The Music Perception Toolbox: Analytical Methods for Pitch and Rhythm Similarity, Consonance, Complexity, and Structure

1.1 One framework, three families of measure

The measures described below fall into three families—expectation tensors and the similarities and entropies built on them, consonance and harmonicity, and the structural features of scales and rhythms—which are usually developed in separate literatures and implemented in separate software. They share three properties, which is what allows them to be used together (Figure 1).

Figure 1

First, the domains are continuous. Pitches and time points enter as real numbers rather than as indices into a chromatic or metrical grid, so microtonal tunings, expressive timing, and spectral peaks measured from a recording are all admitted without quantization. Corpus‑ and learning‑based models of musical expectation work differently: because they typically estimate probabilities over a discrete alphabet of symbols—scale degrees, pitch classes, or categorized inter‑onset intervals—a tuning system and a metrical grid must be fixed before the analysis can begin (e.g., Pearce, 2018). Four features are exceptions, their implementations here requiring integer positions within an equal division of the period: sameness, $n$‑tuple entropy, the circular autocorrelation phase matrix, and Markov prediction.

Second, collections of unequal size can be compared directly. Because each multiset is embedded as a density over a shared domain before any comparison is made, no correspondence need be established between the elements of one collection and those of another. This is what distinguishes the approach from voice‑leading and related metrics, which require a matching between the two chords and so require the cardinalities to be equalized: choosing which elements of the larger collection to match against the smaller often has no unique answer, and, even where the choice is forced, the resulting distance violates the triangle inequality (Milne et al., 2011). Here, a three‑note chord and a seven‑note scale compare as readily as two triads, and a chord’s several dozen partials compare with another’s without any partial having to be paired with a counterpart.

Third, perceptual uncertainty is represented explicitly. A single smoothing‑width parameter $\sigma$ smears each element over a range of nearby values so that near‑coincidences count in proportion to their closeness rather than being either exact matches or complete mismatches. Every measure built on the expectation tensor inherits $\sigma$; among the structural features it appears in $n$‑tuple entropy and, as the concentration parameter of a von Mises kernel, in edge detection. Extending it to the remainder is discussed in Section 8.

Predictors drawn from different families can therefore be applied to the same stimuli within one analysis. A chord’s fit to its context and its consonance considered in isolation are different questions, and Section 5 shows that they receive close to independent answers.

Section 2 presents the expectation tensor framework, Section 3 the consonance and harmonicity measures, and Section 4 the structural features of scales and rhythms. Section 5 is an example of a worked application. Section 6 surveys prior empirical work, Section 7 describes the software design, and Section 8 discusses limitations and future directions.

2.1 Representing pitches and time points

The toolbox operates on collections of pitches or time points, represented as numbers. For pitch, the standard representation is a logarithmic function of frequency: $p=1200{\mathrm{log\hspace{0.17em}}}_2(f/f_{\text{ref}})$, giving values in cents—a unit in which the octave spans 1200 equally spaced steps, each corresponding to 1/100 of an equal‑tempered semitone. Equal frequency ratios then correspond to equal pitch distances, so a perfect fifth (ratio 3:2) is about 702 cents in any register. For rhythmic patterns, positions are typically integers representing pulse locations within a metrical cycle (e.g., onsets at pulses 0, 3, 6, 10, and 12 within a 16‑pulse cycle for the son clave rhythm), though finer representations—such as milliseconds or subdivisions of a pulse—are also supported.

A pitch collection (or time‑point collection) is a multiset of such values—a set whose elements may repeat, such as when a chord tone is doubled by two voices. Each element may carry a non‑negative weight representing its perceptual salience, amplitude, or probability of being perceived. Multisets may be specified from symbolic data (scale degrees, MIDI note numbers, theoretical pitch sets) or extracted from audio: audioPeaks returns the frequencies and amplitudes of a recording’s spectral peaks, which convertPitch—a utility spanning seven pitch and frequency scales (Hz, MIDI, cents, mel, Bark, ERB‑rate, and Greenwood)—converts to cents for use in any toolbox function.

2.2 Definitions

The construction begins from a simple perceptual premise. A listener does not register musical events exactly: a tone is heard at a pitch that could plausibly have been any of a range of nearby pitches and with an onset at a time that could have been any of a range of nearby times—albeit more probably with values close to the true one and less probably those further away. Replacing each event by a probability distribution over nearby values, and adding these distributions together, gives a function over the domain, of pitch or of time, whose value at any point is the expected number of events perceived there (Milne et al., 2011). It is expected in the ordinary statistical sense: a value of $0.6$ means a listener perceives an event there in $60\%$ of hearings. Applied to pairs of events, it gives the expected number of ordered pairs at each combination of two positions and so on upwards. These functions are the expectation tensors.

