology.

1. Chord-frame parameterization in automatic chord recognition

In automatic chord recognition, the standard task is frame-wise chord labelling: for audio frames $x_{1:T}$, predict chord labels $y_{1:T}$, with inference written as

$$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$

Under the usual left-to-right generative assumptions, the posterior is factorized as

$$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$

where $P_A$ is the acoustic model, $P_T$ is the temporal model, and $P(y_t)$ is a chord prior taken as uniform (Korzeniowski et al., 2018).

The motivation for chord-frame parameterization is that frame-wise temporal models are dominated by self-transitions: most frames within a chord segment repeat the same label. The temporal model therefore spends almost all its probability mass on “stay on the same chord” and learns little about changes and chord progressions. The reformulation explicitly distinguishes between chord segments and frame-level evidence attached to those segments through an explicit duration process. The frame-level sequence becomes a discretization of segments, rather than independent chord decisions (Korzeniowski et al., 2018).

A compressed chord sequence is obtained with a compression operator $C(\cdot)$ that removes consecutive duplicates, for example $C(C,C,F,F,G)=(C,F,G)$. This yields a segment-level sequence $\bar y_{1:K}=C(y_{1:T})$, where each $y_{1:T}$0 is the chord symbol for the $y_{1:T}$1-th chord segment. The temporal model is then disentangled into a harmonic LLM $y_{1:T}$2 defined on compressed chord sequences and a duration model $y_{1:T}$3 defined on binary change/stay decisions $y_{1:T}$4 at the frame level (Korzeniowski et al., 2018).

The core factorization is

$y_{1:T}$5

If the chord changes at frame $y_{1:T}$6, temporal probability decomposes into the probability of changing at that frame and the probability of the next chord symbol. If the chord stays the same, temporal probability is purely duration-driven. Frame labels are therefore generated from a latent chord sequence and a binary stay/change process; they are no longer the primitive object of the temporal model (Korzeniowski et al., 2018).

This formulation is also described as a hierarchical process with three parts: frame-level acoustics $y_{1:T}$7, segment-level harmony $y_{1:T}$8, and duration or harmonic rhythm $y_{1:T}$9. A plausible implication is that the term “chord-frame parameterization” in this literature refers less to a specific estimator than to a change of state space: the primary temporal object is the chord-event sequence, while the frame sequence is induced by event durations.

2. Representation, vocabulary, and reconstruction in the music-domain formulation

The frame-level and chord-sequence-level representations share the same vocabulary. The system uses a 25-class chord alphabet focused on major/minor triads plus no-chord: $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$0 Following Cho and Bello (2014), chords containing a minor third are mapped to a minor triad class, and all others to major. This reduced vocabulary ignores extensions, inversions, and seventh chords, but it is used consistently for frame-level recognition and chord-sequence modelling (Korzeniowski et al., 2018).

At frame level, $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$1 assigns one chord symbol per acoustic frame; the details specify an STFT hop of 4410 samples at 44.1 kHz, i.e. $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$2. At segment level, $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$3 assigns one chord symbol per contiguous region of constant chord, and no explicit durations are stored in the LLM. The mapping between the two levels is explicit: consecutive duplicates are removed to produce the compressed sequence, and the full frame sequence can be reconstructed from segment chords and frame-wise change/stay decisions (Korzeniowski et al., 2018).

The reconstruction rule is direct. Starting from some initial chord at $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$4, if $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$5, the current segment index is retained and $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$6; if $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$7, the segment index increments and the new frame label becomes the next segment chord. Thus the parameterization of frame chords is hierarchically determined by segment-level chords and explicit durations (Korzeniowski et al., 2018).

The same idea appears in an HMM realization with explicit duration states. In that formulation, a chord event sequence is combined with a duration distribution so that frame index $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$8 maps to a state $$\hat{y}_{1:T} = \arg\max_{y_{1:T}} P(y_{1:T}\mid x_{1:T}).$$9, and the chord label at frame $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$0 is the current chord, i.e. the first element of the context state. This yields a frame-synchronous representation in which the frame label is a projection of a richer segmental state (Korzeniowski et al., 2018).

