Part IV · Valence
# Consonance and Arithmetic
Not all atomic spectra are equal. Grading them by how nearly their ratios are simple fractions produces, unbidden, the consonance ordering of Western harmony.
> Assumes
> Chapter 6, especially Figure 6.1 and Proposition 6.4.
> Delivers
> The consonance functional (7.1) and kernel (7.2); Proposition 7.1 identifying the dissonance curve.
7.1 Why coherence is not enough
Return to the top two lanes of Figure 6.1. Both are purely atomic, with identical weights, so by Definition 6.1 they have identical coherence $\mathcal{A}=0.222$. Yet the first returns exactly, in sharp full-height revivals every $2\pi$, while the second never quite comes back.
The difference is arithmetic. In the first, the frequencies $\{1,2,3,4,6\}$ are rationally commensurate, so there is a common period $T=2\pi/\gcd$ and the orbit genuinely closes: the symmetry group $G_\Psi$ of (6.2) contains a full lattice $T\mathbb{Z}$. In the second, $\{1,\sqrt2,\sqrt3,\sqrt7,\pi\}$ are rationally independent, the orbit winds densely on a five-torus and never repeats, and $G_\Psi$ is only a Bohr set — relatively dense, but containing no exact periods at all.
Proposition 6.4 cannot see this distinction, because both cases are atomic. But the distinction between a closed orbit and a dense winding is exactly the distinction between a chord and a beating, and no theory of valence can afford to be blind to it.
7.2 The consonance functional
Grade each pair of atoms by how nearly their ratio is a simple rational:
$$ \mathcal{C}[\mu]=\iint \kappa\!\left(\frac{\lambda}{\lambda'}\right)d\mu(\lambda)\,d\mu(\lambda') $$ (7.1)
$$ \kappa(x)=\sum_{\substack{p/q\,\in\,\mathbb{Q}\\ \gcd(p,q)=1}} (pq)^{-\sigma}\,\exp\!\left(-\frac{(x-p/q)^2}{2\delta^2}\right) $$ (7.2)
The kernel $\kappa$ has a spike at every rational, of height $(pq)^{-\sigma}$, smoothed to width $\delta$. Two parameters control it: $\sigma$ sets how fast consonance falls off with arithmetic complexity, and $\delta$ sets the tolerance — how far from a true ratio the ear (or the field) will still accept.
The unison term $p/q=1$ contributes $\kappa(1)=1$ along the diagonal, which reproduces $\mathcal{A}$ exactly. Hence $\mathcal{C}\ge\mathcal{A}$ always, with equality when every off-diagonal ratio is arithmetically hopeless. Consonance contains coherence as its unison term, which is the formal sense in which Chapter 7 refines rather than replaces Chapter 6.
7.3 Thomae, Farey, and the dissonance curve
The unsmoothed kernel is a weighted Thomae function — the "popcorn function" that takes value $1/q$ at each rational $p/q$ in lowest terms and zero at every irrational. Thomae's function is the standard textbook example of a function continuous at every irrational and discontinuous at every rational, and it is a slightly startling object to meet in a theory of pleasure.
> Proposition 7.1 · The dissonance curve
>
> Posited
>
> The empirical two-tone roughness curve of Plomp and Levelt is $1-\kappa$ for a mollified Thomae kernel with $\delta$ set by the critical bandwidth. Its minima therefore occur at the simple ratios in order of $(pq)^{-\sigma}$: unison $(1{:}1)$, octave $(2{:}1)$, fifth $(3{:}2)$, fourth $(4{:}3)$, major third $(5{:}4)$.
That the consonance ordering of Western harmony falls out of a Farey-weighted kernel is either a pleasing coincidence or the point. I take it to be the point. It is also what QRI's consonance–dissonance–noise signature is measuring: decompose a neural spectrum into the part sitting in consonant relationships, the part in dissonant ones, and the aperiodic remainder, and equations (7.1)–(7.2) say precisely how the first two should be weighted.
Note the three-way correspondence this sets up between the spectral vocabulary of Chapter 6 and ordinary musical hearing:
| Spectral measure | Orbit | Heard as | Valence |
| --- | --- | --- | --- |
| Atomic, commensurate | Closed, exactly periodic | A chord | Strongly positive |
| Atomic, incommensurate | Dense winding on a torus | Beating, roughness | Weakly positive or negative |
| Absolutely continuous | Dispersal, no return | Noise | Zero magnitude |
7.4 What this predicts
Because $\sigma$ is a free parameter, (7.2) is a model rather than a derivation, and it earns its place only by being measurable. The prediction is sharp: valence response to two-tone and two-flicker stimuli should show minima at simple ratios with depths decaying as a power of the denominator product $pq$. Fitting that power gives $\sigma$, which the structure of the model expects to lie between 1 and 2. If measured peak heights do not decay as any power of $pq$ — if, say, the octave and the seventh are equally consonant, or the falloff is exponential in $q$ — then (7.2) is simply wrong.
This is Chapter 11, prediction 2, and it is the cheapest experiment in the book.
> ### What Chapter 7 established
>
> - Coherence $\mathcal{A}$ cannot distinguish a closed orbit from a dense torus winding, though the two feel entirely different.
> - The consonance functional (7.1) grades atom pairs by arithmetic simplicity through the kernel (7.2).
> - $\mathcal{C}\ge\mathcal{A}$, with the unison term reproducing coherence exactly — Chapter 7 refines Chapter 6 rather than replacing it.
> - Proposition 7.1 identifies the kernel with a mollified Thomae function and recovers the Plomp–Levelt curve and the standard consonance ordering.
> - The exponent $\sigma$ is measurable, and a non-power-law falloff falsifies the construction.
> ### Exercises
>
> 1. Evaluate the weights $(pq)^{-\sigma}$ for $\sigma=1$ at the ratios 2/1, 3/2, 4/3, 5/4 and 45/32 (the tritone). Confirm the ordering matches musical intuition. [1]
> 2. Show that $\mathcal{C}\ge\mathcal{A}$ for any $\mu$ and any $\kappa\ge 0$ with $\kappa(1)=1$. Where is positivity of $\kappa$ used? [2]
> 3. Prove Thomae's function is continuous at every irrational and discontinuous at every rational. Then explain the role of the mollifier $\delta$ in (7.2) physically, not merely analytically. [2]
> 4. For the incommensurate lane of Figure 6.1, estimate the largest revival height reachable within $s\le 60$ using a simultaneous rational approximation to $(\sqrt2,\sqrt3,\sqrt7,\pi)$. [2]
> 5. Equal temperament makes the fifth $2^{7/12}$, not $3/2$. Compute the resulting reduction in $\kappa$ for $\delta=0.01$ and comment on whether the model predicts a perceptible loss. [2]
> 6. Open. Derive $\sigma$ rather than fitting it, from the density of states of the collective mode. A first-principles value would convert Proposition 7.1 from a model into a prediction. [3]
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