Formally, an expectation tensor is a continuous density over ordered $r$‑tuples—ordered sequences of $r$ elements (an $r$‑ad)—drawn from a weighted multiset, each smoothed by a Gaussian kernel. ‘Tensor’ refers to the discretized form being a rank‑$r$ array indexed by $r$ copies of the pitch or time space—a multidimensional array, not a multilinear map. The term monad denotes $r=1$ (individual elements), dyad denotes $r=2$ (ordered pairs), triad denotes $r=3$ (ordered triples), and so on. Given a multiset $\mathbf{p}=(p_1,\dots ,p_n)$ with weights $\mathbf{w}=(w_1,\dots ,w_n)$, the $r$‑ad expectation tensor density at a query point $\mathbf{x}\in {\mathbb{R}}^d$ is

where the sum ranges over all $J=P(n,r)=r!\left(\genfrac{}{}{0.0pt}{}{n}{r}\right)$ ordered $r$‑tuples drawn without replacement from the multiset; ${\mathbf{c}}_j$ is the $j$‑th tuple (an ordered sequence of $r$ pitch values, treated as a point in ${\mathbb{R}}^d$); $W_j={\prod }_{m=1}^r{w}_{j_m}$ is the product of the constituent weights; $\sigma >0$ models perceptual uncertainty (for pitch, $\sigma$ can be chosen to approximate a just‑noticeable difference; see Milne et al., 2011, Appendix A); and $Q$ is a quadratic form—a function that computes a squared distance.

Two binary flags, $\mathrm{isPer}$ and $\mathrm{isRel}$, together determine the mode, which fixes both the quadratic form $Q$ and the dimensionality $d$: $d=r$ in the absolute case and $d=r-1$ in the relative case.

Non‑periodic/Periodic ($\mathrm{isPer}=\mathrm{false}/\mathrm{true}$). When true, the domain is a circle of circumference $P$ (the period), with distances wrapped to $[-P/2,P/2)$. For pitch, setting $P=1200$ implements pitch‑class equivalence; for rhythm, $P$ is the cycle length.

Absolute/Relative ($\mathrm{isRel}=\mathrm{false}/\mathrm{true}$). When false (absolute), the density depends on the actual positions of the elements: $Q(\mathit{\delta })=\Vert \mathit{\delta }{\Vert }^2$ (the standard Euclidean squared distance, where $\mathit{\delta }=\mathbf{x}-{\mathbf{c}}_j$ is the difference between the query point and a tuple) and the effective dimensionality equals $r$. When true (relative), the density is invariant under transposition—shifting all values by a constant leaves the density unchanged. This is achieved by a quadratic form $Q(\mathit{\delta })={\sum }_i{\delta }_i^2-{({\sum }_i{\delta }_i)}^2/r$, which measures distance in the subspace orthogonal to the all‑ones direction—the direction along which all elements shift together—so projecting out the mean and retaining only the intervals between elements. The effective dimensionality reduces from $r$ to $r-1$, and each $r$‑tuple is represented by its $(r-1)$‑dimensional vector of intervals from one of its own elements (by convention, the first): $(p_{j_2}-p_{j_1},\dots ,p_{j_r}-p_{j_1})$.

The two flags yield four mode combinations, each suited to a different question. Absolute monads ($r=1$) give a density over individual (spectral) pitches, or pitch classes when periodic, whose entropy measures (spectral) pitch complexity (Section 2.5) and whose cosine similarity quantifies perceptual similarity (Section 2.4). Relative dyads ($r=2$) give a density over pairwise intervals, so comparing two such densities compares interval content—Figure 2 uses this to find which generator‑chain tunings best approximate a reference chord (Milne et al., 2011, Examples 6.3–6.5). Relative triads ($r=3$) give a density over interval pairs. Tensor harmonicity builds such a density from a single harmonic series and evaluates it at a triad’s own pair of intervals, thus measuring the density of that pattern within the series (Section 3.3).

Figure 2

The expectation tensor framework applies equally to rhythmic patterns: the same operations, computed over time‑point multisets without spectral enrichment, yield measures of rhythmic similarity and complexity, and, evaluated at a queried pattern of onset intervals, how strongly that pattern is present in the rhythm.

Figure 3 illustrates these mode combinations by displaying the expectation tensor densities for a diatonic scale (or bembe‑12 rhythm; Undy et al., 2026) across different values of $r$: the top two rows show one‑dimensional densities (line plots) for absolute monads ($r=1$) and relative dyads ($r=2$), while the bottom two rows show two‑dimensional densities (heatmaps) for absolute dyads ($r=2$) and relative triads ($r=3$). Each row contrasts the non‑periodic (left) and periodic (right) cases.

Figure 3

2.3 Spectral enrichment

A notated pitch (e.g., middle C) is an abstraction: sounded on a real instrument, it produces not a pure tone but partials at approximately integer multiples of the fundamental. For symbolic input (MIDI note numbers, scale degrees, theoretical pitch sets), addSpectra appends synthetic partials to each pitch, expanding the multiset and weighting it by a spectral envelope. This is unnecessary for spectral peaks extracted from audio via audioPeaks, which already represent the full spectrum.