A common misconception is that this is merely a smoothing strategy. The description in the cited work is more specific: the model separates the decision to change from the choice of the next chord, and assigns the LLM only to the compressed chord sequence. That distinction is the defining structural feature of the parameterization (Korzeniowski et al., 2018).

3. LLMs, duration models, and decoding

The harmonic LLM operates on compressed chord sequences: $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$1 One instantiation uses GRU-based recurrent neural networks. Each chord symbol is embedded in $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$2, the hidden state follows the standard RNN recursion, and a softmax output predicts the next chord symbol. The reported comparisons include GRU-512, GRU-32, and 2-gram and 4-gram models; average log-probability of the correct next chord demonstrates that GRUs assign much higher probability to chord sequences than n-grams (Korzeniowski et al., 2018).

The LLM uses exactly the same 25-class vocabulary as frame-level labels. Training includes key-shift augmentation so that the RNN learns key-invariant harmonic patterns. At a chord-change frame, the chosen segment-level chord from $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$3 directly becomes the new frame label. This direct mapping is a consequence of using a shared vocabulary across levels (Korzeniowski et al., 2018).

The duration model is defined on a binary alphabet $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$4. In practice, it is simplified to $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$5, so that it becomes a sequence model over change/stay indicators and captures harmonic rhythm. One instantiation uses a one-layer GRU with 256 units, trained on 200-step excerpts; Fig. 5 is described as showing peaks of $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$6 at periodic times matching the harmonic rhythm. The comparison baselines are implicit duration models derived from HMM structures: the exponential distribution from a 1-state-per-chord HMM and the negative binomial distribution from a multi-state left-to-right HMM (Korzeniowski et al., 2018).

In the higher-order harmonic language modelling formulation, the explicit duration model is negative binomial: $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$7 where $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$8 is the number of failures and $$P(y_{1:T}\mid x_{1:T}) \propto \prod_{t=1}^T \frac{1}{P(y_t)}\, P_A(y_t\mid x_t)\, P_T(y_t\mid y_{1:t-1}),$$9 is the probability of moving to the next state. The same duration distribution is used for all chords, and $P_A$0 and $P_A$1 are learned by maximum likelihood on chord duration histograms. Higher-order language modelling is implemented as an $P_A$2-gram on the compressed sequence with Lidstone smoothing, and higher-order dependencies are folded into the HMM state space (Korzeniowski et al., 2018).

Because RNN language and duration models depend on all past inputs, exact dynamic-programming-style Viterbi decoding is not tractable in that setting. The reported solution is hashed beam search with beam size $P_A$3, hash length $P_A$4, and per-hash cap $P_A$5. Hashing keeps the beam diverse over compressed chord sequences rather than precise boundary locations. By contrast, for n-gram LLMs and negative-binomial durations, exact dynamic programming remains possible because the temporal model is Markovian with finite memory (Korzeniowski et al., 2018).

This separation of sequence and duration is also the point at which chord-frame parameterization departs most sharply from a first-order HMM. A plausible implication is that the representation is designed to expose two statistically different regularities—harmonic progression and harmonic rhythm—without forcing a single temporal process to absorb both.

4. Empirical consequences in frame-level chord recognition

Evaluation is still performed at frame level. The reported metrics include Weighted Chord Symbol Recall (WCSR),

$P_A$6

where $P_A$7 is the total duration of correctly recognized major/minor chord segments and $P_A$8 is the total annotated major/minor duration, along with root accuracy and segmentation quality (Korzeniowski et al., 2018).

A quantitative comparison reports that a baseline temporal model using a 2-gram LLM with negative binomial duration achieves Root $P_A$9, Maj/min WCSR $P_T$0, and Segmentation $P_T$1, whereas the best model using GRU-512 language and GRU-256 duration achieves Root $P_T$2, Maj/min WCSR $P_T$3, and Segmentation $P_T$4. The improvements are reported as consistent and statistically significant with $P_T$5 (Korzeniowski et al., 2018).

