Prime-Limit Consonance

The prime-limit classification

In just intonation, musical intervals are classified by their prime limit — the largest prime needed to express the frequency ratio in lowest terms:

• 2-limit: Octave (2:1) — only prime 2 required
• 3-limit: Perfect fifth (3:2), perfect fourth (4:3) — primes 2 and 3
• 5-limit: Major third (5:4), minor third (6:5), major sixth (5:3), minor sixth (8:5)
• 7-limit: Harmonic seventh (7:4), septimal minor seventh (7:4)
• 11-limit: Neutral intervals (11:8, 11:6, …)
• 13-limit: Further microtonal territory

This classification — developed from Pythagoras through Ptolemy, codified by Harry Partch in the 20th century — predates our formula by millennia.

The c(p) formula reproduces the hierarchy

The [[connectivity-scaling-law]] gives spectral efficiency c(p) = p·sin(π/p)/(2(p−1)) for each prime p. Applied to the prime-limit classification:

Prime limit	Representative intervals	c(p)
p = 2	Octave (2:1)	1.0000
p = 3	Fifth (3:2), fourth (4:3)	0.6495
p = 5	Major/minor thirds and sixths	0.3674
p = 7	Harmonic seventh (7:4)	0.2531
p = 11	11:8 and other 11-limit intervals	0.1550
p = 13	13:8 and other 13-limit intervals	0.1296

The ordering c(2) > c(3) > c(5) > c(7) > c(11) > c(13) matches the standard consonance ranking of Western (and cross-cultural) music theory: octave is most consonant, fifth and fourth next, thirds and sixths family next, harmonic seventh mildly dissonant to most ears, 11-limit strongly dissonant, and so on.

This is not a psychoacoustic model — no ear model, no critical band theory, no roughness calculation. The formula is derived from the spectral theory of the [[sopfr-binding-energy]] function and verified against thermodynamic computing hardware efficiency. The musical consonance ordering is a consequence, not an input.

Within a prime family, all intervals sharing the same prime limit receive the same c value. The formula operates at the level of prime families, not individual intervals. Finer distinctions (why the major sixth 5:3 sounds more open than the minor third 6:5 despite sharing p=5) require the composite extension of the formula, which is an open problem.

The absolute scale: K₂ = π²/12

The formula is normalised to c(2) = 1 (octave). The absolute spectral scale is:

K₂ = π²/12 = ζ(2)/2

This constant appears in the zeta-regularised Casimir energy of a one-dimensional resonant cavity — the quantum mechanical foundation of a vibrating string in a box. The same constant governs both the prime spectral amplitude and the quantum zero-point energy of a string.

The Farey and Stern-Brocot connection

The Farey sequence F_n lists all fractions p/q ≤ 1 with denominator q ≤ n, in order. The Stern-Brocot tree organises all positive rationals from simplest (low-denominator, high-consonance) to most complex (high-denominator, dissonant). Both structures are fundamental to musical tuning theory: equal temperament (12-TET) is found by seeking the best small-denominator rational approximations to 2^(k/12).

These are also the structures underlying the [[tusk-series]] Ramanujan sum computation. The Ramanujan sum c_d(n) = Σ_{k: gcd(k,d)=1} e^{2πikn/d} is a Farey-weighted Fourier kernel; its coefficient a_sopfr(d) = d/(d−1) gives the spectral weight for degree d.

Schroeder treats Farey fractions and continued fractions in the same chapter as the prime distribution (Chapter 5 of Number Theory in Science and Communication), approaching from signal processing and music theory. We arrived at the same intersection from prime circuit boards and the [[tusk-series]]. The orbit is the same.

Connection to thermodynamic computing

The same formula c(d) = d·sin(π/d)/(2(d−1)) that recovers the musical consonance hierarchy predicts the FLOPs efficiency of a sparse network at connectivity d. At d=16 (Extropic Z1 chip): 1/c(16) = 9.61 ≈ 10×. The hierarchy of musical consonance and the hierarchy of hardware efficiency are governed by the same spectral formula.

This is not coincidence — both systems involve a local-rule interaction at degree d, and both are optimised by choosing d to balance information density against interaction cost. The [[thermodynamic-computing-membrane]] sits at the same live boundary as the resonant string.

See also

[[prime-resonance-computing]] — prime-ratio frequency networks
[[tusk-series]] — the spectral structure from which c(p) is derived
[[connectivity-scaling-law]] — the formula c(d) and its hardware application
[[spectral-fingerprinting]] — frequency-space prime signatures
[[thermodynamic-computing-membrane]] — the hardware application