Five spectral modes are available—harmonic, stretched, frequency‑linear, stiff‑string, and custom—with power‑law ($w_k=1/k^{\rho }$) or geometric ($w_k={\tau }^{k-1}$) weight decay, where $k$ is the harmonic number.

2.4 The pitch similarity family: (S)P(C)S

The cosine similarity of two expectation tensor densities—a standard measure of the alignment of two non‑negative functions, ranging from 0 (no overlap) to 1 (identical shape)—quantifies how similar two weighted multisets are:

where the inner product $⟨f_X,f_Y⟩={\int }_{{\mathbb{R}}^d}{f}_X(\mathbf{x})f_Y(\mathbf{x})\mathrm{d}\mathbf{x}$ is the standard $L^2$ inner product: the integral, over all $\mathbf{x}\in {\mathbb{R}}^d$, of the pointwise product of the two densities. Section 2.6 shows that this integral has a closed form, so that Equation 2 can be evaluated exactly, without a grid. For expectation tensors, it also has a concrete meaning. Were perception perfectly precise, it would be the number of ordered $r$‑ads the two collections have in common, as a proportion of the most they could have had in common—the geometric mean of the numbers each shares with itself—and, with weights read as probabilities of being perceived, these become expected counts (Milne et al., 2011). Smoothing relaxes exact coincidence into graded near‑coincidence, which is the point of it: replacing one pitch of a major triad by a pitch 10 cents away gives a count‑based value of $2/3$ but a smoothed value of $0.93$ at $\sigma =10$ cents. At $r=1$ with spectral enrichment, the count is the proportion of the two sounds’ partials that coincide in pitch (Milne et al., 2016).

In the pitch domain, spectral enrichment and the periodic flag yield a family of four named measures, each Equation 2 applied to a pair of densities of the form of Equation 1, differing only in two binary choices. SPCS is the most familiar: $r=1$, $\mathrm{isRel}=\mathrm{false}$, $\mathrm{isPer}=\mathrm{true}$ with $P=1200$, applied to multisets to which harmonic partials have been added (Milne et al., 2015). Setting $\mathrm{isPer}=\mathrm{false}$ gives SPS, which compares full spectra on an unbounded pitch line (Dean et al., 2019); omitting the partials gives pitch class similarity (PCS); and doing both gives pitch similarity (PS), which compares fundamentals alone (Milne et al., 2011). Enrichment is unnecessary when the input already comes from audio via audioPeaks. SPCS is the most commonly used of the four, having consistently outperformed SPS at modelling perceived tonal fit, which is attributed to the perceptual importance of octave equivalence (Milne et al., 2015).

Where no additional information about the experimental stimuli is available, suitable values are 12–24 harmonics, power‑law rolloff $\rho \approx 1$, and $\sigma$ between about 6 cents (Milne et al., 2015; Milne and Holland, 2016) and 10 cents (Hearne et al., 2026). (The figures and analyses in this article use $\sigma =10$ cents, 16 harmonics, and $\rho =1$ unless stated otherwise.) When experimental conditions are known (e.g., a specific instrumental spectrum), parameters can be matched to those conditions; for example, Hearne et al. (2026) fit $\rho$ to the spectral rolloff of their stimuli. Figure 4 illustrates SPCS profiles for two contrasting scales; Figure 5 shows SPCS for all triads containing a perfect fifth, relative to major and minor reference triads, illustrating the differential similarity structure of the two modes.

Figure 4

Figure 5

2.5 Entropy of expectation tensors

In addition to cosine similarity, the toolbox computes the Shannon entropy of expectation tensor densities via entropyExpTens. The entropy of a density quantifies its disorder: a density concentrated at a few sharp peaks has low entropy, while a diffuse one has high entropy. This gives a complexity measure applicable to any tensor mode and any domain.

The entropy of a relative periodic dyad tensor ($r=2$, $\mathrm{isRel}=\mathrm{true}$, $\mathrm{isPer}=\mathrm{true}$), for example, quantifies the complexity of a pitch set’s interval content: intervals clustering at a few values, as in an equal‑tempered scale, give low entropy; many distinct intervals give high. Spectral entropy (Section 3.1) is one application of this idea—the entropy of a spectrally enriched absolute non‑periodic monad tensor—and higher‑order tensors capture correspondingly higher‑order regularities.

Because the differential entropy of a Gaussian mixture has no known closed form—the logarithm of a sum of Gaussians does not simplify—entropy is computed on a grid, unlike cosine similarity.