A complementary study using exact decoding compares temporal-model configurations on a compound dataset totalling $P_T$6 hours of audio. With $P_T$7 and uniform smoothing $P_T$8, WCSR is reported as $P_T$9 for no temporal model, $P(y_t)$0 for duration-only, $P(y_t)$1 for 2-gram, $P(y_t)$2 for 3-gram, $P(y_t)$3 for 4-gram, and $P(y_t)$4 for 5-gram. The main observations are that duration modelling alone provides most of the improvement over the baseline, and adding $P(y_t)$5-gram LLMs yields small but consistent additional gains (Korzeniowski et al., 2018).

The acoustic model is a VGG-like or VGG-style CNN operating on 1.5 s spectrogram patches and outputting 25-class softmax probabilities. To prevent over-confident posteriors, the work reports label smoothing during training and temperature-scaled softmax at inference, with $P(y_t)$6 in one configuration and $P(y_t)$7 explored in another. One study states that model overconfidence plays only a minor role when exact decoding is possible, although target smoothing still improves the acoustic model (Korzeniowski et al., 2018).

Qualitatively, GRU-based duration models are described as producing change probabilities that peak periodically in time, aligning with the piece’s harmonic rhythm and resulting in more regular, musically plausible segment lengths. Static models produce monotonically increasing probability of change as the current chord gets older, with no adaptation to local rhythmic structure. In frame-level output this manifests as fewer spurious short chords and more regular durations aligned with the learned harmonic rhythm (Korzeniowski et al., 2018).

5. Chord-frame parameterization in digital geometry

In digital geometry, chord-frame parameterization refers to the idea that a circular arc in the digital plane is represented, detected, and parameterized using geometric information anchored on one or more chords and their associated sagittae. The chord provides baseline, orientation, and span, while the sagitta provides radius and center information. The restricted Hough transform is then used for local refinement in a small parameter neighborhood (Bera et al., 2014).

For a Euclidean circle with radius $P(y_t)$8, chord length $P(y_t)$9, and sagitta length $C(\cdot)$0, the classical sagitta formula is

$C(\cdot)$1

The chord property states that for a fixed chord $C(\cdot)$2 in a circle, any point on either arc $C(\cdot)$3 sees the chord under a constant inscribed angle. The paper studies how these properties behave after discretization on $C(\cdot)$4, where exact Euclidean identities become approximate (Bera et al., 2014).

The main analytic statements are two discrete-domain bounds. The first is an angular deviation bound for the chord property: $C(\cdot)$5 where $C(\cdot)$6 is a digital point on the digital arc and $C(\cdot)$7 its real counterpart. This shows that points near the endpoints of the chord have large potential angular deviation, whereas points near the middle have much smaller deviation. The second is a relative radius error bound for the discrete sagitta estimate

$C(\cdot)$8

with

$C(\cdot)$9

The interpretation given is that relative error decreases as the sagitta grows, so very short arcs lead to unreliable radius estimation (Bera et al., 2014).

The algorithmic realization, CSA, proceeds by chord-based detection of circular segments, sagitta-based estimation of center and radius, frame merging, and restricted Hough transform refinement. For each non-straight segment, a central region is defined as the middle third, and the segment is accepted as circular if

$C(C,C,F,F,G)=(C,F,G)$0

with $C(C,C,F,F,G)=(C,F,G)$1. If the test fails, the segment is split recursively until it is too short, straight, or satisfies the chord property. Here the chord acts as a frame in a literal geometric sense: the endpoints define the frame, and the central region is the window in which inscribed-angle constancy is tested (Bera et al., 2014).

Once an arc is accepted, the chord midpoint and an appropriate central point define a discrete sagitta, from which radius and center are estimated. Arcs are merged if the combined segment still satisfies the digital chord property, because a larger frame gives a larger sagitta and lower radius error. The restricted Hough transform then searches only the box

$C(C,C,F,F,G)=(C,F,G)$2

A plausible implication is that “frame” in this usage denotes a validated local coordinate scaffold rather than a temporal index (Bera et al., 2014).