2.6 The analytical inner product

Evaluating Equation 2 requires the inner product of two densities of the form of Equation 1. Because a product of Gaussians is itself Gaussian, that integral has a closed form:

where $\mathcal{P}$ denotes the $r!\left(\genfrac{}{}{0.0pt}{}{n}{r}\right)$ ordered $r$‑tuples of a collection and $\mathcal{C}$ its $\left(\genfrac{}{}{0.0pt}{}{n}{r}\right)$ unordered $r$‑subsets. Because $Q$ is unchanged by permuting the elements within both tuples together, only one side need be ordered; taking combinations on the other is cheaper by a factor of $r!$. The omitted proportionality constant depends only on $\sigma$ and $r$, and is common to all three inner products in Equation 2, so it cancels and the cosine similarity follows directly—a finite double sum over tuples, with no discretization, grid, or resolution parameter and exact to floating‑point precision. This result is due, in its original form, to David Bulger (personal communication, 2016) and was implemented for the cosine similarity in version 1 of the toolbox. Its derivation, for all four modes and including the additional wrapping the relative periodic case requires, is given in the Supplementary Material.

Version 1 instead built the densities themselves on a discrete grid of rounded pitches, an approach that imposes a resolution parameter, rounds every input, and grows with the size of the domain raised to the tensor order: an absolute triad tensor over the octave occupies some $1200^3\approx 1.7\times 10^9$ elements and only $r\le 4$ was implemented. The analytical form has none of these costs and scales with the number of elements in the collection rather than the size of the domain.

3.1 Spectral entropy

Spectral entropy (Milne et al., 2017; Smit et al., 2019) measures the Shannon entropy of a chord’s composite spectrum after Gaussian smoothing. As noted in Section 2.5, this is one specific application of the general entropy‑of‑tensor framework: it is the entropy of an absolute non‑periodic monad expectation tensor density (with spectral enrichment applied). The underlying intuition is one of spectral concentration. Each tone of the chord contributes a series of partials; where partials from different tones land on the same pitch they reinforce one another, and the composite spectrum consists of a few tall peaks separated by empty regions. Where they land on many distinct pitches the spectrum is spread thinly and evenly. Entropy measures exactly this contrast, being low for the concentrated case and high for the diffuse one. It thus approximates the aggregation of all pairwise spectral pitch similarities within a chord, and so models the mutual fit of its tones (Milne, 2024; Milne et al., 2017). Unlike SPCS, which compares two pitch sets, spectral entropy is a property of a single (potentially composite) set. It responds mainly to spacing (0.70) and much less to type (0.17) or inversion (0.01); because the spectrum is referred to its own lowest pitch, register has no effect at all.

3.2 Template harmonicity

Template harmonicity (templateHarmonicity; Milne, 2013; Milne et al., 2016) cross‑correlates a chord’s composite spectrum with a harmonic template. The procedure is: (1) construct the chord’s absolute non‑periodic monad density (its composite spectrum), (2) construct a single harmonic template—the absolute non‑periodic monad density of one complex tone at 0 cents with harmonics, (3) evaluate both densities on a fine grid, and (4) normalize and cross‑correlate.

Informally, the procedure slides a single harmonic series along the pitch line and asks, at each position, how well its partials line up with the chord’s. A chord that is close to a segment of one harmonic series will align well at the position of that series’ fundamental and poorly elsewhere, giving a profile with one dominant peak; a chord that is not will align moderately at several positions. Two summary statistics are extracted: $h_{\text{Max}}$, the maximum of the normalized cross‑correlation (Milne, 2013), which is the cosine similarity at the best‑matching transposition, and $h_{\text{Entropy}}$, the entropy of the cross‑correlation profile (Harrison and Pearce, 2020), which captures whether the chord has one clear virtual fundamental or several competing ones. Both are transposition‑invariant—transposing the chord merely shifts the cross‑correlation profile, leaving its maximum and its entropy unchanged—and so unaffected by register. Otherwise, they diverge: $h_{\text{Max}}$ is the most type‑sensitive of the five measures (0.67), whereas $h_{\text{Entropy}}$ responds to type (0.52) and spacing (0.34) alike. These last two shares are the only ones in this section much affected by the number of harmonics in the template: with 24 rather than 16, they become 0.31 and 0.57, reversing their order.

The related function virtualPitches returns the full cross‑correlation profile, whose peaks indicate candidate fundamentals—the pitches the auditory system infers from a complex of partials (Parncutt, 1988). The template’s spectral rolloff alters the number and prominence of those peaks.

3.3 Tensor harmonicity

Tensor harmonicity (tensorHarmonicity; Smit et al., 2019) measures how probable a chord’s specific ordered intervals are within a single harmonic series.

In the counting terms of Section 2.4, tensor harmonicity is the expected number of ordered $K$‑ads of partials, drawn from a single harmonic series, whose interval pattern is the chord’s own: where the cosine similarity counts the $r$‑ads that two collections share, this counts the $r$‑ads of one collection matching a single query pattern, with $\sigma$ again letting near‑coincidences contribute. Some patterns recur throughout the series and others hardly do at all—the major triad’s intervals, given by 4:5:6 and its multiples, are common; the minor triad’s, 10:12:15, far less so. A chord need not be justly tuned to score well, however. Figure 6a magnifies the landscape around the triads of common practice: the equal‑tempered major triad in root position lies 13.7 cents from the just‑intonation 4:5:6 triad in one interval and 2.0 cents in the other, displacements comparable to $\sigma$ itself, and so sits within that peak rather than outside it. Because the pattern is measured from the lowest tone upwards, the answer depends on how the chord is voiced and not only on which pitch classes it contains. Like $h_{\text{Max}}$, it is driven more by type (0.26) than by spacing (0.17), and, the density being relative, register does not move it at all. It also leaves most of its variance unexplained by any of the four properties singly (0.54), its responses to inversion and spacing depending on the chord type. Figure 6 shows this dependence directly, and it is what produces the distinctive topography of Figure 7c.

Figure 6

Figure 7

Concretely: a harmonic template is built with the template pitch (0 cents) duplicated $K$ times before harmonics are added, where $K$ is the chord cardinality; the relative $K$‑ad expectation tensor of that template is constructed; and it is evaluated at the chord’s ordered interval vector, i.e., the $K-1$ intervals from its lowest tone to each of the others. The duplication is what allows unisons to count as consonant: without it, each position in a tuple must be filled by a different partial, so no partial can contribute to more than one position. The ordered vector distinguishes chords of identical unordered interval content—the major triad $(0,400,700)$ and the minor $(0,300,700)$ both contain the intervals $\{300,400,700\}$, but their ordered vectors are $(400,700)$ and $(300,700)$. Because the chord enters only as a query point, its own perceptual uncertainty is not otherwise represented. Folding it in amounts to integrating the density against a second Gaussian, which is exactly equivalent to widening the template’s kernel: for uncertainties $\sigma$ on the template and $\tau$ on the query, the width becomes $\sqrt{{\sigma }^2+{\tau }^2}$, or $\sigma \sqrt{2}$ when the two are equal.

3.4 Roughness

Sensory roughness (roughness) computes the beating‑based dissonance of close partial pairs using Sethares’s (1993) parameterization of Plomp and Levelt’s (1965) empirical dissonance curve.

The implementation extends the original model in two ways. First, pairwise roughnesses are combined via a configurable $p$‑norm (default $p=1$, a simple sum). Kameoka and Kuriyagawa (1969) and Hutchinson and Knopoff (1978) used different aggregation methods—the former a compressive power law, the latter an amplitude‑weighted average—and both approaches have known limitations (Mashinter, 2006; Parncutt, 2006). The $p$‑norm allows these and other aggregation methods to be explored: higher $p$ gives greater weight to the roughest pairs. Second, optional averaging over the $\left(\genfrac{}{}{0.0pt}{}{K}{2}\right)$ partial pairs gives the expected roughness of a randomly chosen pair, making values comparable across chords with different numbers of partials.

Unlike the other measures, roughness is register‑dependent and so requires frequencies in Hz: the same interval is rougher in lower registers where partials are more closely spaced on the basilar membrane. It is alone in responding to register (0.33), and responds far more to spacing (0.48) than to type (0.09)—though the type effect is real, a semitone cluster being markedly rougher than any triad at the same spacing and register.

Figure 7 compares all five consonance‑related measures across a two‑octave space of triads.

4.1 Fourier‑based measures

The discrete Fourier transform (DFT) of a set of weighted points on a circle (dftCircular) is computed directly from the point positions, in two steps. Each of the $K$ points $p_k$, carrying weight $w_k$ on a circle of circumference $P$, is first mapped to the complex number $w_k\exp (2\pi i{p}_k/P)$, whose argument is the point’s angular position and whose modulus is its weight. The points are then sorted by position and an ordinary length‑$K$ DFT is taken over this sequence of complex numbers, giving $K$ coefficients $F_0,\dots ,F_{K-1}$.

The low‑order coefficients have direct interpretations. The zeroth, $F_0$, is the sum of these complex numbers, so that $|F_0|/\sum w_k$ locates the set’s centre of gravity: greatest when the points cluster to one side of the circle, and zero when that centre of gravity coincides with the circle’s own. The first, $F_1$, is large when successive points are separated by a constant angle, since a constant separation of $P/K$ is exactly the rotation that the first DFT basis vector removes. Balance and evenness, defined below, report these two quantities. Weights have a clear meaning only in the zeroth coefficient, where they scale each point’s contribution to the centre of gravity. Coefficients beyond the first have less obvious interpretations but may nonetheless be perceptually relevant; their potential role is explored in Milne and Herff (2020) and its Supplementary Material.

This construction is distinct from the more common approach in mathematical music theory, where sets are represented as binary indicator vectors—vectors of 0s and 1s over a discretized grid—and the DFT is computed from this indicator (e.g., Amiot, 2016; Quinn, 2007), giving one coefficient per grid position. The direct approach accommodates non‑integer positions and weighted events without discretization.