6. Chord-frame parameterization in the framed Kontsevich integral

In the framed Kontsevich integral, chord-frame parameterization takes the form of a matrix representation of chords on a tangle. A single chord is encoded as a symmetric $C(C,C,F,F,G)=(C,F,G)$3 matrix called a page, and a chord diagram of degree $C(C,C,F,F,G)=(C,F,G)$4 is encoded as an ordered product of pages called a book. The construction is defined after putting the link in Morse position, modifying local maxima with hooks arising from the framing normalization, and slicing the diagram by vertical lines into $C(C,C,F,F,G)=(C,F,G)$5 strips per component (Gauthier, 2010).

If a chord joins component $C(C,C,F,F,G)=(C,F,G)$6 in strip $C(C,C,F,F,G)=(C,F,G)$7 to component $C(C,C,F,F,G)=(C,F,G)$8 in strip $C(C,C,F,F,G)=(C,F,G)$9, it is represented by a matrix $\bar y_{1:K}=C(y_{1:T})$0 whose only nonzero entries are the $\bar y_{1:K}=C(y_{1:T})$1 entry in block $\bar y_{1:K}=C(y_{1:T})$2 and the symmetric $\bar y_{1:K}=C(y_{1:T})$3 entry in block $\bar y_{1:K}=C(y_{1:T})$4. A tangle chord diagram with chords indexed from bottom to top is then represented as

$\bar y_{1:K}=C(y_{1:T})$5

The completed framed Kontsevich integral becomes a sum of such books with coefficients: $\bar y_{1:K}=C(y_{1:T})$6 This is the basic chord parameterization: a chord is specified by component indices and strip indices, while the order of pages encodes vertical order in the tangle (Gauthier, 2010).

Framing enters through the removal of the framing independence relation, the local normalization near extrema, and the insertion of hooks that alter the strip decomposition. Because each local extremum contributes additional strips, the matrix representation carries framing information through block sizes, strip positions, and the presence of isolated chords near extrema. The paper states that orientation changes, Reidemeister moves, and band sum moves are implemented by explicit matrix congruences on pages and therefore on books (Gauthier, 2010).

For a handle slide of component $\bar y_{1:K}=C(y_{1:T})$7 over $\bar y_{1:K}=C(y_{1:T})$8, the induced map is

$\bar y_{1:K}=C(y_{1:T})$9

where $y_{1:T}$00 has ones on the diagonal and the $y_{1:T}$01 block equal to $y_{1:T}$02. Orientation change on component $y_{1:T}$03 is similarly represented by

$y_{1:T}$04

where $y_{1:T}$05 is the identity except for block $y_{1:T}$06, which is $y_{1:T}$07. Framed Reidemeister I moves are also encoded by congruences using matrices that insert or delete strips associated with humps (Gauthier, 2010).

The most important structural statement is that, from $y_{1:T}$08 written in book notation, one can recover the full link $y_{1:T}$09 and determine the corresponding framed Kontsevich integral in the original chord-diagram algebra. This means that the book notation is a faithful coordinate system for framed links, even though the map from the book algebra to the ordinary algebra of chord diagrams is not injective globally (Gauthier, 2010).

7. Comparative perspective

The phrase “chord-frame parameterization” is therefore domain-dependent. In automatic chord recognition, it names a hierarchical probabilistic parameterization in which frame labels are produced from a segment-level chord sequence and a duration process (Korzeniowski et al., 2018). In digital geometry, it denotes a geometric parameterization in which a circular arc is described by a chord-defined frame and a sagitta-derived scale estimate (Bera et al., 2014). In the framed Kontsevich integral, it denotes a linear-algebraic parameterization in which a chord is indexed by components and strips, and framing is encoded by the strip architecture and its congruence transformations (Gauthier, 2010).

Despite the disparity of domains, the same formal theme recurs. A chord-like primitive is separated from the mechanism that places it within a larger structure: duration in the music-domain formulation, central-region and sagitta geometry in the digital-arc formulation, and strip-indexed framing data in the topological formulation. This suggests that the unifying content of the term is not the object “chord” itself, but the explicit modelling of the relationship between a chordal entity and the frame that makes its higher-order organization tractable.