Balance (balanceCircular; Milne et al., 2017) measures how close a set’s centre of gravity is to the circle’s centre and is defined as $1-|F_0|/\sum w_k$. It ranges from 0 (all weight at a single point: the centre of gravity is maximally far from the centre) to 1 (the centre of gravity lies exactly at the centre).

For example, in a 12‑step chromatic universe: the augmented triad $\{0,4,8\}$ has balance 1.000, the major triad $\{0,4,7\}$ has balance 0.827, and the cluster $\{0,1,2\}$ has balance 0.089 (see Figure 8).

Figure 8

A perfectly balanced set—one with balance exactly 1—need not be evenly spaced or regular. Any union of isochronous cycles is perfectly balanced—a three‑against‑four cross‑rhythm, for instance—and stays so when the cycles are rotated independently of one another so that no two events coincide. Others arise only when a negatively weighted cycle is added whose every event coincides with, and so annihilates, an event of the positive ones: nothing negative survives, but the pattern left behind could not have been assembled from positive cycles alone. Such patterns are neither regular nor decomposable into smaller balanced parts and are distinct from the familiar well‑formed (moment‑of‑symmetry) and maximally even classes (Milne et al., 2017).

Evenness (evennessCircular; Milne et al., 2017) is the normalized magnitude of $|F_1|$, computed with uniform weights. It ranges from 0 to 1, equalling 1 only for equally spaced sets.

For example, the whole‑tone scale $\{0,2,4,6,8,10\}$ has evenness 1.000, the diatonic scale $\{0,2,4,5,7,9,11\}$ has evenness 0.989, and a maximally clustered set like $\{0,1,2,3,4,5\}$ has evenness 0.644.

4.2 Interval‑structure measures

The following functions characterize the interval structure of a set of points on a circle. Coherence accepts positions at any resolution (integer or float); the implementations of sameness and $n$‑tuple entropy require integer positions within an equal division of the period (e.g., the 12 pitch classes of 12‑tone equal temperament or the 16 pulses of a 4/4 bar subdivided into semiquavers). More detailed descriptions of these features, including formal definitions, are available in Milne et al. (2023a).

Coherence (coherence; Balzano, 1982; Carey, 2002; Rothenberg, 1977) measures how consistently the ordering of intervals by generic span—the number of scale steps between two events—matches their ordering by specific size—the actual distance between them. A scale is coherent when a larger generic span always implies a larger or equal specific size. Failures are of two kinds: an ambiguity, where the two intervals have the same specific size, and a contradiction, where the larger generic span is strictly the smaller interval. The diatonic scale $\{0,2,4,5,7,9,11\}$ in a 12‑step universe has coherence quotient 0.993, its single failure an ambiguity: the augmented fourth F–B spans three generic steps and the diminished fifth B–F spans four, yet both measure six semitones. The blues scale $\{0,3,5,6,7,10\}$ scores 0.600, and its failures include contradictions: F–G spans two generic steps but measures two semitones, fewer than the three of C–E$♭$, which spans one.

Sameness (sameness; Carey, 2002; Carey, 2007) counts a set’s ambiguities alone, reporting one minus that count as a proportion of the most a set of its size could have. The blues scale therefore scores 0.787 where its coherence was 0.600, the gap being its 10 contradictions; the diatonic scale, having none, scores 0.992 against 0.993.

$n$‑tuple entropy (nTupleEntropy; Milne and Dean, 2016) computes the Shannon entropy of the distribution of consecutive step‑size sequences of length $n$. When $n=1$, this is the step‑size (IOI) entropy; 1‑tuple entropy was empirically validated as a predictor of rhythmic recognizability and liking in Milne and Herff (2020) and applied theoretically as a measure of scale‑step complexity in Milne (2024). Higher values indicate less regularity among the sequences of adjacent steps. For example (1‑tuple entropy), the diatonic scale $\{0,2,4,5,7,9,11\}$ has 1‑tuple entropy of 0.863 bits (two step sizes: 1 and 2, in a 5:2 ratio), the chromatic scale has 1‑tuple entropy of 0.000 bits (one step size), and the son clave $\{0,3,6,10,12\}$ in 16 has 1‑tuple entropy of 1.522 bits (three step sizes: 2, 3, and 4). At $n=2$, the distribution is over adjacent pairs of steps: the diatonic scale’s seven pairs are $(2,2)$ three times and $(2,1)$ and $(1,2)$ twice each, giving 1.557 bits.

Well‑formed patterns minimize $n$‑tuple entropy over all other arrangements of the same steps. Gaussian smoothing is optionally available, incorporating perceptual uncertainty as elsewhere in the toolbox.

4.3 Position‑level features

The following functions return values at every position around the circle. Edges, projected centroid, and mean offset accept event positions at any resolution and evaluate at arbitrary query points; the implementations of the circular autocorrelation phase matrix and Markov prediction require integer event positions. More detailed descriptions are available in Milne et al. (2023a). Figure 9 illustrates all five for the son clave rhythm.

Figure 9

Edge detection (edges; Milne et al., 2023a) identifies sharp transitions between event‑dense and event‑sparse regions of the cycle. It uses the analytic first derivative of the von Mises distribution—the circular analogue of the Gaussian—namely, $f^{\prime }(\theta )\propto -\kappa \sin \theta \exp (\kappa \cos \theta )$, whose concentration parameter $\kappa$ sets the spatial scale: larger $\kappa$ detects finer transitions.

In the son clave $\{0,3,6,10,12\}$ in a 16‑step cycle, large positive values appear at positions 9–10, where the gap at 7–9 gives way to events at 10 and 12, and large negative values at positions 12–13, where events give way to the next gap.

Projected centroid (projCentroid; Milne et al., 2023a) projects the circular centre of gravity onto each angular position, giving a position‑level generalization of the set‑level balance measure.

Mean offset (meanOffset; Milne et al., 2023a) computes, for each position, the net upward arc to all events. For pitch classes, this formalizes and generalizes Huron’s (2008) average pitch height, returning a value at every position and so capturing how ‘bright’ or ‘dark’ a mode sounds from any chromatic vantage point. The related concept of mode height is used by Hearne (2020) and Tymoczko (2023).

For example, in the diatonic scale: the Lydian mode (starting on pitch class 5) has the highest mean offset (brightest), while the Locrian mode (starting on pitch class 11) has the lowest (darkest).

Circular autocorrelation phase matrix (circApm; Milne et al., 2023a) adapts Eck’s (2006) autocorrelation phase matrix (APM) to the periodic case. Standard autocorrelation reveals which periodicities are present but discards phase information; the APM preserves it by storing, for each lag $k$, the autocorrelation energy indexed by phase position so that peaks indicate events recurring at specific intervals from specific phases.

For example, summing across lags assigns each position of the son clave a weight reflecting how strongly it participates in the pattern’s periodic structure. Positions 0, 6, 10, and 12 score highest, sitting on the clave’s dominant even‑lag periodicities, while position 3 scores roughly half that despite carrying an event; non‑event positions score above zero where they align with the underlying structure.

Markov prediction (markovS; Milne et al., 2023a; originally by David Bulger) predicts the event weight at each position from the $S$ positions preceding it. Positions whose preceding patterns of events and non‑events are identical form a group, and each is assigned that group’s mean event weight. The result is the optimal $S$‑th order Markov predictor for the cycle; with binary weights, it is the proportion of a group’s positions that carry an event, and so a probability.

For example, in the son clave with $S=3$, positions 0 and 10 are each preceded by three consecutive non‑events, and, as both carry an event, each takes a predicted value of 1. Positions 2, 5, 8, 12, and 14 are each preceded by non‑event, event, non‑event; only one of the five, position 12, carries an event, so each takes a predicted value of 0.2.

5.1 Stimuli

The context is a four‑part I–IV–V⁷ progression in C major, voiced C3–G3–E4–C5, F2–A3–F4–C5, G2–D4–F4–B4. The context is presented to the measures as the union of the three chords, a 12‑element multiset.

The probes are triads on four scale degrees—the tonic and submediant, which are diatonic to the context, and $♯$IV and $♭$II, which are not—in three qualities (major, minor, diminished) and three inversions, giving 36 probes. Each is voiced in four parts, and the voice leading throughout—within the context, and from the V⁷ to each probe—follows conventional rules, so spacing varies across the probe set rather than being held fixed.

5.2 Predictors

A probe’s contextual fit—how well it accords with the progression—is quantified by the four measures of Section 2.4: SPCS, SPS, PCS, and PS. Its intrinsic consonance is quantified, without reference to the context, by tensor harmonicity, $h_{\text{Max}}$, $h_{\text{Entropy}}$, spectral entropy, and roughness. The fit measures use $\sigma =10$ cents, as in Hearne et al. (2026); the consonance measures use $\sigma =12$ cents, as in Milne et al. (2023b), with tensor harmonicity at $\sigma \sqrt{2}$ for the reason given in Section 3.3. Roughness is computed on frequencies in Hz and averaged over partial pairs. The measures are on different scales, and which transform of a raw density best reflects perception is an open question, so the correlations below are Spearman, unaffected by any monotone rescaling.

5.3 Results

SPCS ranges from $0.84$ for the tonic triad down to $0.17$ for $♯$IV major and correlates at $\rho =0.98$ with PCS and $0.96$ with SPS. PS correlates least with the others ($\rho =0.52$ with SPCS); without partials, it counts coincident pitches, just as PCS counts coincident pitch classes.

Figure 10 shows the relation between the two families. Within the fit block, correlations range from $\rho =0.52$ to $0.98$, and, within the consonance block, range from $0.19$ to $0.84$; across the two blocks, none exceeds $0.37$ in magnitude. Two pairs illustrate this. The tonic and $♭$II minor triads, both in second inversion, are all but identical in consonance ($h_{\text{Max}}=0.69$ against $0.68$, spectral entropy $0.819$ against $0.821$) yet differ nearly fourfold in fit ($\mathrm{SPCS}=0.84$ against $0.22$). Conversely, $♯$IV minor in first inversion and $♭$II major in root position fit the context almost equally well ($0.32$ against $0.33$) while differing sharply in consonance ($h_{\text{Max}}=0.56$ against $0.77$, and tensor harmonicity $0.0008$ against $0.86$). No comparison requires equal cardinality: each probe has four tones, the context 12.

Figure 10

A study predicting responses to chords in context therefore needs predictors from both families since neither carries the other’s information: the contextual (‘horizontal’) and simultaneous (‘vertical’) aspects of musical stability are separately necessary (Hearne et al., 2026; Tenney, 1988).

6.1 Tonal fit and tonal hierarchies

SPCS was introduced as a model of tonal hierarchy in Milne et al. (2015), where it was fitted to Krumhansl and Kessler’s (1982) probe tone data—ratings of how well each of the 12 chromatic pitches fits a previously established major or minor context. Cross‑validated correlations were high ($r_{\text{CV}}(22)=0.96$ for the best variant), using harmonic complex tones with a $1/k^{\rho }$ rolloff and $\sigma \approx 6$ cents. It extends to predicting the tonics of harmonic minor, melodic minor, and harmonic major scales and of microtonal scales for which no empirical data then existed.

Milne et al. (2016) tested SPCS with microtonal melodies and inharmonic spectra. Participants chose which of two melodies—one tuned to match the inharmonic spectrum, the other mismatched—was more ‘natural.’ A logistic regression combining SPCS and template harmonicity significantly predicted their choices.

Milne and Holland (2016) used SPCS to predict perceived triadic distance, finding it outperformed voice‑leading models and predicted a pitch space close to the music‑theoretical Tonnetz. Homer et al. (2024) developed a related spectral model, influenced by SPCS, with broadly comparable performance on the probe tone data.

Hearne et al. (2026) examined tonal hierarchies in microtonal scales (22‑EDO) as well as familiar and unfamiliar 12‑EDO scales, using both ‘fit’ and ‘stability’ as rating terms. SPCS was a strong predictor of ratings for unfamiliar scales—for which corpus‑based predictors (pitch prevalence) are, by definition, unavailable—supporting the claim that acoustical features play a foundational role in tonal function.

6.2 Consonance and affect

Harrison and Pearce (2020), in a large‑scale computational study of simultaneous consonance, introduced the $h_{\text{Entropy}}$ summary statistic extracted from the SPCS cross‑correlation profile (here implemented in templateHarmonicity). Eerola and Lahdelma (2021) evaluated multiple acoustic and cultural predictors of consonance across several published datasets, finding that harmonicity (specifically, the $h_{\text{Entropy}}$ measure of Harrison and Pearce, 2020) and roughness were among the strongest predictors.

Smit et al. (2019) used roughness, spectral entropy, and tensor harmonicity to predict pleasantness and happiness/sadness ratings for chords in the Bohlen–Pierce tuning system—a non‑octave tuning unfamiliar to all participants. Spectral entropy (introduced theoretically in Milne et al., 2017) and tensor harmonicity (introduced theoretically in Smit et al., 2019) were both tested against empirical data for the first time, and all three measures were significant predictors in the expected directions. Further theoretical predictions using spectral entropy—without empirical validation—were subsequently made in Milne (2024).

6.3 Scale and rhythmic structure

Milne and Herff (2020) tested balance, evenness, and 1‑tuple (IOI) entropy as predictors of rhythmic recognizability and liking across a large number of complex rhythms; all three were strongly evidenced.

Milne et al. (2023a) extended this to tapping accuracy, testing a comprehensive set of structural predictors—including edges, projected centroid, mean offset, circular autocorrelation phase matrix, and Markov prediction—against tapping data for a wide variety of long and complex rhythms. Entered into a Bayesian multilevel regression, they explained a substantial proportion of variance in tapping accuracy, with edge detection, projected centroid, and Markov prediction the strongest individually.

Average pitch height—which mean offset formalizes and generalizes—has been tested as a predictor of the perceived emotional valence of diatonic modes. Temperley and Tan (2013) found that perceived happiness of modes generally increases with average pitch height (the Lydian mode bucking this trend, presumably due to other factors). Smit et al. (2022) tested mean pitch as a predictor of perceived happiness cross‑culturally, comparing responses from participants in Sydney and remote communities in Papua New Guinea; mean pitch was a significant predictor for Sydney participants, though the evidence for a universal effect was negligible.